On Singularity of Energy Measures for Symmetric Diffusions with Full Off-Diagonal Heat Kernel Estimates II: Some Borderline Examples

Abstract

We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpiński gaskets and each of which is equipped with a canonical strongly local regular symmetric Dirichlet form. We prove that any fractal K in this family satisfies the full off-diagonal heat kernel estimates with some space-time scale function $\varPsi _{K}$ and the singularity of the associated energy measures with respect to the canonical volume measure (uniform distribution) on K, and also that the decay rate of $r^{-2}\varPsi _{K}(r)$ to 0 as $r\downarrow 0$ can be made arbitrarily slow by suitable choices of K. These results together support the energy measure singularity dichotomy conjecture [Ann. Probab. 48 (2020), no. 6, 2920–2951, Conjecture 2.15] stating that, if the full off-diagonal heat kernel estimates with space-time scale function $\varPsi$ satisfying $\lim _{r\downarrow 0}r^{-2}\varPsi (r)=0$ hold for a strongly local regular symmetric Dirichlet space with complete metric, then the associated energy measures are singular with respect to the reference measure of the Dirichlet space.

1 Introduction

This paper is a follow-up of the author’s recent joint work [26] with Mathav Murugan on singularity of energy measures associated with a strongly local regular symmetric Dirichlet space $(K,d,m,\mathcal {E},\mathcal {F})$ satisfying full off-diagonal heat kernel estimates. The $\mathcal {E}$-energy measure $\mu _{\langle u\rangle }$ of $u\in \mathcal {F}$ is a Borel measure on K which plays, in the theory of regular symmetric Dirichlet forms as presented in [13, 17], the same roles as the classical energy integral measure $|\nabla u|^{2}\,dx$ on $\mathbb {R}^{N}$. It is defined for $u\in \mathcal {F}\cap L^{\infty }(K,m)$ as the unique Borel measure on K such that

$$\begin{aligned} \int \limits _{K}f\,d\mu _{\langle u\rangle }=\mathcal {E}(u,fu)-\frac{1}{2}\mathcal {E}(u^{2},f) \quad \text {for any}\,\, f\in \mathcal {F}\cap \mathcal {C}_{\text {c}}(K), \end{aligned}$$

(1.1)

where $\mathcal {C}_{\text {c}}(K)$ denotes the space of $\mathbb {R}$-valued continuous functions on K with compact supports, and then for $u\in \mathcal {F}$ by $\mu _{\langle u\rangle }(A):=\lim _{n\rightarrow \infty }\mu _{\langle (-n)\vee (u\wedge n)\rangle }(A)$ for each Borel subset A of K; see [17, Theorem 1.4.2-(ii),(iii), (3.2.13), (3.2.14) and (3.2.15)] for the details of this definition.

The main results of [26] concern the singularity and the absolute continuity of the $\mathcal {E}$-energy measures $\mu _{\langle u\rangle }$ with respect to the reference measure $m$. While $\mu _{\langle u\rangle }$ is easily identified as $\langle \nabla u,\nabla u\rangle _{x}\,dm(x)$ if $\mathcal {E}=\int _{K}\langle \nabla \cdot ,\nabla \cdot \rangle _{x}\,dm(x)$ for some linear differential operator $\nabla$ satisfying the Leibniz rule and some measurable field $\langle \cdot ,\cdot \rangle _{x}$ of non-negative definite symmetric bilinear forms, there is no simple expression of $\mu _{\langle u\rangle }$ and the nature of $\mu _{\langle u\rangle }$ is a deep mystery when K is a fractal. The question of whether $\mu _{\langle u\rangle }$ is singular with respect to $m$ is probably the most fundamental one toward better understanding of $\mu _{\langle u\rangle }$ in such cases, had been answered affirmatively for essentially all known examples of self-similar Dirichlet forms on self-similar fractals in [11, 23, 24, 30, 31], but had been studied only under the assumption of exact self-similarity until [26]. As the main results of [26], it has been now proved that the $\mathcal {E}$-energy measures $\mu _{\langle u\rangle }$ are singular or absolutely continuous with respect to $m$ according to whether the behavior of the associated heat kernel $p_{t}(x,y)$ in infinitesimal scale is “sufficiently sub-Gaussian” or “Gaussian”, as stated in the following theorem. Recall that a family $\{p_{t}\}_{t\in (0,\infty )}$ of $[-\infty ,\infty ]$-valued Borel measurable functions on $K\times K$ is called a heat kernel of $(K,d,m,\mathcal {E},\mathcal {F})$ if and only if the symmetric Markovian semigroup $\{T_{t}\}_{t\in (0,\infty )}$ on $L^{2}(K,m)$ associated with $(\mathcal {E},\mathcal {F})$ is expressed as $T_{t}u=\int \nolimits _{K}p_{t}(\cdot ,y)u(y)\,dm(y)$ $m$-a.e. for any $t\in (0,\infty )$ and any $u\in L^{2}(K,m)$. We set $\mathop {{\text {diam}}}(K,d):=\sup _{x,y\in K}d(x,y)$ and $B_{d}(x,r):=\{y\in K\mid d(x,y)<r\}$ for $(x,r)\in K\times (0,\infty )$.

Theorem 1.1

(A simplification of [26, Theorem 2.13]) Let $(K,d,m,\mathcal {E},\mathcal {F})$ be a metric measure Dirichlet space, i.e., a strongly local regular symmetric Dirichlet space with $(K,d)$ complete and K containing at least two elements, so that $\mathop {{\text {diam}}}(K,d)\in (0,\infty ]$. Let $\varPsi :[0,\infty )\rightarrow [0,\infty )$ be a homeomorphism satisfying

$$\begin{aligned} c_{\varPsi }^{-1}\biggl (\frac{R}{r}\biggr )^{\beta _{0}}\le \frac{\varPsi (R)}{\varPsi (r)} \le c_{\varPsi }\biggl (\frac{R}{r}\biggr )^{\beta _{1}} \quad \text {for any}\,\, r,R\in (0,\infty )\, \text {with}\,\, r\le R \end{aligned}$$

(1.2)

for some $c_{\varPsi },\beta _{0},\beta _{1}\in [1,\infty )$ with $1<\beta _{0}\le \beta _{1}$, and define $\Phi _{\varPsi }:[0,\infty )\rightarrow [0,\infty )$ by $\Phi _{\varPsi }(s):=\sup _{r\in (0,\infty )}(s/r-1/\varPsi (r))$. Suppose further that $(K,d,m,\mathcal {E},\mathcal {F})$ satisfies the full off-diagonal heat kernel estimates$fHKE (\varPsi )$, i.e., that there exist a heat kernel $\{p_{t}\}_{t\in (0,\infty )}$ of $(K,d,m,\mathcal {E},\mathcal {F})$ and $c_{1},c_{2},c_{3},c_{4}\in (0,\infty )$ such that

$$\begin{aligned} \frac{c_{1}\exp \bigl (-c_{2}t\Phi _{\varPsi }(d(x,y)/t)\bigr )}{m\bigl (B_{d}(x,\varPsi ^{-1}(t))\bigr )}&\le p_{t}(x,y)\\&\le \frac{c_{3}\exp \bigl (-c_{4}t\Phi _{\varPsi }(d(x,y)/t)\bigr )}{m\bigl (B_{d}(x,\varPsi ^{-1}(t))\bigr )} \quad \quad fHKE (\varPsi ) \end{aligned}$$

for $m$-a.e. $x,y\in K$ for each $t\in (0,\infty )$. Then the following hold:

• (1) ($fHKE (\varPsi )$ with “sufficiently sub-Gaussian” $\varPsi$ implies singularity) If

  $$\begin{aligned} \liminf _{\lambda \rightarrow \infty }\liminf _{r\downarrow 0}\frac{\lambda ^{2}\varPsi (r/\lambda )}{\varPsi (r)}=0, \end{aligned}$$

  (1.3)

  then $\mu _{\langle u\rangle }$ is singular with respect to $m$ for any $u\in \mathcal {F}$.

• (2) ($fHKE (\varPsi )$ with “Gaussian” $\varPsi$ implies absolute continuity) If

  $$\begin{aligned} \limsup _{r\downarrow 0}\frac{\varPsi (r)}{r^{2}}>0, \end{aligned}$$

  (1.4)

  then $m(A)=0$ if and only if $\sup _{u\in \mathcal {F}}\mu _{\langle u\rangle }(A)=0$ for each Borel subset A of K, thus in particular $\mu _{\langle u\rangle }$ is absolutely continuous with respect to $m$ for any $u\in \mathcal {F}$, and there exist $r_{1}\in (0,\mathop {{\text {diam}}}(K,d))$ and $c_{5}\in [1,\infty )$ such that

  $$\begin{aligned} c_{5}^{-1}r^{2}\le \varPsi (r)\le c_{5}r^{2}\quad {for\,\, any}\,\,r\in (0,r_{1}). \end{aligned}$$

  (1.5)

Remark 1.2

Let $\varPsi :[0,\infty )\rightarrow [0,\infty )$ be a homeomorphism satisfying (1.2) for some $c_{\varPsi },\beta _{0},\beta _{1}\in [1,\infty )$ with $1<\beta _{0}\le \beta _{1}$, and let $(K,d,m,\mathcal {E},\mathcal {F})$ be a metric measure Dirichlet space satisfying $fHKE (\varPsi )$.

• (1) It is known that in this situation $(K,d,m,\mathcal {E},\mathcal {F})$ satisfies the assumptions of [26, Theorem 2.13], namely $VD$, $PI (\varPsi )$, $CS (\varPsi )$ and the chain condition for $(K,d)$. Indeed, $VD$ follows in the same way as [8, Proof of Lemma 5.1-(i)] by integrating the lower inequality in $fHKE (\varPsi )$ on $B_{d}(x,2\varPsi ^{-1}(t))$ with respect to $m$ and applying the upper bound on $\Phi _{\varPsi }(R,t):=t\Phi _{\varPsi }(R/t)$ in [20, (5.13)], (1.2) and the inequality $\int _{B_{d}(x,2\varPsi ^{-1}(t))}p_{t}(x,y)\,dm(y)\le \int _{K}p_{t}(x,y)\,dm(y)\le 1$ for $m$-a.e. $x\in K$. Then $VD$ and $fHKE (\varPsi )$ imply $PI (\varPsi )$ and $CS (\varPsi )$ by the results in [1, 6, 7, 19] as summarized in [32, Theorem 3.2] and [26, Theorem 2.8 and Remark 2.9], and $fHKE (\varPsi )$ also implies the chain condition for $(K,d)$ by [33, Theorem 2.11].

• (2) If $\varPsi _{0}:[0,\infty )\rightarrow [0,\infty )$ is a homeomorphism and $\varPsi _{0}(r)/\varPsi (r)\in [c_{0}^{-1},c_{0}]$ for any $r\in (0,\infty )$ for some $c_{0}\in [1,\infty )$, then $(K,d,m,\mathcal {E},\mathcal {F})$ satisfies $fHKE (\varPsi _{0})$. Indeed, this is immediate from $fHKE (\varPsi )$, $VD$, which is implied by $fHKE (\varPsi )$ as noted in (1) above, and the elementary observation based on (1.2) that $\Phi _{\varPsi _{0}}(s)/\Phi _{\varPsi }(s)\in \bigl [(c_{0}c_{\varPsi })^{-\frac{1}{\beta _{0}-1}},(c_{0}c_{\varPsi })^{\frac{1}{\beta _{0}-1}}\bigr ]$ for any $s\in (0,\infty )$.

Note that, if $\varPsi (r)=r^{\beta }$ for any $r\in [0,\infty )$ for some $\beta \in (1,\infty )$, then $\Phi (s)=\beta ^{-\frac{\beta }{\beta -1}}(\beta -1)s^{\frac{\beta }{\beta -1}}$ for any $s\in [0,\infty )$, so that $fHKE (\varPsi )$ with this $\varPsi$ is the typical form of heat kernel estimates known to hold widely; see, e.g., [18, 35,36,37] and references therein for the studies on the case of $\beta =2$ and [4, 5, 10, 16, 29] for known results with $\beta >2$ for self-similar fractals. For this class of $\varPsi$, the classification by (1.3) and (1.4) gives a complete dichotomy between $\beta >2$ and $\beta \le 2$, with the latter identified further as $\beta =2$ by (1.5). On the other hand, (1.3) and (1.4) do not give a complete classification of general $\varPsi$ since there are examples of $\varPsi$, like $\varPsi (r)=r^{2}/\log (e-1+r^{-1})$, satisfying (1.2) but neither (1.3) nor (1.4), and it is not clear under $fHKE (\varPsi )$ with such $\varPsi$ whether the $\mathcal {E}$-energy measures $\mu _{\langle u\rangle }$ are singular or absolutely continuous with respect to the reference measure $m$. In view of Theorem 1.1, one might expect the following conjecture to hold.

Conjecture 1.3

(Energy measure singularity dichotomy; a simplification of [26, Conjecture 2.15]) Theorem 1.1-(1) with (1.3) replaced by

$$\begin{aligned} \lim _{r\downarrow 0}\frac{\varPsi (r)}{r^{2}}=0 \end{aligned}$$

(1.6)

($fHKE (\varPsi )$ with “however weakly sub-Gaussian” $\varPsi$ implies singularity) holds.

As announced already in [26, Remark 2.14], this paper is aimed at giving firm evidence that Conjecture 1.3 should be true, by presenting concrete examples of metric measure Dirichlet spaces satisfying both the singularity of the energy measures and $fHKE (\varPsi )$ for some $\varPsi$, whose decay rate at 0 can be made arbitrarily close to $r^{2}$. Their state spaces are certain fractals, which we call the (two-dimensional) thin scale irregular Sierpiński gaskets (see Fig. 2), obtained by modifying the construction of the scale irregular (or homogeneous random) Sierpiński gaskets studied in [9, 21, 22] (see also [28, Chap. 24]) so as to make them look very much like one-dimensional frames in infinitesimal scale. An arbitrarily slow decay rate of $\varPsi (r)/r^{2}$ as $r\downarrow 0$ can be then realized by choosing suitably the parameters defining the fractal to make its infinitesimal geometry arbitrarily close to being one-dimensional, which is an idea suggested to the author by Martin T. Barlow in [3]. An important point here is to allow infinitely many patterns of cell subdivisions to be present in the construction of the fractal, in contrast to that of the usual scale irregular Sierpiński gaskets considered in [9, 21, 22, 28], each of which involves only finitely many patterns of cell subdivisions and typically falls within the scope of Theorem 1.1-(1) as illustrated in [26, Sect. 5]. We remark that the singularity of the energy measures has been proved also in [25] for a class of (two-dimensional) spatially inhomogeneous Sierpiński gaskets, which typically do not satisfy the volume doubling property $VD$ and are thereby beyond the scope of [26, Theorem 2.13].

The rest of this paper is organized as follows. In Sect. 2 we define the thin scale irregular Sierpiński gaskets and construct the canonical Dirichlet forms (resistance forms) on them, and we verify in Sect. 3 that they satisfy $fHKE (\varPsi )$ with $\varPsi$ explicit in terms of their defining parameters (Theorem 3.3). In Sect. 4 we prove the singularity of the energy measures for the canonical Dirichlet form on any thin scale irregular Sierpiński gasket (Theorem 4.3), and Sect. 5 is devoted to stating and proving our last main result that an arbitrarily slow decay rate of $\varPsi (r)/r^{2}$ can be realized by some thin scale irregular Sierpiński gasket (Theorem 5.1 and Proposition 5.2).

Notation

In this paper, we adopt the following notation and conventions.

• (1) The symbols $\subset$ and $\supset$ for set inclusion allow the case of the equality.

• (2) $\mathbb {N}:=\{n\in \mathbb {Z}\mid n>0\}$, i.e., $0\not \in \mathbb {N}$.

• (3) The cardinality (the number of elements) of a set A is denoted by $\#A$.

• (4) We set $a\vee b:=\max \{a,b\}$, $a\wedge b:=\min \{a,b\}$, $a^{+}:=a\vee 0$, $a^{-}:=-(a\wedge 0)$ and $\lfloor a\rfloor :=\max \{n\in \mathbb {Z}\mid n\le a\}$ for $a,b\in \mathbb {R}$, and we use the same notation also for $\mathbb {R}$-valued functions and equivalence classes of them. All numerical functions in this paper are assumed to be $[-\infty ,\infty ]$-valued.

• (5) The Euclidean inner product and norm on $\mathbb {R}^{2}$ are denoted by $\langle \cdot ,\cdot \rangle$ and $|\cdot |$, respectively.

• (6) Let K be a non-empty set. We define $\mathop {{\text {id}}}_{K}: K\rightarrow K$ by $\mathop {{\text {id}}}_{K}(x):=x$, $\mathbbm {1}_{A}=\mathbbm {1}_{A}^{K}\in \mathbb {R}^{K}$ for $A\subset K$ by $\mathbbm {1}_{A}(x):=\mathbbm {1}_{A}^{K}(x):=\bigl \{{\begin{matrix}1 &{} \text {if}\,\, x\in A,\\ 0 &{} \text {if}\,\, x\not \in A,\end{matrix}}$ set $\mathbbm {1}_{x}:=\mathbbm {1}_{x}^{K}:=\mathbbm {1}_{\{x\}}$ for $x\in K$ and $\Vert u\Vert _{\sup }:=\Vert u\Vert _{\sup ,K}:=\sup _{x\in K}|u(x)|$ for $u: K\rightarrow \mathbb {R}$.

• (7) Let K be a topological space. We set $\mathcal {C}(K):=\{{u}\in \mathbb {R}^{K}\mid u \,\, \text {is continuous}\}$, and the closure of $K\setminus u^{-1}(0)$ in K is denoted by $\mathop {{\text {supp}}}_{K}[u]$ for each $u\in \mathcal {C}(K)$. The Borel $\sigma$-algebra of K is denoted by $\mathscr {B}(K)$.

• (8) Let $(K,d)$ be a metric space. We set $B_{d}(x,r):=\{y\in K\mid d(x,y)<r\}$ for $(x,r)\in K\times (0,\infty )$.

• (9) Let $(K,\mathscr {B})$ be a measurable space and let $\mu ,\nu$ be measures on $(K,\mathscr {B})$. We write $\nu \ll \mu$ and $\nu \perp \mu$ to mean that $\nu$ is absolutely continuous and singular, respectively, with respect to $\mu$.

2 The Examples: Thin Scale Irregular Sierpiński Gaskets

In this section, we introduce the (two-dimensional) thin scale irregular Sierpiński gaskets, and construct the canonical Dirichlet forms (resistance forms) on them by applying the standard method developed in [27, Chaps. 2 and 3]. We closely follow [26, Sect. 5] for the presentation of this section.

Fig. 1

The level-l (self-similar) thin Sierpiński gaskets $K^{l}$ ($l=5,6,7,8$)

To start with, the thin scale irregular Sierpiński gaskets are defined as follows.

Definition 2.1

(Thin scale irregular Sierpiński gasket) Let $q_{0},q_{1},q_{2}\in \mathbb {R}^{2}$ satisfy $|q_{j}-q_{k}|=1$ for any $j,k\in \{0,1,2\}$ with $j\not =k$, so that the convex hull $\triangle$ of $V_{0}:=\{q_{0},q_{1},q_{2}\}$ in $\mathbb {R}^{2}$ is a closed equilateral triangle with side length 1. For each $l\in \mathbb {N}\setminus \{1,2,3,4\}$, we set

$$\begin{aligned} S_{l}:=\bigl \{(i_{1},i_{2})\in (\mathbb {N}\cup \{0\})^{2} \bigm |{i_{1}+i_{2}\le l-1, i_{1}i_{2}(l-1-i_{1}-i_{2})=0}\bigr \}, \end{aligned}$$

(2.1)

and for each $i=(i_{1},i_{2})\in S_{l}$ set $q^{l}_{i}:=q_{0}+\sum _{k=1}^{2}(i_{k}/l)(q_{k}-q_{0})$ and define $f^{l}_{i}:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}$ by $f^{l}_{i}(x):=q^{l}_{i}+l^{-1}(x-q_{0})$. Let $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$, set $W^{\boldsymbol{l}}_{n}:=\prod _{k=1}^{n}S_{l_{k}}$ for $n\in \mathbb {N}\cup \{0\}$, $W^{\boldsymbol{l}}_{*}:=\bigcup _{n=0}^{\infty }W^{\boldsymbol{l}}_{n}$, $|w|:=n$ and $f^{\boldsymbol{l}}_{w}:=f^{l_{1}}_{w_{1}}\circ \cdots \circ f^{l_{n}}_{w_{n}}$ for $n\in \mathbb {N}\cup \{0\}$ and $w=w_{1}\ldots w_{n}\in W^{\boldsymbol{l}}_{n}$, where $W^{\boldsymbol{l}}_{0}$ is defined as the singleton $\{\emptyset \}$ of the empty word $\emptyset$ and $f^{\boldsymbol{l}}_{\emptyset }:=\mathop {{\text {id}}}_{\mathbb {R}^{2}}$. Noting that $\bigl \{\bigcup _{w\in W^{\boldsymbol{l}}_{n}}f^{\boldsymbol{l}}_{w}(\triangle )\bigr \}_{n=0}^{\infty }$ is a strictly decreasing sequence of non-empty compact subsets of $\triangle$, we define the (two-dimensional) level- $\boldsymbol{l}$ thin scale irregular Sierpiński gasket $K^{\boldsymbol{l}}$ as the non-empty compact subset of $\triangle$ given by

$$\begin{aligned} K^{\boldsymbol{l}}:=\bigcap _{n=0}^{\infty }\bigcup _{w\in W^{\boldsymbol{l}}_{n}}f^{\boldsymbol{l}}_{w}(\triangle ) \end{aligned}$$

(2.2)

(see Fig. 2), and set $K^{\boldsymbol{l}}_{w}:=K^{\boldsymbol{l}}\cap f^{\boldsymbol{l}}_{w}(\triangle )$ and $F^{\boldsymbol{l}}_{w}:=f^{\boldsymbol{l}}_{w}|_{K^{\boldsymbol{l}^{|w|}}}$ for $w\in W^{\boldsymbol{l}}_{*}$, where $\boldsymbol{l}^{k}:=(l_{n+k})_{n=1}^{\infty }$ for $k\in \mathbb {N}\cup \{0\}$. We also set $V^{\boldsymbol{l}}_{n}:=\bigcup _{w\in W^{\boldsymbol{l}}_{n}}f^{\boldsymbol{l}}_{w}(V_{0})$ for $n\in \mathbb {N}\cup \{0\}$ and $V^{\boldsymbol{l}}_{*}:=\bigcup _{n=0}^{\infty }V^{\boldsymbol{l}}_{n}$, so that $V^{\boldsymbol{l}}_{0}=V_{0}$, $\{V^{\boldsymbol{l}}_{n}\}_{n=0}^{\infty }$ is a strictly increasing sequence of finite subsets of $K^{\boldsymbol{l}}$, and $V^{\boldsymbol{l}}_{*}$ is dense in $K^{\boldsymbol{l}}$.

In particular, for each $l\in \mathbb {N}\setminus \{1,2,3,4\}$ we let $\boldsymbol{l}_{l}:=(l)_{n=1}^{\infty }$ denote the constant sequence with value l, set $K^{l}:=K^{\boldsymbol{l}_{l}}$ and $V^{l}_{n}:=V^{\boldsymbol{l}_{l}}_{n}$ for $n\in \mathbb {N}\cup \{0\}$, and call $K^{l}$ the (two-dimensional) level-l thin Sierpiński gasket, which is exactly self-similar in the sense that $K^{l}=\bigcup _{i\in S_{l}}f^{l}_{i}(K^{l})$ (see Fig. 1 and, e.g., [27, Sect. 1.1]).

Fig. 2

A level-$\boldsymbol{l}$ thin scale irregular Sierpiński gasket $K^{\boldsymbol{l}}$ ($\boldsymbol{l}=(5,7,6,12,\ldots )$)

We fix an arbitrary $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$ in the rest of this section. The following proposition is immediate from Definition 2.1.

Proposition 2.2

• (1) Let $w=w_{1}\ldots w_{|w|},v=v_{1}\ldots v_{|v|}\in W^{\boldsymbol{l}}_{*}\setminus \{\emptyset \}$ satisfy $w_{k}\not =v_{k}$ for some $k\in \{1,\ldots ,|w|\wedge |v|\}$. Then $\#(K^{\boldsymbol{l}}_{w}\cap K^{\boldsymbol{l}}_{v})\le 1$ and

  $$\begin{aligned} f^{\boldsymbol{l}}_{w}(\triangle )\cap f^{\boldsymbol{l}}_{v}(\triangle ) =K^{\boldsymbol{l}}_{w}\cap K^{\boldsymbol{l}}_{v} =F^{\boldsymbol{l}}_{w}(V_{0})\cap F^{\boldsymbol{l}}_{v}(V_{0}). \end{aligned}$$

  (2.3)

• (2) $K^{\boldsymbol{l}}=\bigcup _{w\in W^{\boldsymbol{l}}_{n}}K^{\boldsymbol{l}}_{w}$ for any $n\in \mathbb {N}\cup \{0\}$, and $F^{\boldsymbol{l}}_{w}(K^{\boldsymbol{l}^{|w|}})=K^{\boldsymbol{l}}_{w}$ for any $w\in W^{\boldsymbol{l}}_{*}$.

• (3) $V^{\boldsymbol{l}}_{n+k}=\bigcup _{w\in W^{\boldsymbol{l}}_{n}}F^{\boldsymbol{l}}_{w}(V^{\boldsymbol{l}^{n}}_{k})$ and $V^{\boldsymbol{l}}_{*}=\bigcup _{w\in W^{\boldsymbol{l}}_{n}}F^{\boldsymbol{l}}_{w}(V^{\boldsymbol{l}^{n}}_{*})$ for any $n,k\in \mathbb {N}\cup \{0\}$.

In exactly the same way as in [9, 21, 22] (see also [28, Part 4]), we can define a canonical strongly local regular symmetric Dirichlet space $(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ over $K^{\boldsymbol{l}}$. First, the metric $d_{\boldsymbol{l}}$ on $K^{\boldsymbol{l}}$ is defined as follows.

Definition 2.3

We define $d_{\boldsymbol{l}}: K^{\boldsymbol{l}}\times K^{\boldsymbol{l}}\rightarrow [0,\infty ]$ by

$$\begin{aligned} d_{\boldsymbol{l}}(x,y):=\inf \{\ell _{\mathbb {R}^{2}}(\gamma )\mid {\gamma :[0,1]\rightarrow K^{\boldsymbol{l}}, \gamma \,\,\text {is continuous}, \gamma (0)=x, \gamma (1)=y}\}, \end{aligned}$$

(2.4)

where $\ell _{\mathbb {R}^{2}}(\gamma )$ denotes the Euclidean length of $\gamma$, i.e., the total variation of the $\mathbb {R}^{2}$-valued map $\gamma$ with respect to the Euclidean norm $|\cdot |$. We also set $L^{\boldsymbol{l}}_{n}:=l_{1}\cdots l_{n}$ ($L^{\boldsymbol{l}}_{0}:=1$) for $n\in \mathbb {N}\cup \{0\}$.

Proposition 2.4

$d_{\boldsymbol{l}}$ is a metric on $K^{\boldsymbol{l}}$, and it is geodesic, i.e., for any $x,y\in K^{\boldsymbol{l}}$ there exists $\gamma :[0,1]\rightarrow K^{\boldsymbol{l}}$ such that $\gamma (0)=x$, $\gamma (1)=y$ and $d_{\boldsymbol{l}}(\gamma (s),\gamma (t))=|s-t|d_{\boldsymbol{l}}(x,y)$ for any $s,t\in [0,1]$. Moreover,

$$\begin{aligned} |x-y|\le d_{\boldsymbol{l}}(x,y)\le 6|x-y|\quad \text {for any}\,\,x,y\in K^{\boldsymbol{l}}. \end{aligned}$$

(2.5)

Proof

This proof is similar to [9, Proof of Lemma 2.4], but some additional argument is required to take care of the possible unboundedness of $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }$. It is immediate from (2.4) that $|x-y|\le d_{\boldsymbol{l}}(x,y)<\infty$ for any $x,y\in K^{\boldsymbol{l}}$ and thereby that $d_{\boldsymbol{l}}$ is a metric on $K^{\boldsymbol{l}}$, which is also geodesic by [12, Proposition 2.5.19]; indeed, the infimum in (2.4) is easily seen to be attained for each $x,y\in K^{\boldsymbol{l}}$, by choosing a sequence $\{\gamma _{n}\}_{n=1}^{\infty }$ of continuous maps as in (2.4) with $\lim _{n\rightarrow \infty }\ell _{\mathbb {R}^{2}}(\gamma _{n})=d_{\boldsymbol{l}}(x,y)$, reparameterizing them by arc length on the basis of [12, Proposition 2.5.9], and applying to them the Arzelà–Ascoli theorem [12, Theorem 2.5.14] and the lower semi-continuity [12, Proposition 2.3.4-(iv)] of $\ell _{\mathbb {R}^{2}}$ with respect to pointwise convergence.

Thus it remains to prove the upper inequality in (2.5) for any $x,y\in K^{\boldsymbol{l}}$ with $x\not =y$. First, for any $w\in W^{\boldsymbol{l}}_{*}$ and any $x\in K^{\boldsymbol{l}}_{w}$, we easily see that

$$\begin{aligned} \max _{k\in \{0,1,2\}}d_{\boldsymbol{l}}(F^{\boldsymbol{l}}_{w}(q_{k}),x) \le \sum _{n=|w|+1}^{\infty }\frac{\frac{3}{2}l_{n}-\frac{5}{2}}{L^{\boldsymbol{l}}_{n}} \le \frac{\frac{3}{2}l_{|w|+1}-\frac{5}{2}+\sum _{n=0}^{\infty }\frac{3}{2}(\frac{1}{5})^{n}}{L^{\boldsymbol{l}}_{|w|}l_{|w|+1}} <\frac{\frac{3}{2}}{L^{\boldsymbol{l}}_{|w|}}, \end{aligned}$$

(2.6)

from which it further follows that for any $j,k\in \{0,1,2\}$ with $j\not =k$,

$$\begin{aligned} d_{\boldsymbol{l}}(F^{\boldsymbol{l}}_{w}(q_{k}),x)\le 5|\langle x-F^{\boldsymbol{l}}_{w}(q_{k}),e_{k,j}\rangle |, \end{aligned}$$

(2.7)

where $e_{k,j}:=q_{j}-q_{k}$. Now let $x,y\in K^{\boldsymbol{l}}$ satisfy $x\not =y$ and set $n_{0}:=\min \{n\in \mathbb {N}\mid \{x,y\}\not \subset K^{\boldsymbol{l}}_{w}\,\, \text {for any}\,\,w\in W^{\boldsymbol{l}}_{n}\}$, so that $x,y\in K^{\boldsymbol{l}}_{w}$ for a unique $w\in W^{\boldsymbol{l}}_{n_{0}-1}$ by Proposition 2.2-(1). If $K^{\boldsymbol{l}}_{wi_{x}}\cap K^{\boldsymbol{l}}_{wi_{y}}\not =\emptyset$ for some $i_{x},i_{y}\in S_{l_{n_{0}}}$ with $x\in K^{\boldsymbol{l}}_{wi_{x}}$ and $y\in K^{\boldsymbol{l}}_{wi_{y}}$, then $i_{x}\not =i_{y}$ by the definition of $n_{0}$, $q_{x,y}=F^{\boldsymbol{l}}_{wi_{x}}(q_{k})=F^{\boldsymbol{l}}_{wi_{y}}(q_{j})$ for the unique element $q_{x,y}$ of $K^{\boldsymbol{l}}_{wi_{x}}\cap K^{\boldsymbol{l}}_{wi_{y}}$ and some $j,k\in \{0,1,2\}$ with $j\not =k$ by Proposition 2.2-(1), and from (2.7) we obtain

$$\begin{aligned} d_{\boldsymbol{l}}(x,y)&\le d_{\boldsymbol{l}}(x,q_{x,y})+d_{\boldsymbol{l}}(q_{x,y},y) \\&\le 5|\langle x-q_{x,y},e_{k,j}\rangle |+5|\langle q_{x,y}-y,e_{k,j}\rangle | =5|\langle x-y,e_{k,j}\rangle | \le 5|x-y|. \end{aligned}$$

On the other hand, if $K^{\boldsymbol{l}}_{wi_{x}}\cap K^{\boldsymbol{l}}_{wi_{y}}=\emptyset$ for any $i_{x},i_{y}\in S_{l_{k}}$ with $x\in K^{\boldsymbol{l}}_{wi_{x}}$ and $y\in K^{\boldsymbol{l}}_{wi_{y}}$, then setting

$$\text{IMAGE}$$

we have $2\le n_{1}\le \frac{3}{2}l_{n_{0}}-\frac{5}{2}$, $L^{\boldsymbol{l}}_{n_{0}}d_{\boldsymbol{l}}(x,y)\le \frac{3}{2}+(n_{1}-1)+\frac{3}{2} =n_{1}+2\le \frac{3}{2}l_{n_{0}}$ by (2.6), $\frac{2}{\sqrt{3}}L^{\boldsymbol{l}}_{n_{0}}|x-y|\ge (\frac{1}{2}l_{n_{0}}-1)\wedge \lfloor \frac{1}{2}n_{1}\rfloor$, and thus $d_{\boldsymbol{l}}(x,y)/|x-y|\le \frac{10}{\sqrt{3}}<6$.    $\square$

Next, the canonical volume measure $m_{\boldsymbol{l}}$ on $K^{\boldsymbol{l}}$ is defined as follows.

Definition 2.5

We define $m_{\boldsymbol{l}}$ as the unique Borel measure on $K^{\boldsymbol{l}}$ such that

$$\begin{aligned} m_{\boldsymbol{l}}(K^{\boldsymbol{l}}_{w})=\frac{1}{M^{\boldsymbol{l}}_{|w|}} \quad \text {for any}\,\,w\in W^{\boldsymbol{l}}_{*}, \end{aligned}$$

(2.8)

where $M^{\boldsymbol{l}}_{n}:=(\# S_{l_{1}})\cdots (\# S_{l_{n}})=\prod _{k=1}^{n}(3l_{k}-3)$ ($M^{\boldsymbol{l}}_{0}:=1$) for $n\in \mathbb {N}\cup \{0\}$.

The measure $m_{\boldsymbol{l}}$ can be considered as the “uniform distribution on $K^{\boldsymbol{l}}$”. Its uniqueness stated in Definition 2.5 is immediate from the Dynkin class theorem (see, e.g., [15, Appendixes, Theorem 4.2]). It is also easily seen to be obtained as $m_{\boldsymbol{l}}=\bigl (\prod _{n=1}^{\infty }\mathop {{\text {unif}}}(S_{l_{n}})\bigr )(\pi _{\boldsymbol{l}}^{-1}(\cdot ))$, where $\mathop {{\text {unif}}}(S_{l_{n}})$ denotes the uniform distribution on $S_{l_{n}}$, $\prod _{n=1}^{\infty }\mathop {{\text {unif}}}(S_{l_{n}})$ their product probability measure on $\prod _{n=1}^{\infty }S_{l_{n}}$ (see, e.g., [14, Theorem 8.2.2] for its unique existence) and $\pi _{\boldsymbol{l}}:\prod _{n=1}^{\infty }S_{l_{n}}\rightarrow K^{\boldsymbol{l}}$ the continuous surjection given by $\{\pi _{\boldsymbol{l}}((\omega _{n})_{n=1}^{\infty })\}:=\bigcap _{n=1}^{\infty }K^{\boldsymbol{l}}_{\omega _{1}\ldots \omega _{n}}$.

Now we turn to the construction of the canonical Dirichlet form (resistance form) $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ on $K^{\boldsymbol{l}}$, which is achieved by taking the “inductive limit” of a certain canonical sequence of discrete Dirichlet forms on the finite sets $\{V^{\boldsymbol{l}}_{n}\}_{n=0}^{\infty }$ via the standard method presented in [27, Chaps. 2 and 3] (see also [2, Sects. 6 and 7]). The whole construction is based on the following definition and lemma.

Definition 2.6

Recalling that $V^{\boldsymbol{l}}_{0}=V_{0}$, we define a non-negative definite symmetric bilinear form $\mathcal {E}^{0}:\mathbb {R}^{V_{0}}\times \mathbb {R}^{V_{0}}\rightarrow \mathbb {R}$ on $\mathbb {R}^{V_{0}}=\mathbb {R}^{V^{\boldsymbol{l}}_{0}}$ by

$$\begin{aligned} \mathcal {E}^{0}(u,v) :=\frac{1}{2}\sum _{j,k=0}^{2}(u(q_{j})-u(q_{k}))(v(q_{j})-v(q_{k})), \quad u,v\in \mathbb {R}^{V_{0}}, \end{aligned}$$

(2.9)

and set $r_{l}:=(\frac{2}{3}l+\frac{1}{9})^{-1}$ for each $l\in \mathbb {N}\setminus \{1,2,3,4\}$.

The value of $r_{l}$ is specifically chosen in order for the following lemma to hold.

Lemma 2.7

Let $l\in \mathbb {N}\setminus \{1,2,3,4\}$. Then for any $u\in \mathbb {R}^{V_{0}}$,

$$\begin{aligned} \min \Biggl \{\sum _{i\in S_{l}}\mathcal {E}^{0}\bigl (v\circ F^{l}_{i}|_{V_{0}},v\circ F^{l}_{i}|_{V_{0}}\bigr ) \Biggm |{v\in \mathbb {R}^{V^{l}_{1}}, v|_{V_{0}}=u}\Biggr \} =r_{l}\mathcal {E}^{0}(u,u). \end{aligned}$$

(2.10)

Proof

This is immediate from a direct calculation using the $\Delta$–Y transform (see, e.g., [27, Lemma 2.1.15]).    $\square$

We would like to define a bilinear form $\mathcal {E}^{\boldsymbol{l},n}$ on $\mathbb {R}^{V^{\boldsymbol{l}}_{n}}$ for each $n\in \mathbb {N}$ as the sum of the copies of (2.9) on $\{F^{\boldsymbol{l}}_{w}(V_{0})\}_{w\in W^{\boldsymbol{l}}_{n}}$ and then to take their limit as $n\rightarrow \infty$, which is enabled by introducing the scaling factors $R^{\boldsymbol{l}}_{n}$ suggested by Lemma 2.7 as in the following definition.

Definition 2.8

For each $n\in \mathbb {N}\cup \{0\}$, we define a non-negative definite symmetric bilinear form $\mathcal {E}^{\boldsymbol{l},n}:\mathbb {R}^{V^{\boldsymbol{l}}_{n}}\times \mathbb {R}^{V^{\boldsymbol{l}}_{n}}\rightarrow \mathbb {R}$ on $\mathbb {R}^{V^{\boldsymbol{l}}_{n}}$ by

$$\begin{aligned} \mathcal {E}^{\boldsymbol{l},n}(u,v) :=\frac{1}{R^{\boldsymbol{l}}_{n}}\sum _{w\in W^{\boldsymbol{l}}_{n}}\mathcal {E}^{0}\bigl (u\circ F^{\boldsymbol{l}}_{w}|_{V_{0}},v\circ F^{\boldsymbol{l}}_{w}|_{V_{0}}\bigr ), \quad u,v\in \mathbb {R}^{V^{\boldsymbol{l}}_{n}}, \end{aligned}$$

(2.11)

where $R^{\boldsymbol{l}}_{n}:=r_{l_{1}}\cdots r_{l_{n}}=\prod _{k=1}^{n}(\frac{2}{3}l_{k}+\frac{1}{9})^{-1}$ ($R^{\boldsymbol{l}}_{0}:=1$), so that $\mathcal {E}^{\boldsymbol{l},0}=\mathcal {E}^{0}$.

Proposition 2.9

The sequence $\{\mathcal {E}^{\boldsymbol{l},n}\}_{n=0}^{\infty }$ of forms is compatible, i.e., for any $n,k\in \mathbb {N}\cup \{0\}$ and any $u\in \mathbb {R}^{V^{\boldsymbol{l}}_{n}}$,

$$\begin{aligned} \min \bigl \{\mathcal {E}^{\boldsymbol{l},n+k}(v,v)\bigm |{v\in \mathbb {R}^{V^{\boldsymbol{l}}_{n+k}}, v|_{V^{\boldsymbol{l}}_{n}}=u}\bigr \} =\mathcal {E}^{\boldsymbol{l},n}(u,u). \end{aligned}$$

(2.12)

Proof

This is immediate from an induction on k based on Lemma 2.7.    $\square$

Proposition 2.9 allows us to take the “inductive limit” of $\{\mathcal {E}^{\boldsymbol{l},n}\}_{n=0}^{\infty }$ as in the following definition. Note that $\{\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},u|_{V^{\boldsymbol{l}}_{n}})\}_{n=0}^{\infty }\subset [0,\infty )$ is non-decreasing by (2.12) and hence has a limit in $[0,\infty ]$ for any $u\in \mathbb {R}^{V^{\boldsymbol{l}}_{*}}$.

Definition 2.10

We define a linear subspace $\mathcal {F}_{\boldsymbol{l}}$ of $\mathbb {R}^{V^{\boldsymbol{l}}_{*}}$ and a non-negative definite symmetric bilinear form $\mathcal {E}^{\boldsymbol{l}}:\mathcal {F}_{\boldsymbol{l}}\times \mathcal {F}_{\boldsymbol{l}}\rightarrow \mathbb {R}$ on $\mathcal {F}_{\boldsymbol{l}}$ by

$$\begin{aligned} \mathcal {F}_{\boldsymbol{l}}&:=\Bigl \{u\in \mathbb {R}^{V^{\boldsymbol{l}}_{*}}\Bigm |\lim _{n\rightarrow \infty }\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},u|_{V^{\boldsymbol{l}}_{n}})<\infty \Bigr \}, \end{aligned}$$

(2.13)

$$\begin{aligned} \mathcal {E}^{\boldsymbol{l}}(u,v)&:=\lim _{n\rightarrow \infty }\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},v|_{V^{\boldsymbol{l}}_{n}})\in \mathbb {R}, \quad u,v\in \mathcal {F}_{\boldsymbol{l}}. \end{aligned}$$

(2.14)

Then applying [27, Lemma 2.2.2, Proposition 2.2.4, Lemma 2.2.5 and Theorem 2.2.6] on the basis of Proposition 2.9, we obtain the following proposition. See [27, Definition 2.3.1] or [28, Definition 3.1] for the notion of resistance forms.

Proposition 2.11

$(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is a resistance form on $V^{\boldsymbol{l}}_{*}$, i.e., the following hold:

• (RF1) $\{u\in \mathcal {F}_{\boldsymbol{l}}\mid \mathcal {E}^{\boldsymbol{l}}(u,u)=0\}=\mathbb {R}\mathbbm {1}_{V^{\boldsymbol{l}}_{*}}$.

• (RF2) $(\mathcal {F}_{\boldsymbol{l}}/\mathbb {R}\mathbbm {1}_{V^{\boldsymbol{l}}_{*}},\mathcal {E}^{\boldsymbol{l}})$ is a Hilbert space.

• (RF3) $\{u|_{V}\mid u\in \mathcal {F}_{\boldsymbol{l}}\}=\mathbb {R}^{V}$ for any non-empty finite subset V of $V^{\boldsymbol{l}}_{*}$.

• (RF4) $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y):=\sup \biggl \{\dfrac{|u(x)-u(y)|^{2}}{\mathcal {E}^{\boldsymbol{l}}(u,u)}\biggm |u\in \mathcal {F}_{\boldsymbol{l}}\setminus \mathbb {R}\mathbbm {1}_{V^{\boldsymbol{l}}_{*}}\biggr \}<\infty$ for any $x,y\in V^{\boldsymbol{l}}_{*}$.

• (RF5) $u^{+}\wedge 1\in \mathcal {F}_{\boldsymbol{l}}$ and $\mathcal {E}^{\boldsymbol{l}}(u^{+}\wedge 1,u^{+}\wedge 1)\le \mathcal {E}^{\boldsymbol{l}}(u,u)$ for any $u\in \mathcal {F}_{\boldsymbol{l}}$.

Moreover, $R_{\mathcal {E}^{\boldsymbol{l}}}: V^{\boldsymbol{l}}_{*}\times V^{\boldsymbol{l}}_{*}\rightarrow [0,\infty )$ is a metric on $V^{\boldsymbol{l}}_{*}$, called the resistance metric of $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$, and for any $u\in \mathcal {F}_{\boldsymbol{l}}$ and any $x,y\in V^{\boldsymbol{l}}_{*}$,

$$\begin{aligned} |u(x)-u(y)|^{2}\le R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\mathcal {E}^{\boldsymbol{l}}(u,u). \end{aligned}$$

(2.15)

Recalling Proposition 2.2-(3), we also see from the above construction that the following (non-exact) self-similarity of $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ holds.

Proposition 2.12

Let $n\in \mathbb {N}\cup \{0\}$. Then

$$\begin{aligned} \mathcal {F}_{\boldsymbol{l}}&=\bigl \{u\in \mathbb {R}^{V^{\boldsymbol{l}}_{*}}\bigm |{u\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}\in \mathcal {F}_{\boldsymbol{l}^{n}}\quad for\,any\, \,w\in W^{\boldsymbol{l}}_{n}}\bigr \}, \end{aligned}$$

(2.16)

$$\begin{aligned} \mathcal {E}^{\boldsymbol{l}}(u,v)&=\frac{1}{R^{\boldsymbol{l}}_{n}}\sum _{w\in W^{\boldsymbol{l}}_{n}}\mathcal {E}^{\boldsymbol{l}^{n}}\bigl (u\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}},v\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}\bigr ) \quad \text {for any u,v}\in \mathcal {F}_{\boldsymbol{l}}. \end{aligned}$$

(2.17)

Proof

It follows from Proposition 2.2-(3) and (2.11) that for each $n,k\in \mathbb {N}\cup \{0\}$,

$$\begin{aligned} \mathcal {E}^{\boldsymbol{l},n+k}(u,v) =\frac{1}{R^{\boldsymbol{l}}_{n}}\sum _{w\in W^{\boldsymbol{l}}_{n}}\mathcal {E}^{\boldsymbol{l}^{n},k}\bigl (u\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{k}},v\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{k}}\bigr ) \quad \text {for any}\,\,u,v\in \mathbb {R}^{V^{\boldsymbol{l}}_{n+k}}, \end{aligned}$$

which together with (2.13) and (2.14) immediately yields (2.16) and (2.17).    $\square$

Lemma 2.13

For any $w\in W^{\boldsymbol{l}}_{*}$ and any $x,y\in V^{\boldsymbol{l}^{|w|}}_{*}$,

$$\begin{aligned} R_{\mathcal {E}^{\boldsymbol{l}}}(F^{\boldsymbol{l}}_{w}(x),F^{\boldsymbol{l}}_{w}(y)) \le R^{\boldsymbol{l}}_{|w|}R_{\mathcal {E}^{\boldsymbol{l}^{|w|}}}(x,y). \end{aligned}$$

(2.18)

Proof

This is immediate from Proposition 2.12 and Proposition 2.11-(RF4).    $\square$

Later we will use the following definition and proposition several times.

Definition 2.14

Let $h\in \mathbb {R}^{V^{\boldsymbol{l}}_{*}}$ and $n\in \mathbb {N}\cup \{0\}$. We say that h is $\mathcal {E}^{\boldsymbol{l}}$-harmonic off $V^{\boldsymbol{l}}_{n}$ if and only if $h\in \mathcal {F}_{\boldsymbol{l}}$ and

$$\begin{aligned} \begin{aligned} \mathcal {E}^{\boldsymbol{l}}(h,h)&=\min \{\mathcal {E}^{\boldsymbol{l}}(v,v)\mid {v\in \mathcal {F}_{\boldsymbol{l}}, v|_{V^{\boldsymbol{l}}_{n}}=h|_{V^{\boldsymbol{l}}_{n}}}\}, \\ \text {or equivalently,}\quad \mathcal {E}^{\boldsymbol{l}}(h,v)&=0 \quad \text {for any}\,\,v\in \mathcal {F}_{\boldsymbol{l}}\,\text {with}\, v|_{V^{\boldsymbol{l}}_{n}}=0. \end{aligned} \end{aligned}$$

(2.19)

We set $\mathcal {H}_{\boldsymbol{l},n}:=\{h\in \mathbb {R}^{V^{\boldsymbol{l}}_{*}}\mid$  h  $\text {is}\,\, \mathcal {E}^{\boldsymbol{l}} \text {-harmonic off}\,\, V^{\boldsymbol{l}}_{n}\}$, which is a linear subspace of $\mathcal {F}_{\boldsymbol{l}}$.

Proposition 2.15

Let $n\in \mathbb {N}\cup \{0\}$. Then for each $h\in \mathbb {R}^{V^{\boldsymbol{l}}_{*}}$, the following four conditions (1), (2), (3) and (4) are equivalent to each other and imply (5) below:

• (1) $h\in \mathcal {H}_{\boldsymbol{l},n}$.

• (2) $\sum _{y\in V^{\boldsymbol{l}}_{n+k},\,L^{\boldsymbol{l}}_{n+k}d_{\boldsymbol{l}}(x,y)=1}(h(y)-h(x))=0$ for any $k\in \mathbb {N}$ and any $x\in V^{\boldsymbol{l}}_{n+k}\setminus V^{\boldsymbol{l}}_{n}$.

• (3) $h\in \mathcal {F}_{\boldsymbol{l}}$ and $\mathcal {E}^{\boldsymbol{l}}(h,h)=\mathcal {E}^{\boldsymbol{l},n}(h|_{V^{\boldsymbol{l}}_{n}},h|_{V^{\boldsymbol{l}}_{n}})$.

• (4) $h\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}\in \mathcal {H}_{\boldsymbol{l}^{n},0}$ for any $w\in W^{\boldsymbol{l}}_{n}$.

• (5) (Maximum principle) For any $w\in W^{\boldsymbol{l}}_{n}$ and any $x\in F^{\boldsymbol{l}}_{w}(V^{\boldsymbol{l}^{n}}_{*})$,

  $$\begin{aligned} \min _{q\in F^{\boldsymbol{l}}_{w}(V_{0})}h(q)\le h(x)\le \max _{q\in F^{\boldsymbol{l}}_{w}(V_{0})}h(q). \end{aligned}$$

  (2.20)

Also, for each $u\in \mathbb {R}^{V^{\boldsymbol{l}}_{n}}$ there exists a unique $h^{\boldsymbol{l}}_{n}(u)\in \mathcal {H}_{\boldsymbol{l},n}$ with $h^{\boldsymbol{l}}_{n}(u)|_{V^{\boldsymbol{l}}_{n}}=u$, and the map $h^{\boldsymbol{l}}_{n}:\mathbb {R}^{V^{\boldsymbol{l}}_{n}}\rightarrow \mathcal {H}_{\boldsymbol{l},n}$ is a linear isomorphism.

Proof

The assertions for $h^{\boldsymbol{l}}_{n}$ and the equivalence of (1), (2) and (3) follow from Proposition 2.9, [27, Lemma 2.2.2] and (2.11). Moreover, noting that $\mathcal {E}^{\boldsymbol{l}^{n}}(u,u)\ge \mathcal {E}^{0}(u|_{V_{0}},u|_{V_{0}})$ for any $u\in \mathcal {F}_{\boldsymbol{l}^{n}}$, we easily see from (2.16), (2.17) and (2.11) that (3) holds if and only if $h\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}\in \mathcal {F}_{\boldsymbol{l}^{n}}$ and $\mathcal {E}^{\boldsymbol{l}^{n}}\bigl (h\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}},h\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}\bigr ) =\mathcal {E}^{0}\bigl (h\circ F^{\boldsymbol{l}}_{w}|_{V_{0}},h\circ F^{\boldsymbol{l}}_{w}|_{V_{0}}\bigr )$ for any $w\in W^{\boldsymbol{l}}_{n}$, which is equivalent to (4) by the equivalence of (3) and (1) with $h\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}},\boldsymbol{l}^{n},0$ in place of $h,\boldsymbol{l},n$. Lastly, (4) implies (5) by [27, Lemma 2.2.3] applied to $h\circ F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}$ for each $w\in W^{\boldsymbol{l}}_{n}$.    $\square$

Note that at this stage the domain $\mathcal {F}_{\boldsymbol{l}}$ of $\mathcal {E}^{\boldsymbol{l}}$ is only a linear subspace of $\mathbb {R}^{V^{\boldsymbol{l}}_{*}}$, unlike that of a regular symmetric Dirichlet form on $L^{2}(K^{\boldsymbol{l}},m_{\boldsymbol{l}})$, which is a linear subspace of $L^{2}(K^{\boldsymbol{l}},m_{\boldsymbol{l}})$ including a dense subalgebra of $(\mathcal {C}(K^{\boldsymbol{l}}),\Vert \cdot \Vert _{\sup })$. As the last step of the construction of the canonical Dirichlet form on $K^{\boldsymbol{l}}$, we now fill this gap by proving that $\mathop {{\text {id}}}_{V^{\boldsymbol{l}}_{*}}:(V^{\boldsymbol{l}}_{*},d_{\boldsymbol{l}}|_{V^{\boldsymbol{l}}_{*}\times V^{\boldsymbol{l}}_{*}}) \rightarrow (V^{\boldsymbol{l}}_{*},R_{\mathcal {E}^{\boldsymbol{l}}})$ is uniformly continuous with uniformly continuous inverse and consequently that each $u\in \mathcal {F}_{\boldsymbol{l}}$ uniquely extends to an element of $\mathcal {C}(K^{\boldsymbol{l}})$ by virtue of (2.15).

Proposition 2.16

For any $x,y\in V^{\boldsymbol{l}}_{*}$ and any $n\in \mathbb {N}$, the following hold:

• (1) If $d_{\boldsymbol{l}}(x,y)<1/L^{\boldsymbol{l}}_{n}$, then $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\le 4R^{\boldsymbol{l}}_{n}$.

• (2) If $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<\frac{1}{6}R^{\boldsymbol{l}}_{n}$, then $d_{\boldsymbol{l}}(x,y)\le 3/L^{\boldsymbol{l}}_{n}$.

In particular, $R_{\mathcal {E}^{\boldsymbol{l}}}$ uniquely extends to $\overline{R}_{\mathcal {E}^{\boldsymbol{l}}}\in \mathcal {C}(K^{\boldsymbol{l}}\times K^{\boldsymbol{l}})$, $\overline{R}_{\mathcal {E}^{\boldsymbol{l}}}$ is a metric on $K^{\boldsymbol{l}}$ compatible with the original (Euclidean) topology of $K^{\boldsymbol{l}}$, and $\bigl ((K^{\boldsymbol{l}},\overline{R}_{\mathcal {E}^{\boldsymbol{l}}}),\mathop {{\text {id}}}_{V^{\boldsymbol{l}}_{*}}\bigr )$ is the completion of $(V^{\boldsymbol{l}}_{*},R_{\mathcal {E}^{\boldsymbol{l}}})$.

Proof

We essentially follow [28, Chap. 22], but the possible unboundedness of $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }$ requires some additional care. First, since

$$\begin{aligned} R_{\mathcal {E}^{\boldsymbol{l}}}(q_{j},q_{k}) =\bigl (\min \{\mathcal {E}^{0}(u,u)\mid {u\in \mathbb {R}^{{V}_{0}}, u(q_{j})=1, u(q_{k})=0}\}\bigr )^{-1} =\frac{2}{3} \end{aligned}$$

for any $j,k\in \{0,1,2\}$ with $j\not =k$ by [27, (2.2.3) and Lemma 2.2.5], it follows from Lemma 2.13 that for any $w\in W^{\boldsymbol{l}}_{*}$ and any $j,k\in \{0,1,2\}$,

$$\begin{aligned} R_{\mathcal {E}^{\boldsymbol{l}}}(F^{\boldsymbol{l}}_{w}(q_{j}), F^{\boldsymbol{l}}_{w}(q_{k})) \le \frac{2}{3}R^{\boldsymbol{l}}_{|w|}. \end{aligned}$$

(2.21)

Recalling that $R_{\mathcal {E}^{\boldsymbol{l}}}$ is a metric on $V^{\boldsymbol{l}}_{*}$ as stated in Proposition 2.11, we easily see from (2.21) and the triangle inequality for $R_{\mathcal {E}^{\boldsymbol{l}}}$ that for any $x\in V^{\boldsymbol{l}}_{*}$,

$$\begin{aligned} \max _{k\in \{0,1,2\}}R_{\mathcal {E}^{\boldsymbol{l}}}(q_{k},x) \le \sum _{n=1}^{\infty }\frac{3l_{n}-5}{2}\cdot \frac{2}{3}R^{\boldsymbol{l}}_{n} \le \frac{l_{1}-\frac{5}{3}+\sum _{n=2}^{\infty }\frac{3}{2}(\frac{9}{31})^{n-2}}{\frac{2}{3}l_{1}+\frac{1}{9}} <2, \end{aligned}$$

(2.22)

which together with Lemma 2.13 further implies that for any $w\in W^{\boldsymbol{l}}_{*}$ and any $x\in F^{\boldsymbol{l}}_{w}(V^{\boldsymbol{l}^{|w|}}_{*})$,

$$\begin{aligned} \max _{k\in \{0,1,2\}}R_{\mathcal {E}^{\boldsymbol{l}}}(F^{\boldsymbol{l}}_{w}(q_{k}),x)<2R^{\boldsymbol{l}}_{|w|}. \end{aligned}$$

(2.23)

To see (1) and (2), let $x,y\in V^{\boldsymbol{l}}_{*}$, $n\in \mathbb {N}$, choose $w\in W^{\boldsymbol{l}}_{n}$ so that $x\in K^{\boldsymbol{l}}_{w}$, and set $\Lambda _{n,w}:=\{v\in W^{\boldsymbol{l}}_{n}\mid K^{\boldsymbol{l}}_{w}\cap K^{\boldsymbol{l}}_{v}\not =\emptyset \}$ and $U_{n,w}:=\bigcup _{v\in \Lambda _{n,w}}K^{\boldsymbol{l}}_{v}$. It holds that

$$\begin{aligned} \text {if}\,\, y\in U_{n,w}, \quad \text {then}\,\, d_{\boldsymbol{l}}(x,y)<\frac{3}{L^{\boldsymbol{l}}_{n}} \, \,\text {and}\, \, R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<4R^{\boldsymbol{l}}_{n} \end{aligned}$$

(2.24)

by (2.3), the triangle inequality for $d_{\boldsymbol{l}}$ and $R_{\mathcal {E}^{\boldsymbol{l}}}$, (2.6) and (2.23). On the other hand, if $y\not \in U_{n,w}$, then clearly $d_{\boldsymbol{l}}(x,y)\ge 1/L^{\boldsymbol{l}}_{n}$ by (2.3) and (2.4), and recalling Proposition 2.15 and setting $h_{n,w}:=h^{\boldsymbol{l}}_{n}(\mathbbm {1}_{F^{\boldsymbol{l}}_{w}(V_{0})})$, we have $h_{n,w}|_{F^{\boldsymbol{l}}_{w}(V^{\boldsymbol{l}^{n}}_{*})}=1$, $h_{n,w}|_{F^{\boldsymbol{l}}_{v}(V^{\boldsymbol{l}^{n}}_{*})}=0$ for any $v\in W^{\boldsymbol{l}}_{n}\setminus \Lambda _{n,w}$, $\mathcal {E}^{\boldsymbol{l}}(h_{n,w},h_{n,w}) =\mathcal {E}^{\boldsymbol{l},n}(\mathbbm {1}_{F^{\boldsymbol{l}}_{w}(V_{0})},\mathbbm {1}_{F^{\boldsymbol{l}}_{w}(V_{0})})$, and therefore

$$\begin{aligned} R_{\mathcal {E}^{\boldsymbol{l}}}(x,y) \ge \frac{|h_{n,w}(x)-h_{n,w}(y)|^{2}}{\mathcal {E}^{\boldsymbol{l}}(h_{n,w},h_{n,w})} =\frac{1}{\mathcal {E}^{\boldsymbol{l},n}(\mathbbm {1}_{F^{\boldsymbol{l}}_{w}(V_{0})},\mathbbm {1}_{F^{\boldsymbol{l}}_{w}(V_{0})})} =\frac{\frac{1}{2}R^{\boldsymbol{l}}_{n}}{\#\Lambda _{n,w}-1} \ge \frac{R^{\boldsymbol{l}}_{n}}{6} \end{aligned}$$

by Proposition 2.11-(RF4), (2.11), (2.9), Proposition 2.2-(1) and $\#\Lambda _{n,w}\le 4$. It follows that, if either $d_{\boldsymbol{l}}(x,y)<1/L^{\boldsymbol{l}}_{n}$ or $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<\frac{1}{6}R^{\boldsymbol{l}}_{n}$, then $y\in U_{n,w}$, hence $d_{\boldsymbol{l}}(x,y)<3/L^{\boldsymbol{l}}_{n}$ and $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<4R^{\boldsymbol{l}}_{n}$ by (2.24), proving (1) and (2), which in turn immediately imply the existence and the stated properties of $\overline{R}_{\mathcal {E}^{\boldsymbol{l}}}$.    $\square$

Definition 2.17

Throughout the rest of this paper, we identify $\mathcal {F}_{\boldsymbol{l}}$ with the linear subspace of $\mathcal {C}(K^{\boldsymbol{l}})$ given by

$$\begin{aligned} \{u\in \mathcal {C}(K^{\boldsymbol{l}})\mid u|_{V^{\boldsymbol{l}}_{*}}\in \mathcal {F}_{\boldsymbol{l}}\} =\Bigl \{u\in \mathcal {C}(K^{\boldsymbol{l}})\Bigm |\lim _{n\rightarrow \infty }\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},u|_{V^{\boldsymbol{l}}_{n}})<\infty \Bigr \} \end{aligned}$$

(2.25)

through the mapping $u\mapsto u|_{V^{\boldsymbol{l}}_{*}}$, which is a linear isomorphism from (2.25) to $\mathcal {F}_{\boldsymbol{l}}$ since each $u\in \mathcal {F}_{\boldsymbol{l}}$ uniquely extends to an element of $\mathcal {C}(K^{\boldsymbol{l}})$ by Proposition 2.16 and (2.15). The pair $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is then called the canonical resistance form on $K^{\boldsymbol{l}}$.

Theorem 2.18

• (1) $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is a resistance form on $K^{\boldsymbol{l}}$ with resistance metric $\overline{R}_{\mathcal {E}^{\boldsymbol{l}}}$, which is hereafter denoted as $R_{\mathcal {E}^{\boldsymbol{l}}}$ for simplicity of the notation.

• (2) $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is regular, i.e., $\mathcal {F}_{\boldsymbol{l}}$ is a dense subalgebra of $(\mathcal {C}(K^{\boldsymbol{l}}),\Vert \cdot \Vert _{\sup })$.

• (3) $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is strongly local, i.e., $\mathcal {E}^{\boldsymbol{l}}(u,v)=0$ for any $u,v\in \mathcal {F}_{\boldsymbol{l}}$ that satisfy $\mathop {{\text {supp}}}_{K}[u-a\mathbbm {1}_{K^{\boldsymbol{l}}}]\cap \mathop {{\text {supp}}}_{K}[v]=\emptyset$ for some $a\in \mathbb {R}$.

Proof

(1) follows from Propositions 2.11, 2.16, Definition 2.17, [27, Lemma 2.3.9 and Theorem 2.3.10], (2) from (1), the compactness of $(K^{\boldsymbol{l}},R_{\mathcal {E}^{\boldsymbol{l}}})$, [28, Corollary 6.4 and Lemma 6.5], and (3) from $\mathbbm {1}_{K^{\boldsymbol{l}}}\in \mathcal {F}_{\boldsymbol{l}}$, $\mathcal {E}^{\boldsymbol{l}}(\mathbbm {1}_{K^{\boldsymbol{l}}},\mathbbm {1}_{K^{\boldsymbol{l}}})=0$ and (2.17).    $\square$

Remark 2.19

• (1) To be explicit, Theorem 2.18-(1) means the following: Proposition 2.11-(RF1), (RF2), (RF3), (RF5) with $K^{\boldsymbol{l}}$ in place of $V^{\boldsymbol{l}}_{*}$ hold and $\overline{R}_{\mathcal {E}^{\boldsymbol{l}}}(x,y)=\sup \bigl \{|u(x)-u(y)|^{2}/\mathcal {E}^{\boldsymbol{l}}(u,u) \bigm |u\in \mathcal {F}_{\boldsymbol{l}}\setminus \mathbb {R}\mathbbm {1}_{K^{\boldsymbol{l}}}\bigr \}$ for any $x,y\in K^{\boldsymbol{l}}$.

• (2) Under the conventions introduced in Definition 2.17 and Theorem 2.18-(1), we easily get the following, which we will utilize below without further notice:

  • (2.15) for any $u\in \mathcal {F}_{\boldsymbol{l}}$ and any $x,y\in K^{\boldsymbol{l}}$;

  • Proposition 2.12 with $\mathcal {C}(K^{\boldsymbol{l}}),F^{\boldsymbol{l}}_{w}$ in place of $\mathbb {R}^{V^{\boldsymbol{l}}_{*}},F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}}$;

  • Lemma 2.13 with $K^{\boldsymbol{l}^{|w|}}$ in place of $V^{\boldsymbol{l}^{|w|}}_{*}$;

  • Proposition 2.15 with $\mathcal {C}(K^{\boldsymbol{l}}),F^{\boldsymbol{l}}_{w},K^{\boldsymbol{l}}_{w}$ in place of $\mathbb {R}^{V^{\boldsymbol{l}}_{*}},F^{\boldsymbol{l}}_{w}|_{V^{\boldsymbol{l}^{n}}_{*}},F^{\boldsymbol{l}}_{w}(V^{\boldsymbol{l}^{n}}_{*})$;

  • Proposition 2.16-(1), (2) for any $x,y\in K^{\boldsymbol{l}}$ and any $n\in \mathbb {N}$;

  • (2.23) for any $w\in W^{\boldsymbol{l}}_{*}$ and any $x\in K^{\boldsymbol{l}}_{w}$.

Finally, we can now consider $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ as an irreducible, strongly local, regular symmetric Dirichlet form over $K^{\boldsymbol{l}}$ as follows. See [17, Sects. 1.1 and 1.6] or [13, Sects. 1.1, 1.3 and 2.1] for the definitions of the relevant notions.

Theorem 2.20

Let $\mu$ be a Radon measure on $K^{\boldsymbol{l}}$ with full support, i.e., a Borel measure on $K^{\boldsymbol{l}}$ with $\mu (K^{\boldsymbol{l}})<\infty$ and $\mu (K^{\boldsymbol{l}}_{w})>0$ for any $w\in W^{\boldsymbol{l}}_{*}$. Then $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is an irreducible, strongly local regular symmetric Dirichlet form on $L^{2}(K^{\boldsymbol{l}},\mu )$.

Proof

Since $\mathcal {C}(K^{\boldsymbol{l}})$ is dense in $L^{2}(K^{\boldsymbol{l}},\mu )$ by [34, Theorem 3.14], $\mathcal {F}_{\boldsymbol{l}}$ is also dense in $L^{2}(K^{\boldsymbol{l}},\mu )$ by Theorem 2.18-(2), and then $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is a regular symmetric Dirichlet form on $L^{2}(K^{\boldsymbol{l}},\mu )$ by Proposition 2.11-(RF2), (2.15), Proposition 2.11-(RF5) and Theorem 2.18-(2), strongly local by Theorem 2.18-(3), and irreducible by Proposition 2.11-(RF1) and [13, Theorem 2.1.11].    $\square$

3 Space-Time Scale Function $\varPsi _{\boldsymbol{l}}$ and $fHKE (\varPsi _{\boldsymbol{l}})$

In this section, we continue to fix an arbitrary $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$, define a space-time scale function $\varPsi _{\boldsymbol{l}}$ explicitly in terms of $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }$, and show that $(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ satisfies $fHKE (\varPsi _{\boldsymbol{l}})$. First, $\varPsi _{\boldsymbol{l}}$ is defined in a way analogous to [26, (5.11)] for the usual scale irregular Sierpiński gaskets but modified so as to take the “asymptotically one-dimensional” nature of $K^{\boldsymbol{l}}$ into account, as follows.

Definition 3.1

We define a homeomorphism $\varPsi _{\boldsymbol{l}}:[0,\infty )\rightarrow [0,\infty )$ by

$$\begin{aligned} \varPsi _{\boldsymbol{l}}(s)&:=\biggl (\frac{1}{M^{\boldsymbol{l}}_{n}} +\frac{s-1/L^{\boldsymbol{l}}_{n}}{1/L^{\boldsymbol{l}}_{n-1}-1/L^{\boldsymbol{l}}_{n}} \biggl (\frac{1}{M^{\boldsymbol{l}}_{n-1}}-\frac{1}{M^{\boldsymbol{l}}_{n}}\biggr )\biggr ) \nonumber \\&\cdot \biggl (R^{\boldsymbol{l}}_{n} +\frac{s-1/L^{\boldsymbol{l}}_{n}}{1/L^{\boldsymbol{l}}_{n-1}-1/L^{\boldsymbol{l}}_{n}} (R^{\boldsymbol{l}}_{n-1}-R^{\boldsymbol{l}}_{n})\biggr ) \nonumber \\&=\frac{1}{T^{\boldsymbol{l}}_{n}}\biggl (1+\frac{3l_{n}-4}{l_{n}-1}(L^{\boldsymbol{l}}_{n}s-1)\biggr ) \biggl (1+\frac{\frac{2}{3}l_{n}-\frac{8}{9}}{l_{n}-1}(L^{\boldsymbol{l}}_{n}s-1)\biggr ) \end{aligned}$$

(3.1)

for $n\in \mathbb {N}$ and $s\in [1/L^{\boldsymbol{l}}_{n},1/L^{\boldsymbol{l}}_{n-1}]$ and $\varPsi _{\boldsymbol{l}}(s):=s^{\beta _{\boldsymbol{l},0}}$ for $s\in \{0\}\cup [1,\infty )$, where $T^{\boldsymbol{l}}_{n}:=M^{\boldsymbol{l}}_{n}/R^{\boldsymbol{l}}_{n} =(\# S_{l_{1}}/r_{l_{1}})\cdots (\# S_{l_{n}}/r_{l_{n}}) =\prod _{k=1}^{n}(2l_{k}^{2}-\frac{5}{3}l_{k}-\frac{1}{3})$ ($T^{\boldsymbol{l}}_{0}:=1$) and $\beta _{\boldsymbol{l},0}:=\inf _{n\in \mathbb {N}}\beta _{l_{n}}$ with $\beta _{l}:=\log _{l}(\# S_{l}/r_{l})=\log _{l}(2l^{2}-\frac{5}{3}l-\frac{1}{3})\in (2,2+\log _{5}2)$ for $l\in \mathbb {N}\setminus \{1,2,3,4\}$; note that $\{\beta _{l}\}_{l=5}^{\infty }$ is strictly decreasing and converges to 2. We also set $\beta _{\boldsymbol{l},1}:=\max _{n\in \mathbb {N}}\beta _{l_{n}}$, so that $2\le \beta _{\boldsymbol{l},0}\le \beta _{\boldsymbol{l},1}\le \beta _{5}<2+\log _{5}2$.

Lemma 3.2

$\varPsi _{\boldsymbol{l}}$ satisfies (1.2) with $c_{\varPsi }=81$, $\beta _{0}=\beta _{\boldsymbol{l},0}$ and $\beta _{1}=\beta _{\boldsymbol{l},1}$.

Proof

Let $r,R\in (0,\infty )$ satisfy $r\le R$. If $r\ge 1$, then $\varPsi _{\boldsymbol{l}}(R)/\varPsi _{\boldsymbol{l}}(r)=(R/r)^{\beta _{\boldsymbol{l},0}}\le (R/r)^{\beta _{\boldsymbol{l},1}}$. Next, if $r,R\in [1/L^{\boldsymbol{l}}_{n},1/L^{\boldsymbol{l}}_{n-1}]$ for some $n\in \mathbb {N}$, then we easily see from (3.1), $1\le L^{\boldsymbol{l}}_{n}r\le L^{\boldsymbol{l}}_{n}R\le l_{n}$ and $l_{n}^{\beta _{l_{n}}-2}=2-\frac{5}{3}l_{n}^{-1}-\frac{1}{3}l_{n}^{-2}<2$ that

$$\begin{aligned} \frac{1}{9}\biggl (\frac{R}{r}\biggr )^{\beta _{\boldsymbol{l},0}} <\frac{2}{9}l_{n}^{2-\beta _{l_{n}}}\biggl (\frac{R}{r}\biggr )^{\beta _{l_{n}}} \le \frac{2}{9}\biggl (\frac{R}{r}\biggr )^{2} \le \frac{\varPsi _{\boldsymbol{l}}(R)}{\varPsi _{\boldsymbol{l}}(r)} \le \frac{9}{2}\biggl (\frac{R}{r}\biggr )^{2} \le \frac{9}{2}\biggl (\frac{R}{r}\biggr )^{\beta _{\boldsymbol{l},1}}. \end{aligned}$$

(3.2)

Lastly, if $r<1$ and no such $n\in \mathbb {N}$ exists, then we can choose $j,k\in \mathbb {N}\cup \{0\}$ with $j\le k$ so that $r\in [1/L^{\boldsymbol{l}}_{k+1},1/L^{\boldsymbol{l}}_{k})$ and $R\in [1/L^{\boldsymbol{l}}_{j},1/L^{\boldsymbol{l}}_{j-1})$, where $1/L^{\boldsymbol{l}}_{-1}:=\infty$, and by (3.1) and the definitions of $\beta _{\boldsymbol{l},0}$ and $\beta _{\boldsymbol{l},1}$ we have

$$\begin{aligned} \frac{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{j})}{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{k})} =\frac{T^{\boldsymbol{l}}_{k}}{T^{\boldsymbol{l}}_{j}} =\prod _{n=j+1}^{k}\frac{\# S_{l_{n}}}{r_{l_{n}}} =\prod _{n=j+1}^{k}l_{n}^{\beta _{l_{n}}} \in \biggl [\biggl (\frac{1/L^{\boldsymbol{l}}_{j}}{1/L^{\boldsymbol{l}}_{k}}\biggr )^{\beta _{\boldsymbol{l},0}}, \biggl (\frac{1/L^{\boldsymbol{l}}_{j}}{1/L^{\boldsymbol{l}}_{k}}\biggr )^{\beta _{\boldsymbol{l},1}}\biggr ], \end{aligned}$$

which together with (3.2) and the equality

$$\begin{aligned} \frac{\varPsi _{\boldsymbol{l}}(R)}{\varPsi _{\boldsymbol{l}}(r)} =\frac{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{k})}{\varPsi _{\boldsymbol{l}}(r)} \frac{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{j})}{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{k})} \frac{\varPsi _{\boldsymbol{l}}(R)}{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{j})} \end{aligned}$$

immediately yields (1.2) for $\varPsi _{\boldsymbol{l}}$ with $c_{\varPsi }=81$, $\beta _{0}=\beta _{\boldsymbol{l},0}$ and $\beta _{1}=\beta _{\boldsymbol{l},1}$.    $\square$

The main result of this section is the following theorem.

Theorem 3.3

$(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ satisfies $fHKE (\varPsi _{\boldsymbol{l}})$.

The rest of this section is devoted to the proof of Theorem 3.3. We will conclude it from [28, Theorem 15.10] by proving that $(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ satisfies the conditions $(DM1) _{\varPsi _{\boldsymbol{l}},d_{\boldsymbol{l}}}$ and $(DM2) _{\varPsi _{\boldsymbol{l}},d_{\boldsymbol{l}}}$ defined in [28, Definition 15.9-(3),(4)], which are the central assumptions in [28, Theorem 15.10]. A similar argument is given in [28, Chap. 24] for a large class of scale irregular Sierpiński gaskets, but the possible unboundedness of $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }$ requires some additional care.

The core of the proof of Theorem 3.3 is to establish the following proposition, which is an extension of (the proof of) Proposition 2.16 to the case where $n\in \mathbb {N}$, $k\in \{1,\ldots ,l_{n}\}$ and either $d_{\boldsymbol{l}}(x,y)<k/L^{\boldsymbol{l}}_{n}$ or $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<\frac{1}{12}kR^{\boldsymbol{l}}_{n}$.

Definition 3.4

Let $n\in \mathbb {N}$ and $w\in W^{\boldsymbol{l}}_{n}$. For each $k\in \{0,\ldots ,l_{n}\}$, we define

(3.3)

($\Lambda _{n,w}^{(0)}:=\{w\}$) and $U_{n,w}^{(k)}:=\bigcup _{v\in \Lambda _{n,w}^{(k)}}K^{\boldsymbol{l}}_{v}$, so that $2k+1\le \#\Lambda _{n,w}^{(k)}\le (6k)\vee 1$.

Proposition 3.5

Let $n\in \mathbb {N}$, $w\in W^{\boldsymbol{l}}_{n}$, $x\in K^{\boldsymbol{l}}_{w}$ and $k\in \{1,\ldots ,l_{n}\}$.

• (1) If $y\in U_{n,w}^{(k)}$, then $d_{\boldsymbol{l}}(x,y)<(k+2)/L^{\boldsymbol{l}}_{n}$ and $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<(\frac{2}{3}k+\frac{10}{3})R^{\boldsymbol{l}}_{n}$.

• (2) If $y\in K^{\boldsymbol{l}}\setminus U_{n,w}^{(k)}$, then $d_{\boldsymbol{l}}(x,y)\ge k/L^{\boldsymbol{l}}_{n}$ and $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\ge \frac{1}{12}kR^{\boldsymbol{l}}_{n}$.

• (3) $B_{d_{\boldsymbol{l}}}(x,k/L^{\boldsymbol{l}}_{n})\subset U_{n,w}^{(k)}\subset B_{d_{\boldsymbol{l}}}(x,(k+2)/L^{\boldsymbol{l}}_{n})$.

• (4) $B_{R_{\mathcal {E}^{\boldsymbol{l}}}}(x,\frac{1}{12}kR^{\boldsymbol{l}}_{n})\subset U_{n,w}^{(k)}\subset B_{R_{\mathcal {E}^{\boldsymbol{l}}}}(x,(\frac{2}{3}k+\frac{10}{3})R^{\boldsymbol{l}}_{n})$.

Proof

(1) is immediate from (2.3), the triangle inequality for $d_{\boldsymbol{l}}$ and $R_{\mathcal {E}^{\boldsymbol{l}}}$, (2.6), (2.21) and (2.23). To see (2), let $y\in K^{\boldsymbol{l}}\setminus U_{n,w}^{(k)}$. For $d_{\boldsymbol{l}}(x,y)$, by Proposition 2.4 we can take $\gamma :[0,1]\rightarrow K^{\boldsymbol{l}}$ such that $\gamma (0)=x$, $\gamma (1)=y$ and $d_{\boldsymbol{l}}(\gamma (s),\gamma (t))=|s-t|d_{\boldsymbol{l}}(x,y)$ for any $s,t\in [0,1]$, and it then follows from (2.3) and $y\not \in U_{n,w}^{(k)}$ that $\#(\gamma ^{-1}(V^{\boldsymbol{l}}_{n})\cap (0,1))\ge k+1$, which yields $d_{\boldsymbol{l}}(x,y)=\ell _{\mathbb {R}^{2}}(\gamma )\ge k/L^{\boldsymbol{l}}_{n}$. For $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)$, recalling Proposition 2.15, define $u\in \mathbb {R}^{V^{\boldsymbol{l}}_{n}}$ by

$$\begin{aligned} u(z):= {\left\{ \begin{array}{ll} 1 &{} \text {if}\,\,z\in K^{\boldsymbol{l}}_{w}=U_{n,w}^{(0)}, \\ 1-\frac{j}{k} &{} \text {if}\,\,j\in \{1,\ldots ,k\}\,\, \text {and}\,\, z\in U_{n,w}^{(j)}\setminus U_{n,w}^{(j-1)}, \\ 0 &{} \text {if}\,\,z\in K^{\boldsymbol{l}}\setminus U_{n,w}^{(k)} \end{array}\right. } \end{aligned}$$

(3.4)

for each $z\in V^{\boldsymbol{l}}_{n}$ and set $h_{n,w}^{(k)}:=h^{\boldsymbol{l}}_{n}(u)$, so that $\mathcal {E}^{\boldsymbol{l}}(h_{n,w}^{(k)},h_{n,w}^{(k)})=\mathcal {E}^{\boldsymbol{l},n}(u,u)$, $h_{n,w}^{(k)}|_{K^{\boldsymbol{l}}_{w}}=1$, $h_{n,w}^{(k)}|_{K^{\boldsymbol{l}}_{v}}=0$ for any $v\in W^{\boldsymbol{l}}_{n}\setminus \Lambda _{n,w}^{(k)}$, and $u(F^{\boldsymbol{l}}_{v}(V_{0}))\subset \{1-\frac{j-1}{k},1-\frac{j}{k}\}$ for any $j\in \{1,\ldots ,k\}$ and any $v\in \Lambda _{n,w}^{(j)}\setminus \Lambda _{n,w}^{(j-1)}$. Then combining these properties with Proposition 2.11-(RF4), (2.11), (2.9) and $\#\Lambda _{n,w}^{(k)}\le 6k$, we obtain

$$\begin{aligned} R_{\mathcal {E}^{\boldsymbol{l}}}(x,y) \ge \frac{|h_{n,w}^{(k)}(x)-h_{n,w}^{(k)}(y)|^{2}}{\mathcal {E}^{\boldsymbol{l}}(h_{n,w}^{(k)},h_{n,w}^{(k)})} =\frac{1}{\mathcal {E}^{\boldsymbol{l},n}(u,u)} \ge \frac{\frac{1}{2}k^{2}R^{\boldsymbol{l}}_{n}}{\#\Lambda _{n,w}^{(k)}-1} \ge \frac{kR^{\boldsymbol{l}}_{n}}{12}, \end{aligned}$$

which proves (2). Lastly, we also get (3) and (4) since the conjunction of (3) and (4) is clearly equivalent to that of (1) and (2).    $\square$

As an easy consequence of Proposition 3.5, we further obtain the following proposition, which contains $(DM2) _{\varPsi _{\boldsymbol{l}},d_{\boldsymbol{l}}}$ as defined in [28, Definition 15.9-(4)].

Proposition 3.6

Let $x,y\in K^{\boldsymbol{l}}$, $s\in (0,\infty )$, $n\in \mathbb {N}$ and $k\in \{2,\ldots ,l_{n}\}$.

• (1) If $s\in [(k-1)/L^{\boldsymbol{l}}_{n},k/L^{\boldsymbol{l}}_{n}]$, then

  $$\begin{aligned} \frac{1}{18}\frac{k^{2}}{T^{\boldsymbol{l}}_{n}} \le \varPsi _{\boldsymbol{l}}(s)\le \frac{9}{2}\frac{k^{2}}{T^{\boldsymbol{l}}_{n}} {and} \frac{7}{36}\frac{k}{M^{\boldsymbol{l}}_{n}} \le m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s)) \le 6\frac{k}{M^{\boldsymbol{l}}_{n}}, \end{aligned}$$

  (3.5)

  whereas if $s\in [1,3]$, then $1\le \varPsi _{\boldsymbol{l}}(s)\le 14$ and $\frac{7}{12}\le m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s))\le 1$.

• (2) If $d_{\boldsymbol{l}}(x,y)\in [(k-1)/L^{\boldsymbol{l}}_{n},k/L^{\boldsymbol{l}}_{n})$, then

  $$\begin{aligned} \frac{1}{48}kR^{\boldsymbol{l}}_{n}\le R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\le \frac{7}{3}kR^{\boldsymbol{l}}_{n}, \end{aligned}$$

  (3.6)

  whereas $d_{\boldsymbol{l}}(x,y)<3$, $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<4$, and if $d_{\boldsymbol{l}}(x,y)\ge 1$ then $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\ge \frac{1}{14}$.

• (3) If $x\not =y$, then

  $$\begin{aligned} 6^{-4}\frac{\varPsi _{\boldsymbol{l}}(d_{\boldsymbol{l}}(x,y))}{m_{\boldsymbol{l}}\bigl (B_{d_{\boldsymbol{l}}}(x,d_{\boldsymbol{l}}(x,y))\bigr )} \le R_{\mathcal {E}^{\boldsymbol{l}}}(x,y) \le 2^{8}\frac{\varPsi _{\boldsymbol{l}}(d_{\boldsymbol{l}}(x,y))}{m_{\boldsymbol{l}}\bigl (B_{d_{\boldsymbol{l}}}(x,d_{\boldsymbol{l}}(x,y))\bigr )}. \end{aligned}$$

  (3.7)

Proof

• (1) Assume that $s\in [(k-1)/L^{\boldsymbol{l}}_{n},k/L^{\boldsymbol{l}}_{n}]$. By (3.1) and (3.2) we have

  $$\begin{aligned} \begin{aligned} \frac{1}{18}\frac{k^{2}}{T^{\boldsymbol{l}}_{n}} \le \frac{2}{9}\frac{(k-1)^{2}}{T^{\boldsymbol{l}}_{n}} =\frac{2}{9}(k-1)^{2}\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})&\le \varPsi _{\boldsymbol{l}}((k-1)/L^{\boldsymbol{l}}_{n}) \\ \le \varPsi _{\boldsymbol{l}}(s) \le \varPsi _{\boldsymbol{l}}(k/L^{\boldsymbol{l}}_{n})&\le \frac{9}{2}k^{2}\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n}) =\frac{9}{2}\frac{k^{2}}{T^{\boldsymbol{l}}_{n}}. \end{aligned} \end{aligned}$$

  (3.8)

  For $m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s))$, choosing $w\in W^{\boldsymbol{l}}_{n}$ so that $x\in K^{\boldsymbol{l}}_{w}$, we see from Proposition 3.5-(3), (2.8) and $2j+1\le \#\Lambda _{n,w}^{(j)}\le 6j$ for $j\in \{1,\ldots ,l_{n}\}$ that

  $$\begin{aligned} m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s)) \le m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,k/L^{\boldsymbol{l}}_{n})) \le m_{\boldsymbol{l}}(U_{n,w}^{(k)}) =\frac{\#\Lambda _{n,w}^{(k)}}{M^{\boldsymbol{l}}_{n}} \le 6\frac{k}{M^{\boldsymbol{l}}_{n}} \end{aligned}$$

  (3.9)

  and that, provided $k\ge 4$,

  $$\begin{aligned} m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s)) \ge m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,(k-1)/L^{\boldsymbol{l}}_{n})) \ge m_{\boldsymbol{l}}(U_{n,w}^{(k-3)}) =\frac{\#\Lambda _{n,w}^{(k-3)}}{M^{\boldsymbol{l}}_{n}} \ge \frac{2k-5}{M^{\boldsymbol{l}}_{n}}. \end{aligned}$$

  (3.10)

  If $k\in \{2,3\}$, then choosing $v\in W^{\boldsymbol{l}}_{n+1}$ so that $x\in K^{\boldsymbol{l}}_{v}$, by Proposition 3.5-(3), (2.8) and $\#\Lambda _{n+1,v}^{(l_{n+1}-2)}\ge 2l_{n+1}-3$ we get

  $$\begin{aligned} \begin{aligned} m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s))&\ge m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,1/L^{\boldsymbol{l}}_{n})) = m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,l_{n+1}/L^{\boldsymbol{l}}_{n+1})) \\&\ge m_{\boldsymbol{l}}(U_{n+1,v}^{(l_{n+1}-2)}) =\frac{\#\Lambda _{n+1,v}^{(l_{n+1}-2)}}{M^{\boldsymbol{l}}_{n+1}} \ge \frac{2l_{n+1}-3}{(3l_{n+1}-3)M^{\boldsymbol{l}}_{n}} \ge \frac{7}{12}\frac{1}{M^{\boldsymbol{l}}_{n}}. \end{aligned} \end{aligned}$$

  (3.11)

  (3.8), (3.9), (3.10) and (3.11) together yield (3.5). On the other hand, if $s\in [1,3]$, then $\varPsi _{\boldsymbol{l}}(s)=s^{\beta _{\boldsymbol{l},0}}\in [1,3^{\beta _{5}}]\subset [1,14]$ and $1=m_{\boldsymbol{l}}(K^{\boldsymbol{l}}) \ge m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s)) \ge m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,1)) \ge \frac{7}{12}$ by (3.11) with $n=0$.

• (2) Assume that $d_{\boldsymbol{l}}(x,y)\in [(k-1)/L^{\boldsymbol{l}}_{n},k/L^{\boldsymbol{l}}_{n})$. Then by Proposition 3.5-(3), (4), it follows from $d_{\boldsymbol{l}}(x,y)<k/L^{\boldsymbol{l}}_{n}$ that $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\le (\frac{2}{3}k+\frac{10}{3})R^{\boldsymbol{l}}_{n}\le \frac{7}{3}kR^{\boldsymbol{l}}_{n}$, from $d_{\boldsymbol{l}}(x,y)\ge (k-1)/L^{\boldsymbol{l}}_{n}$ that $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\ge \frac{1}{12}(k-3)R^{\boldsymbol{l}}_{n}\ge \frac{1}{48}kR^{\boldsymbol{l}}_{n}$ provided $k\ge 4$, and from $d_{\boldsymbol{l}}(x,y)\ge 1/L^{\boldsymbol{l}}_{n}=l_{n+1}/L^{\boldsymbol{l}}_{n+1}$ that, provided $k\in \{2,3\}$,

  $$\begin{aligned} R_{\mathcal {E}^{\boldsymbol{l}}}(x,y) \ge \frac{1}{12}(l_{n+1}-2)R^{\boldsymbol{l}}_{n+1} =\frac{1}{12}\frac{l_{n+1}-2}{\frac{2}{3}l_{n+1}+\frac{1}{9}}R^{\boldsymbol{l}}_{n} \ge \frac{9}{124}R^{\boldsymbol{l}}_{n} >\frac{kR^{\boldsymbol{l}}_{n}}{48}, \end{aligned}$$

  (3.12)

  proving (3.6).

  On the other hand, $d_{\boldsymbol{l}}(x,y)<3$ by (2.6), $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)<4$ by (2.23), and if $d_{\boldsymbol{l}}(x,y)\ge 1$ then $R_{\mathcal {E}^{\boldsymbol{l}}}(x,y)\ge \frac{9}{124}>\frac{1}{14}$ by (3.12) with $n=0$.

• (3) This is immediate from (1) and (2).    $\square$

We need the following definition and lemma for the proof of the other condition $(DM1) _{{\varPsi }_{\boldsymbol{l}},d_{\boldsymbol{l}}}$ required to apply [28, Theorem 15.10].

Definition 3.7

We define homeomorphisms $\varPsi ^{\textrm{M}}_{\boldsymbol{l}},\varPsi ^{\textrm{R}}_{\boldsymbol{l}}:[0,\infty )\rightarrow [0,\infty )$ by

$$\begin{aligned} \varPsi ^{\textrm{M}}_{\boldsymbol{l}}(s)&:=\frac{1}{M^{\boldsymbol{l}}_{n}} +\frac{(s-1/L^{\boldsymbol{l}}_{n})(1/M^{\boldsymbol{l}}_{n-1}-1/M^{\boldsymbol{l}}_{n})}{1/L^{\boldsymbol{l}}_{n-1}-1/L^{\boldsymbol{l}}_{n}} =\frac{1}{M^{\boldsymbol{l}}_{n}}\biggl (1+\frac{3l_{n}-4}{l_{n}-1}(L^{\boldsymbol{l}}_{n}s-1)\biggr ), \nonumber \\ \varPsi ^{\textrm{R}}_{\boldsymbol{l}}(s)&:=R^{\boldsymbol{l}}_{n} +\frac{(s-1/L^{\boldsymbol{l}}_{n})(R^{\boldsymbol{l}}_{n-1}-R^{\boldsymbol{l}}_{n})}{1/L^{\boldsymbol{l}}_{n-1}-1/L^{\boldsymbol{l}}_{n}} =R^{\boldsymbol{l}}_{n}\biggl (1+\frac{\frac{2}{3}l_{n}-\frac{8}{9}}{l_{n}-1}(L^{\boldsymbol{l}}_{n}s-1)\biggr ) \end{aligned}$$

(3.13)

for $n\in \mathbb {N}$ and $s\in [1/L^{\boldsymbol{l}}_{n},1/L^{\boldsymbol{l}}_{n-1}]$ and $\varPsi ^{\textrm{M}}_{\boldsymbol{l}}(s):=s^{\beta ^{\textrm{M}}_{\boldsymbol{l},0}}$ and $\varPsi ^{\textrm{R}}_{\boldsymbol{l}}(s):=s^{\beta ^{\textrm{R}}_{\boldsymbol{l},1}}$ for $s\in \{0\}\cup [1,\infty )$, where $\beta ^{\textrm{M}}_{\boldsymbol{l},0}:=\inf _{n\in \mathbb {N}}\log _{l_{n}}\# S_{l_{n}}$ and $\beta ^{\textrm{R}}_{\boldsymbol{l},1}:=-\inf _{n\in \mathbb {N}}\log _{l_{n}}r_{l_{n}}$ (note that $\{\log _{l}\# S_{l}\}_{l=5}^{\infty }$ and $\{\log _{l}r_{l}\}_{l=5}^{\infty }$ are strictly decreasing), so that $\varPsi _{\boldsymbol{l}}=\varPsi ^{\textrm{M}}_{\boldsymbol{l}}\varPsi ^{\textrm{R}}_{\boldsymbol{l}}$. We also set $\beta ^{\textrm{M}}_{\boldsymbol{l},1}:=\max _{n\in \mathbb {N}}\log _{l_{n}}\# S_{l_{n}}$ and $\beta ^{\textrm{R}}_{\boldsymbol{l},0}:=-\max _{n\in \mathbb {N}}\log _{l_{n}}r_{l_{n}}$.

Lemma 3.8

• (1) $\varPsi ^{\textrm{M}}_{\boldsymbol{l}}$ satisfies (1.2) with $c_{\varPsi }=81$, $\beta _{0}=\beta ^{\textrm{M}}_{\boldsymbol{l},0}$ and $\beta _{1}=\beta ^{\textrm{M}}_{\boldsymbol{l},1}$.

• (2) $\varPsi ^{\textrm{R}}_{\boldsymbol{l}}$ satisfies (1.2) with $c_{\varPsi }=6$, $\beta _{0}=\beta ^{\textrm{R}}_{\boldsymbol{l},0}$ and $\beta _{1}=\beta ^{\textrm{R}}_{\boldsymbol{l},1}$.

Proof

These are proved in exactly the same way as Lemma 3.2.    $\square$

Finally, $(DM1) _{{\varPsi }_{\boldsymbol{l}},d_{\boldsymbol{l}}}$ as defined in [28, Definition 15.9-(3)] is deduced as follows.

Proposition 3.9

Let $x,y\in K^{\boldsymbol{l}}$ and $s\in (0,3]$. Then

$$\begin{aligned} \frac{1}{16}\varPsi ^{\textrm{M}}_{\boldsymbol{l}}(s) \le m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s)) \le 12\varPsi ^{\textrm{M}}_{\boldsymbol{l}}(s), \frac{1}{12}\varPsi ^{\textrm{R}}_{\boldsymbol{l}}(s) \le \frac{\varPsi _{\boldsymbol{l}}(s)}{m_{\boldsymbol{l}}(B_{d_{\boldsymbol{l}}}(x,s))} \le 16\varPsi ^{\textrm{R}}_{\boldsymbol{l}}(s), \end{aligned}$$

(3.14)

$$\begin{aligned} 2^{-14}\varPsi ^{\textrm{R}}_{\boldsymbol{l}}(d_{\boldsymbol{l}}(x,y)) \le R_{\mathcal {E}^{\boldsymbol{l}}}(x,y) \le 2^{12}\varPsi ^{\textrm{R}}_{\boldsymbol{l}}(d_{\boldsymbol{l}}(x,y)). \end{aligned}$$

(3.15)

In particular, if $x\not =y$, then for any $\lambda \in (0,1]$,

$$\begin{aligned} \frac{6^{-4}\lambda ^{\beta ^{\textrm{R}}_{\boldsymbol{l},1}}\varPsi _{\boldsymbol{l}}(d_{\boldsymbol{l}}(x,y))}{m_{\boldsymbol{l}}\bigl (B_{d_{\boldsymbol{l}}}(x,d_{\boldsymbol{l}}(x,y))\bigr )} \le \frac{\varPsi _{\boldsymbol{l}}(\lambda d_{\boldsymbol{l}}(x,y))}{m_{\boldsymbol{l}}\bigl (B_{d_{\boldsymbol{l}}}(x,\lambda d_{\boldsymbol{l}}(x,y))\bigr )} \le \frac{6^{4}\lambda ^{\beta ^{\textrm{R}}_{\boldsymbol{l},0}}\varPsi _{\boldsymbol{l}}(d_{\boldsymbol{l}}(x,y))}{m_{\boldsymbol{l}}\bigl (B_{d_{\boldsymbol{l}}}(x,d_{\boldsymbol{l}}(x,y))\bigr )}. \end{aligned}$$

(3.16)

Proof

If $n\in \mathbb {N}$, $k\in \{2,\ldots ,l_{n}\}$ and $s\in [(k-1)/L^{\boldsymbol{l}}_{n},k/L^{\boldsymbol{l}}_{n}]$ then we easily see from (3.13) that $\frac{1}{2}k/M^{\boldsymbol{l}}_{n}\le \varPsi ^{\textrm{M}}_{\boldsymbol{l}}(s)\le 3k/M^{\boldsymbol{l}}_{n}$, and if $s\in [1,3]$ then $\varPsi ^{\textrm{M}}_{\boldsymbol{l}}(s)=s^{\beta ^{\textrm{M}}_{\boldsymbol{l},0}}\in [1,3^{\log _{5}\# S_{5}}]\subset [1,6]$. These facts, Proposition 3.6-(1) and $\varPsi _{\boldsymbol{l}}=\varPsi ^{\textrm{M}}_{\boldsymbol{l}}\varPsi ^{\textrm{R}}_{\boldsymbol{l}}$ together imply (3.14), which in turn in combination with Proposition 3.6-(2) and Lemma 3.8-(2), respectively, yields (3.15) and (3.16) since $d_{\boldsymbol{l}}(x,y)\in [0,3)$ by (2.6).    $\square$

Proof of Theorem 3.3 By Propositions 2.4, 2.16 and Theorem 2.18, $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is a strongly local regular resistance form on $K^{\boldsymbol{l}}$ whose resistance metric $R_{\mathcal {E}^{\boldsymbol{l}}}$ gives the same topology as the geodesic metric $d_{\boldsymbol{l}}$. We also have (ACC) as defined in [28, Definition 7.4] by [28, Proposition 7.6], $(DM1) _{\varPsi _{\boldsymbol{l}},d_{\boldsymbol{l}}}$ by (3.16) and $\beta ^{\textrm{R}}_{\boldsymbol{l},0}>0$, and $(DM2) _{\varPsi _{\boldsymbol{l}},d_{\boldsymbol{l}}}$ by Proposition 3.6-(3). Thus [28, Theorem 15.10, Cases 1 and 2] are applicable to $(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ and imply that it satisfies $fHKE (\varPsi _{\boldsymbol{l}})$.    $\square$

4 Singularity of the Energy Measures

As in the previous two sections, we fix an arbitrary $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$ throughout this section. We first recall the definition of the $\mathcal {E}^{\boldsymbol{l}}$-energy measures.

Definition 4.1

($\mathcal {E}^{\boldsymbol{l}}$-energy measure; [17, (3.2.14)]) Let $u\in \mathcal {F}_{\boldsymbol{l}}$. We define the $\mathcal {E}^{\boldsymbol{l}}$-energy measure $\mu ^{\boldsymbol{l}}_{\langle u\rangle }$ of u as the unique Borel measure on $K^{\boldsymbol{l}}$ such that

$$\begin{aligned} \int \limits _{K^{\boldsymbol{l}}}f\,d\mu ^{\boldsymbol{l}}_{\langle u\rangle } =\mathcal {E}^{\boldsymbol{l}}(u,fu)-\frac{1}{2}\mathcal {E}^{\boldsymbol{l}}(u^{2},f) \quad \text {for any}\,\, f\in \mathcal {F}_{\boldsymbol{l}}; \end{aligned}$$

(4.1)

since $\mathcal {F}_{\boldsymbol{l}}$ is a dense subalgebra of $(\mathcal {C}(K^{\boldsymbol{l}}),\Vert \cdot \Vert _{\sup })$ by Theorem 2.18-(2) and

$$\begin{aligned} 0\le \mathcal {E}^{\boldsymbol{l}}(u,f^{+}u)-\frac{1}{2}\mathcal {E}^{\boldsymbol{l}}(u^{2},f^{+}) \le \Vert f\Vert _{\sup }\mathcal {E}^{\boldsymbol{l}}(u,u) \qquad \text {for any}\,\,f\in \mathcal {F}_{\boldsymbol{l}} \end{aligned}$$

(4.2)

by (2.9), (2.11) and (2.14), such $\mu ^{\boldsymbol{l}}_{\langle u\rangle }$ exists and is unique by the Riesz(–Markov–Kakutani) representation theorem (see, e.g., [34, Theorems 2.14 and 2.18]).

Proposition 2.12 yields the following alternative characterization of $\mu ^{\boldsymbol{l}}_{\langle u\rangle }$.

Proposition 4.2

Let $u\in \mathcal {F}_{\boldsymbol{l}}$. Then $\mu ^{\boldsymbol{l}}_{\langle u\rangle }(\{x\})=0$ for any $x\in K^{\boldsymbol{l}}$. Moreover, $\mu ^{\boldsymbol{l}}_{\langle u\rangle }$ is the unique Borel measure on $K^{\boldsymbol{l}}$ such that

$$\begin{aligned} \mu ^{\boldsymbol{l}}_{\langle u\rangle }(K^{\boldsymbol{l}}_{w}) =\frac{1}{R^{\boldsymbol{l}}_{|w|}}\mathcal {E}^{\boldsymbol{l}^{|w|}}(u\circ F^{\boldsymbol{l}}_{w},u\circ F^{\boldsymbol{l}}_{w}) \qquad \text {for any}\,\,w\in W^{\boldsymbol{l}}_{*}. \end{aligned}$$

(4.3)

Proof

Since $(\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ is a strongly local regular symmetric Dirichlet form on $L^{2}(K^{\boldsymbol{l}},m_{\boldsymbol{l}})$ by Theorem 2.20, the Borel measure $\mu ^{\boldsymbol{l}}_{\langle u\rangle }(u^{-1}(\cdot ))$ on $\mathbb {R}$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb {R}$ by [13, Theorem 4.3.8], and therefore $\mu ^{\boldsymbol{l}}_{\langle u\rangle }(\{x\})\le \mu ^{\boldsymbol{l}}_{\langle u\rangle }(u^{-1}(u(x)))=0$ for any $x\in K^{\boldsymbol{l}}$.

The uniqueness of a Borel measure on $K^{\boldsymbol{l}}$ satisfying (4.3) is immediate from (2.17) and the Dynkin class theorem (see, e.g., [15, Appendixes, Theorem 4.2]). To show that $\mu ^{\boldsymbol{l}}_{\langle u\rangle }$ has the property (4.3), let $n,k\in \mathbb {N}\cup \{0\}$, $w\in W^{\boldsymbol{l}}_{n}$ and set $f_{k}:=h^{\boldsymbol{l}}_{n+k}(\mathbbm {1}_{K^{\boldsymbol{l}}_{w}\cap V^{\boldsymbol{l}}_{n+k}})$, so that $\mathbbm {1}_{K^{\boldsymbol{l}}_{w}}\le f_{k}\le \mathbbm {1}_{K^{\boldsymbol{l}}_{w}\cup \bigcup _{q\in F^{\boldsymbol{l}}_{w}(V_{0})}B_{d_{\boldsymbol{l}}}(q,2/L^{\boldsymbol{l}}_{n+k})}$ by Proposition 2.15 and (2.6). Then from (4.2), (2.17) and (4.1) we obtain

$$\begin{aligned} \frac{\mathcal {E}^{\boldsymbol{l}^{n}}(u\circ F^{\boldsymbol{l}}_{w},u\circ F^{\boldsymbol{l}}_{w})}{R^{\boldsymbol{l}}_{n}}&=\frac{1}{R^{\boldsymbol{l}}_{n}}\Bigl (\mathcal {E}^{\boldsymbol{l}^{n}}(u\circ F^{\boldsymbol{l}}_{w},(f_{k}u)\circ F^{\boldsymbol{l}}_{w}) -\frac{1}{2}\mathcal {E}^{\boldsymbol{l}^{n}}((u^{2})\circ F^{\boldsymbol{l}}_{w},f_{k}\circ F^{\boldsymbol{l}}_{w})\Bigr ) \\&\le \mathcal {E}^{\boldsymbol{l}}(u,f_{k}u)-\frac{1}{2}\mathcal {E}^{\boldsymbol{l}}(u^{2},f_{k}) =\int \limits _{K^{\boldsymbol{l}}}f_{k}\,d\mu ^{\boldsymbol{l}}_{\langle u\rangle } \\&\le \mu ^{\boldsymbol{l}}_{\langle u\rangle }\Bigl (K^{\boldsymbol{l}}_{w}\cup \bigcup \nolimits _{q\in F^{\boldsymbol{l}}_{w}(V_{0})}B_{d_{\boldsymbol{l}}}(q,2/L^{\boldsymbol{l}}_{n+k})\Bigr ) \xrightarrow {k\rightarrow \infty }\mu ^{\boldsymbol{l}}_{\langle u\rangle }(K^{\boldsymbol{l}}_{w}) \end{aligned}$$

and hence $\mathcal {E}^{\boldsymbol{l}^{n}}(u\circ F^{\boldsymbol{l}}_{w},u\circ F^{\boldsymbol{l}}_{w})/R^{\boldsymbol{l}}_{n} \le \mu ^{\boldsymbol{l}}_{\langle u\rangle }(K^{\boldsymbol{l}}_{w})$, where the equality necessarily holds since the sum over $w\in W^{\boldsymbol{l}}_{n}$ of each side of this inequality is equal to $\mathcal {E}^{\boldsymbol{l}}(u,u)=\mu ^{\boldsymbol{l}}_{\langle u\rangle }(K^{\boldsymbol{l}})$ by (2.17), (2.3), $\mu ^{\boldsymbol{l}}_{\langle u\rangle }(V^{\boldsymbol{l}}_{n})=0$ and (4.1) with $f=\mathbbm {1}_{K^{\boldsymbol{l}}}$.    $\square$

The purpose of this section is to prove the following theorem.

Theorem 4.3

$\mu ^{\boldsymbol{l}}_{\langle u\rangle }\perp m_{\boldsymbol{l}}$ for any $u\in \mathcal {F}_{\boldsymbol{l}}$.

The rest of this section is devoted to the proof of Theorem 4.3. First, we observe that the proof is reduced to the case of $u\in \bigcup _{n=0}^{\infty }\mathcal {H}_{\boldsymbol{l},n}$ by the following two lemmas.

Lemma 4.4

Let $u\in \mathcal {F}_{\boldsymbol{l}}$ and set $u_{n}:=h^{\boldsymbol{l}}_{n}(u|_{V^{\boldsymbol{l}}_{n}})$ for each $n\in \mathbb {N}\cup \{0\}$ (recall Proposition 2.15). Then $\mathcal {E}^{\boldsymbol{l}}(u-u_{n},u-u_{n}) =\mathcal {E}^{\boldsymbol{l}}(u,u)-\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},u|_{V^{\boldsymbol{l}}_{n}})$ for any $n\in \mathbb {N}\cup \{0\}$. In particular, $\lim _{n\rightarrow \infty }\mathcal {E}^{\boldsymbol{l}}(u-u_{n},u-u_{n})=0$.

Proof

We follow [27, Proof of Lemma 3.2.17]. Let $n\in \mathbb {N}\cup \{0\}$. Then $\mathcal {E}^{\boldsymbol{l}}(u_{n},u) =\mathcal {E}^{\boldsymbol{l}}(u_{n},u_{n}) =\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},u|_{V^{\boldsymbol{l}}_{n}})$ by $u_{n}\in \mathcal {H}_{\boldsymbol{l},n}$, (2.19) and Proposition 2.15 and thus $\mathcal {E}^{\boldsymbol{l}}(u-u_{n},u-u_{n}) =\mathcal {E}^{\boldsymbol{l}}(u,u)-\mathcal {E}^{\boldsymbol{l},n}(u|_{V^{\boldsymbol{l}}_{n}},u|_{V^{\boldsymbol{l}}_{n}})$, which converges to 0 as $n\rightarrow \infty$ by (2.14).    $\square$

Lemma 4.5

If a Borel measure $\mu$ on $K^{\boldsymbol{l}}$, $\{u_{n}\}_{n=1}^{\infty }\subset \mathcal {F}_{\boldsymbol{l}}$ and $u\in \mathcal {F}_{\boldsymbol{l}}$ satisfy $\lim _{n\rightarrow \infty }\mathcal {E}^{\boldsymbol{l}}(u-u_{n},u-u_{n})=0$ and $\mu ^{\boldsymbol{l}}_{\langle u_{n}\rangle }\perp \mu$ for any $n\in \mathbb {N}$, then $\mu ^{\boldsymbol{l}}_{\langle u\rangle }\perp \mu$.

Proof

This is a special case of [26, Lemma 3.7-(b)], whose proof works for any regular symmetric Dirichlet space.    $\square$

To prove that $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\perp m_{\boldsymbol{l}}$ for any $h\in \bigcup _{n=0}^{\infty }\mathcal {H}_{\boldsymbol{l},n}$, noting that $\mathcal {H}_{\boldsymbol{l},0}\subset \mathcal {H}_{\boldsymbol{l},1}$ and recalling Proposition 2.15, in the following lemma we calculate explicitly the matrix representation of the linear maps $\mathbb {R}^{{V}_{0}}\ni u\mapsto h^{\boldsymbol{l}}_{0}(u)\circ F^{\boldsymbol{l}}_{i}|_{V_{0}}\in \mathbb {R}^{{V}_{0}}$, $i\in W^{\boldsymbol{l}}_{1}=S_{l_{1}}$, which we identify with the linear maps $\mathcal {H}_{\boldsymbol{l},0}\ni h\mapsto h\circ F^{\boldsymbol{l}}_{i}\in \mathcal {H}_{\boldsymbol{l}^{1},0}$.

Lemma 4.6

Set $l:=l_{1}$, $a_{l}:=\frac{1}{9}r_{l}=(6l+1)^{-1}$, and for each $i\in S_{l}$ let $A^{l}_{i}$ denote the matrix representation of the linear map $\mathbb {R}^{{V}_{0}}\ni u\mapsto h^{\boldsymbol{l}}_{0}(u)\circ F^{\boldsymbol{l}}_{i}|_{V_{0}}\in \mathbb {R}^{{V}_{0}}$ with respect to the basis $(\mathbbm {1}_{q_{0}},\mathbbm {1}_{q_{1}},\mathbbm {1}_{q_{2}})$ of $\mathbb {R}^{{V}_{0}}$. Then for any $k\in \{2,\ldots ,l-3\}$,

$$\begin{aligned} \begin{aligned} A^{l}_{(k,0)}&= \begin{pmatrix} 1-(6k+3)a_{l} &{} (6k-2)a_{l} &{} 5a_{l} \\ 1-(6k+9)a_{l} &{} (6k+4)a_{l} &{} 5a_{l} \\ 1-(6k+6)a_{l} &{} (6k+1)a_{l} &{} 5a_{l} \end{pmatrix}, \\ A^{l}_{(0,k)}&= \begin{pmatrix} 1-(6k+3)a_{l} &{} 5a_{l} &{} (6k-2)a_{l} \\ 1-(6k+6)a_{l} &{} 5a_{l} &{} (6k+1)a_{l} \\ 1-(6k+9)a_{l} &{} 5a_{l} &{} (6k+4)a_{l} \end{pmatrix}, \\ A^{l}_{(l-1-k,k)}&= \begin{pmatrix} 5a_{l} &{} 1-(6k+6)a_{l} &{} (6k+1)a_{l} \\ 5a_{l} &{} 1-(6k+3)a_{l} &{} (6k-2)a_{l} \\ 5a_{l} &{} 1-(6k+9)a_{l} &{} (6k+4)a_{l} \end{pmatrix}. \end{aligned} \end{aligned}$$

(4.4)

Proof

This follows by solving the linear equation in $(h(x))_{x\in V^{l}_{1}\setminus V_{0}}$ for $h=h^{\boldsymbol{l}}_{0}(u)$ from Proposition 2.15-(2) with $(n,k)=(0,1)$ under $h(q)=u(q)$ for $q\in V_{0}$.    $\square$

Our proof that $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\perp m_{\boldsymbol{l}}$ for $h\in \bigcup _{n=0}^{\infty }\mathcal {H}_{\boldsymbol{l},n}$ is based on the following fact.

Theorem 4.7

([23, Theorem 4.1]) Let $(\Omega ,\mathscr {F},\mathbb {P})$ be a probability space and let $\{\mathscr {F}_{n}\}_{n=0}^{\infty }$ be a non-decreasing sequence of $\sigma$-algebras in $\Omega$ such that $\bigcup _{n=0}^{\infty }\mathscr {F}_{n}$ generates $\mathscr {F}$. Let $\widetilde{\mathbb {P}}$ be a probability measure on $(\Omega ,\mathscr {F})$ such that $\widetilde{\mathbb {P}}|_{\mathscr {F}_{n}}\ll \mathbb {P}|_{\mathscr {F}_{n}}$ for any $n\in \mathbb {N}\cup \{0\}$, and for each $n\in \mathbb {N}$ define $\alpha _{n}\in L^{1}(\Omega ,\mathscr {F}_{n},\mathbb {P}|_{\mathscr {F}_{n}})$ by

$$\begin{aligned} \alpha _{n}:= {\left\{ \begin{array}{ll} \dfrac{d(\widetilde{\mathbb {P}}|_{\mathscr {F}_{n}})/d(\mathbb {P}|_{\mathscr {F}_{n}})}{d(\widetilde{\mathbb {P}}|_{\mathscr {F}_{n-1}})/d(\mathbb {P}|_{\mathscr {F}_{n-1}})} &{} \text {on}\,\,\{d(\widetilde{\mathbb {P}}|_{\mathscr {F}_{n-1}})/d(\mathbb {P}|_{\mathscr {F}_{n-1}})>0\}, \\ 0 &{} \text {on}\,\, \{d(\widetilde{\mathbb {P}}|_{\mathscr {F}_{n-1}})/d(\mathbb {P}|_{\mathscr {F}_{n-1}})=0\}, \end{array}\right. } \end{aligned}$$

(4.5)

so that $\mathbb {E}[\sqrt{\alpha _{n}}\mid \mathscr {F}_{n-1}]\le 1$ $\mathbb {P}|_{\mathscr {F}_{n-1}}$-a.s. by conditional Jensen’s inequality, where $\mathbb {E}[\cdot \mid \mathscr {F}_{n-1}]$ denotes the conditional expectation given $\mathscr {F}_{n-1}$ with respect to $\mathbb {P}$. If

$$\begin{aligned} \sum _{n=1}^{\infty }(1-\mathbb {E}[\sqrt{\alpha _{n}}\mid \mathscr {F}_{n-1}])=\infty \mathbb {P}\text {-a.s.,} \end{aligned}$$

(4.6)

then $\widetilde{\mathbb {P}}\perp \mathbb {P}$.

We will apply Theorem 4.7 under the setting of the following lemma with $\mathbb {P}=m_{\boldsymbol{l}}$.

Lemma 4.8

Set $\Omega :=K^{\boldsymbol{l}}$, $\mathscr {F}:=\mathscr {B}(K^{\boldsymbol{l}})$ and let $\mathbb {P},\widetilde{\mathbb {P}}$ be probability measures on $(\Omega ,\mathscr {F})$ such that $\mathbb {P}(K^{\boldsymbol{l}}_{w})>0$ for any $w\in W^{\boldsymbol{l}}_{*}$ and $\mathbb {P}(V^{\boldsymbol{l}}_{*})=\widetilde{\mathbb {P}}(V^{\boldsymbol{l}}_{*})=0$. Set $\mathscr {F}_{n}:=\{A\cup \bigcup _{w\in \Lambda }(K^{\boldsymbol{l}}_{w}\setminus V^{\boldsymbol{l}}_{n}) \mid \Lambda \subset W^{\boldsymbol{l}}_{n}, A\subset V^{\boldsymbol{l}}_{n}\}$ for each $n\in \mathbb {N}\cup \{0\}$, so that $\{\mathscr {F}_{n}\}_{n=0}^{\infty }$ is a non-decreasing sequence of $\sigma$-algebras in $\Omega$ by (2.3), $\bigcup _{n=0}^{\infty }\mathscr {F}_{n}$ generates $\mathscr {F}$, and $\widetilde{\mathbb {P}}|_{\mathscr {F}_{n}}\ll \mathbb {P}|_{\mathscr {F}_{n}}$ for any $n\in \mathbb {N}\cup \{0\}$. Let $n\in \mathbb {N}$ and define $\alpha _{n}\in L^{1}(\Omega ,\mathscr {F}_{n},\mathbb {P}|_{\mathscr {F}_{n}})$ by (4.5). Then for each $w\in W^{\boldsymbol{l}}_{n-1}$,

$$\begin{aligned} \mathbb {E}[\sqrt{\alpha _{n}}\mid \mathscr {F}_{n-1}]|_{K^{\boldsymbol{l}}_{w}\setminus V^{\boldsymbol{l}}_{n-1}}= {\left\{ \begin{array}{ll} \displaystyle \sum _{i\in S_{l_{n}}}\sqrt{\frac{\widetilde{\mathbb {P}}(K^{\boldsymbol{l}}_{wi})}{\widetilde{\mathbb {P}}(K^{\boldsymbol{l}}_{w})}} \sqrt{\frac{\mathbb {P}(K^{\boldsymbol{l}}_{wi})}{\mathbb {P}(K^{\boldsymbol{l}}_{w})}} &{} \text {if}\quad \widetilde{\mathbb {P}}(K^{\boldsymbol{l}}_{w})>0, \\ 0 &{} \text {if}\quad \widetilde{\mathbb {P}}(K^{\boldsymbol{l}}_{w})=0. \end{array}\right. } \end{aligned}$$

(4.7)

Proof

This follows easily by direct calculations based on (4.5) and (2.3).    $\square$

The following proposition is the key step of the proof of Theorem 4.3.

Proposition 4.9

Let $k\in \mathbb {N}\cup \{0\}$, $h\in \mathcal {H}_{\boldsymbol{l},k}$, $x\in K^{\boldsymbol{l}}\setminus V^{\boldsymbol{l}}_{*}$, and let $\omega ^{x}=(\omega ^{x}_{n})_{n=1}^{\infty }$ be the element of $\prod _{n=1}^{\infty }S_{l_{n}}$, unique by (2.3), such that $\{x\}=\bigcap _{n=1}^{\infty }K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n}}$. Let $n\in \mathbb {N}\cap [k+2,\infty )$ and assume that $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )>0$ and that $\omega ^{x}_{n-1}\in S_{l_{n-1},1}$, where $S_{l,1}:=\{(i_{1},i_{2})\in S_{l}\mid i_{1}\vee i_{2}\in \{2,\ldots ,l-3\}\}$ for $l\in \mathbb {N}\setminus \{1,2,3,4\}$. Then

$$\begin{aligned} \sum _{i\in S_{l_{n}}}\sqrt{\frac{\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}i}\bigr )}{\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )}} \sqrt{\frac{m_{\boldsymbol{l}}\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}i}\bigr )}{m_{\boldsymbol{l}}\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )}} \le \sqrt{\frac{361}{372}}. \end{aligned}$$

(4.8)

Proof

Set $v:=\omega ^{x}_{1}\ldots \omega ^{x}_{n-2}$ ($v:=\emptyset$ if $n=2$) and $w:=\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}=v\omega ^{x}_{n-1}$. By $h\in \mathcal {H}_{\boldsymbol{l},k}\subset \mathcal {H}_{\boldsymbol{l},n-2}$, Proposition 2.15 and (4.3) we have $h\circ F^{\boldsymbol{l}}_{v}\in \mathcal {H}_{\boldsymbol{l}^{n-2},0}$, $h\circ F^{\boldsymbol{l}}_{w} =(h\circ F^{\boldsymbol{l}}_{v})\circ F^{\boldsymbol{l}^{n-2}}_{\omega ^{x}_{n-1}} \in \mathcal {H}_{\boldsymbol{l}^{n-1},0}$ and $\mathcal {E}^{0}(h\circ F^{\boldsymbol{l}}_{w}|_{V_{0}},h\circ F^{\boldsymbol{l}}_{w}|_{V_{0}}) =R^{\boldsymbol{l}}_{n-1}\mu ^{\boldsymbol{l}}_{\langle h\rangle }(K^{\boldsymbol{l}}_{w})>0$, and therefore $h\circ F^{\boldsymbol{l}}_{w}(V_{0})=\{c-b,c,c+b\}$ for some $b,c\in \mathbb {R}$ with $b>0$ by Lemma 4.6 with $\boldsymbol{l}^{n-2}$ in place of $\boldsymbol{l}$ applied to $i=\omega ^{x}_{n-1}\in S_{l_{n-1},1}$ and $u=h\circ F^{\boldsymbol{l}}_{v}|_{V_{0}}$. Then we see from Lemma 4.6 with $\boldsymbol{l}^{n-1}$ in place of $\boldsymbol{l}$ applied to $i\in S_{l_{n},1}$ and $u=h\circ F^{\boldsymbol{l}}_{w}|_{V_{0}}$ that $h\circ F^{\boldsymbol{l}}_{wi}(V_{0})=(h\circ F^{\boldsymbol{l}}_{w})\circ F^{\boldsymbol{l}^{n-1}}_{i}(V_{0})$ is equal to $\{c_{i}-3a_{l_{n}}b,c_{i},c_{i}+3a_{l_{n}}b\}$ for some $c_{i}\in \mathbb {R}$ for $2(l_{n}-4)$ elements i of $S_{l_{n},1}$ and to $\{c_{i}-6a_{l_{n}}b,c_{i},c_{i}+6a_{l_{n}}b\}$ for some $c_{i}\in \mathbb {R}$ for the other $l_{n}-4$ elements i of $S_{l_{n},1}$. It follows by combining this fact with (2.8), (4.3), $h\in \mathcal {H}_{\boldsymbol{l},n-1}\subset \mathcal {H}_{\boldsymbol{l},n}$, Proposition 2.15, (2.9) and (2.17) that

$$\begin{aligned}&\sum _{i\in S_{l_{n}}}\sqrt{\frac{\mu ^{\boldsymbol{l}}_{\langle h\rangle }(K^{\boldsymbol{l}}_{wi})}{\mu ^{\boldsymbol{l}}_{\langle h\rangle }(K^{\boldsymbol{l}}_{w})}} \sqrt{\frac{m_{\boldsymbol{l}}(K^{\boldsymbol{l}}_{wi})}{m_{\boldsymbol{l}}(K^{\boldsymbol{l}}_{w})}} =\sum _{i\in S_{l_{n}}}\sqrt{\frac{\mathcal {E}^{\boldsymbol{l}^{n}}(h\circ F^{\boldsymbol{l}}_{wi},h\circ F^{\boldsymbol{l}}_{wi})}{(r_{l_{n}}\# S_{l_{n}})\mathcal {E}^{\boldsymbol{l}^{n-1}}(h\circ F^{\boldsymbol{l}}_{w},h\circ F^{\boldsymbol{l}}_{w})}} \\&\le (l_{n}-4)\frac{2\cdot 3\sqrt{6}a_{l_{n}}b+6\sqrt{6}a_{l_{n}}b}{\sqrt{r_{l_{n}}\# S_{l_{n}}}\cdot \sqrt{6}b} +3\sqrt{\frac{\sum _{i\in S_{l_{n}}\setminus S_{l_{n},1}}\mathcal {E}^{\boldsymbol{l}^{n}}(h\circ F^{\boldsymbol{l}}_{wi},h\circ F^{\boldsymbol{l}}_{wi})}{(r_{l_{n}}\# S_{l_{n}})\cdot 6b^{2}}} \\&=\frac{4(l_{n}-4)}{\sqrt{\# S_{l_{n}}/a_{l_{n}}}} +3\sqrt{\frac{r_{l_{n}}\mathcal {E}^{\boldsymbol{l}^{n-1}}(h\circ F^{\boldsymbol{l}}_{w},h\circ F^{\boldsymbol{l}}_{w})-\sum _{i\in S_{l_{n},1}}\mathcal {E}^{\boldsymbol{l}^{n}}(h\circ F^{\boldsymbol{l}}_{wi},h\circ F^{\boldsymbol{l}}_{wi})}{6b^{2}r_{l_{n}}\# S_{l_{n}}}} \\&=\frac{4(l_{n}-4)}{\sqrt{\# S_{l_{n}}/a_{l_{n}}}} +3\sqrt{\frac{6b^{2}r_{l_{n}}-(l_{n}-4)(2\cdot 54a_{l_{n}}^{2}b^{2}+216a_{l_{n}}^{2}b^{2})}{6b^{2}r_{l_{n}}\# S_{l_{n}}}} \\&=\frac{4(l_{n}-4)}{\sqrt{\# S_{l_{n}}/a_{l_{n}}}} +3\sqrt{\frac{a_{l_{n}}^{-1}-6(l_{n}-4)}{\# S_{l_{n}}/a_{l_{n}}}} =\frac{4l_{n}-1}{\sqrt{(3l_{n}-3)(6l_{n}+1)}} \le \sqrt{\frac{361}{372}}, \end{aligned}$$

proving (4.8).    $\square$

Proof of Theorem 4.3 Let $k\in \mathbb {N}\cup \{0\}$ and $h\in \mathcal {H}_{\boldsymbol{l},k}$. In view of Lemmas 4.4 and 4.5 it suffices to prove, for any such k and h, that $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\perp m_{\boldsymbol{l}}$, which is obvious if $\mu ^{\boldsymbol{l}}_{\langle h\rangle }(K^{\boldsymbol{l}})=0$. Assume that $\mu ^{\boldsymbol{l}}_{\langle h\rangle }(K^{\boldsymbol{l}})>0$, set $(\Omega ,\mathscr {F},\mathbb {P}):=(K^{\boldsymbol{l}},\mathscr {B}(K^{\boldsymbol{l}}),m_{\boldsymbol{l}})$, let $\{\mathscr {F}_{n}\}_{n=0}^{\infty }$ denote the non-decreasing sequence of $\sigma$-algebras in $\Omega$ with $\bigcup _{n=0}^{\infty }\mathscr {F}_{n}$ generating $\mathscr {F}$ as defined in Lemma 4.8, and set $\widetilde{\mathbb {P}}:=\mu ^{\boldsymbol{l}}_{\langle h\rangle }(K^{\boldsymbol{l}})^{-1}\mu ^{\boldsymbol{l}}_{\langle h\rangle }$, so that $\mathbb {P}(K^{\boldsymbol{l}}_{w})>0$ for any $w\in W^{\boldsymbol{l}}_{*}$ and $\mathbb {P}(V^{\boldsymbol{l}}_{*})=\widetilde{\mathbb {P}}(V^{\boldsymbol{l}}_{*})=0$ by (2.8) and Proposition 4.2. In particular, $\widetilde{\mathbb {P}}|_{\mathscr {F}_{n}}\ll \mathbb {P}|_{\mathscr {F}_{n}}$ for any $n\in \mathbb {N}\cup \{0\}$, and define $\alpha _{n}\in L^{1}(\Omega ,\mathscr {F}_{n},\mathbb {P}|_{\mathscr {F}_{n}})$ by (4.5) for each $n\in \mathbb {N}$. Now let $\omega ^{x}=(\omega ^{x}_{n})_{n=1}^{\infty }\in \prod _{n=1}^{\infty }S_{l_{n}}$ for $x\in K^{\boldsymbol{l}}\setminus V^{\boldsymbol{l}}_{*}$ and $S_{l,1}\subset S_{l}$ for $l\in \mathbb {N}\setminus \{1,2,3,4\}$ be as in Proposition 4.9. Then by (2.8), the $\mathbb {P}$-a.s. defined Borel measurable maps $K^{\boldsymbol{l}}\setminus V^{\boldsymbol{l}}_{*}\ni x\mapsto \omega ^{x}_{n}\in S_{l_{n}}$, $n\in \mathbb {N}$, form a sequence of independent random variables on $(\Omega ,\mathscr {F},\mathbb {P})$ and satisfy

$$\begin{aligned} \sum _{n=1}^{\infty }\mathbb {P}\bigl (\{x\in K^{\boldsymbol{l}}\setminus V^{\boldsymbol{l}}_{*}\mid \omega ^{x}_{n}\in S_{l_{n},1}\}\bigr ) =\sum _{n=1}^{\infty }\frac{\# S_{l_{n},1}}{\# S_{l_{n}}} =\sum _{n=1}^{\infty }\frac{3l_{n}-12}{3l_{n}-3} \ge \sum _{n=1}^{\infty }\frac{1}{4}=\infty , \end{aligned}$$

and hence the second Borel–Cantelli lemma implies that

$$\begin{aligned} \#\{n\in \mathbb {N}\mid \omega ^{x}_{n}\in S_{l_{n},1}\}=\infty \quad \text {for}\,\,\mathbb {P}\text {-a.s.,}\ x\in K^{\boldsymbol{l}}\setminus V^{\boldsymbol{l}}_{*}. \end{aligned}$$

(4.9)

On the other hand, for each $x\in K^{\boldsymbol{l}}\setminus V^{\boldsymbol{l}}_{*}$, Lemma 4.8 and Proposition 4.9 imply that $\mathbb {E}[\sqrt{\alpha _{n}}\mid \mathscr {F}_{n-1}](x)=0$ for any $n\in \mathbb {N}$ with $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )=0$ and that

$$\begin{aligned} \mathbb {E}[\sqrt{\alpha _{n}}\mid \mathscr {F}_{n-1}](x) =\sum _{i\in S_{l_{n}}}\sqrt{\frac{\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}i}\bigr )}{\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )}} \sqrt{\frac{m_{\boldsymbol{l}}\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}i}\bigr )}{m_{\boldsymbol{l}}\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )}} \le \sqrt{\frac{361}{372}} \end{aligned}$$

for any $n\in \mathbb {N}\cap [k+2,\infty )$ with $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\bigl (K^{\boldsymbol{l}}_{\omega ^{x}_{1}\ldots \omega ^{x}_{n-1}}\bigr )>0$ and $\omega ^{x}_{n-1}\in S_{l_{n-1},1}$, whence

$$\begin{aligned} \sum _{n=1}^{\infty }\bigl (1-\mathbb {E}[\sqrt{\alpha _{n}}\mid \mathscr {F}_{n-1}](x)\bigr ) \ge \delta \#\{n\in \mathbb {N}\cap [k+2,\infty )\mid \omega ^{x}_{n-1}\in S_{l_{n-1},1}\}, \end{aligned}$$

(4.10)

where $\delta :=1-\sqrt{\frac{361}{372}}\in (0,1)$. Combining (4.9) and (4.10), we obtain (4.6), so that Theorem 4.7 is applicable and yields $\widetilde{\mathbb {P}}\perp \mathbb {P}$, namely $\mu ^{\boldsymbol{l}}_{\langle h\rangle }\perp m_{\boldsymbol{l}}$.    $\square$

5 Realizing Arbitrarily Slow Decay Rates of $\varPsi (r)/r^{2}$

In this last section, we show that an arbitrarily slow decay rate of $\varPsi (r)/r^{2}$ for a homeomorphism $\varPsi :[0,\infty )\rightarrow [0,\infty )$ satisfying (1.2) and (1.6) can be realized by $\varPsi _{\boldsymbol{l}}$ (recall Definition 3.1) for some $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$. We achieve this by providing in Theorem 5.1 a simple sufficient condition for $\varPsi$ to be comparable to $\varPsi _{\boldsymbol{l}}$ for some $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$ with $\sum _{n=1}^{\infty }l_{n}^{-1}<\infty$ and proving in Proposition 5.2 that the decay rate of $\varPsi (r)/r^{2}$ for such $\varPsi$ can be arbitrarily slow. We also give criteria for verifying this sufficient condition for concrete examples of $\varPsi$ in Proposition 5.3 and apply them to the case where $\varPsi (r)/r^{2}$ is a multiple composition of the function $r\mapsto 1/\log (e-1+(r\wedge 1)^{-1})$ in Example 5.4.

Theorem 5.1

Let $\eta :[0,1]\rightarrow [0,1]$ be a homeomorphism with $\eta (0)=0$ such that

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{\eta ^{-1}(2^{-n})}{\eta ^{-1}(2^{1-n})}<\infty , \end{aligned}$$

(5.1)

and define a homeomorphism $\varPsi _{\eta }:[0,\infty )\rightarrow [0,\infty )$ by $\varPsi _{\eta }(r):=r^{2}\eta (r\wedge 1)$. Then there exists $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$ with $\sum _{n=1}^{\infty }l_{n}^{-1}<\infty$ such that $\varPsi _{\eta }(r)/\varPsi _{\boldsymbol{l}}(r)\in [c^{-1},c]$ for any $r\in (0,\infty )$ for some $c\in [1,\infty )$, and consequently, $(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ satisfies $fHKE (\varPsi _{\eta })$.

Proof

Set $c_{\eta }:=\inf _{n\in \mathbb {N}}\eta ^{-1}(2^{1-n})/\eta ^{-1}(2^{-n})$, so that $c_{\eta }\in (1,\infty )$ since the sequence $\{\eta ^{-1}(2^{1-n})/\eta ^{-1}(2^{-n})\}_{n=1}^{\infty }$ is $(1,\infty )$-valued and tends to $\infty$ by (5.1). Then for any $r,R\in (0,1]$ with $r\le R$, taking $j,k\in \mathbb {N}$ such that $\eta (r)\in (2^{-k},2^{1-k}]$ and $\eta (R)\in (2^{-j},2^{1-j}]$, we have $j\le k$, hence

$$\begin{aligned} \frac{R}{r}=\frac{\eta ^{-1}(\eta (R))}{\eta ^{-1}(\eta (r))} \ge \frac{\eta ^{-1}(2^{-j})}{\eta ^{-1}(2^{1-k})}\vee 1 \ge c_{\eta }^{k-j-1}=2^{(k-j-1)/\beta _{\eta }} \ge \biggl (\frac{\eta (R)}{4\eta (r)}\biggr )^{1/\beta _{\eta }} \end{aligned}$$

by the definition of $c_{\eta }$, where $\beta _{\eta }:=(\log _{2}c_{\eta })^{-1}\in (0,\infty )$, and therefore

$$\begin{aligned} \frac{\eta (R)}{\eta (r)}\le 4\biggl (\frac{R}{r}\biggr )^{\beta _{\eta }}. \end{aligned}$$

(5.2)

Recalling that $\lim _{n\rightarrow \infty }\eta ^{-1}(2^{1-n})/\eta ^{-1}(2^{-n})=\infty$ by (5.1), choose $n_{0}\in \mathbb {N}$ so that $\eta ^{-1}(2^{1-n})/\eta ^{-1}(2^{-n})\ge 5$ for any $n\in \mathbb {N}$ with $n\ge n_{0}$, set $l_{0}:=1$, and define $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in \mathbb {N}^{\mathbb {N}}$ inductively by

$$\begin{aligned} l_{n}:=\bigg \lfloor \frac{\eta ^{-1}(2^{-n_{0}})}{(l_{0}\cdots l_{n-1})\eta ^{-1}(2^{-n-n_{0}})}\biggr \rfloor , \qquad n\in \mathbb {N}. \end{aligned}$$

(5.3)

Then an induction on n based on (5.3) and the choice of $n_{0}$ immediately shows that $\boldsymbol{l}=(l_{n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$ and that for any $n\in \mathbb {N}\cup \{0\}$,

$$\begin{aligned} \frac{\eta ^{-1}(2^{-n-n_{0}})}{\eta ^{-1}(2^{-n_{0}})} \le \frac{1}{L^{\boldsymbol{l}}_{n}} \le \frac{6}{5}\frac{\eta ^{-1}(2^{-n-n_{0}})}{\eta ^{-1}(2^{-n_{0}})}, \end{aligned}$$

(5.4)

which together with (5.1) implies in particular that

$$\begin{aligned} \sum _{n=1}^{\infty }l_{n}^{-1} =\sum _{n=1}^{\infty }\frac{1/L^{\boldsymbol{l}}_{n}}{1/L^{\boldsymbol{l}}_{n-1}} \le \frac{6}{5}\sum _{n=1}^{\infty }\frac{\eta ^{-1}(2^{-n-n_{0}})}{\eta ^{-1}(2^{1-n-n_{0}})} <\infty . \end{aligned}$$

(5.5)

We claim that $\varPsi _{\eta }(r)/\varPsi _{\boldsymbol{l}}(r)\in [c^{-1},c]$ for any $r\in (0,\infty )$ for some $c\in [1,\infty )$. Indeed, recalling Definition 3.1, we have $\beta _{\boldsymbol{l},0}=\inf _{n\in \mathbb {N}}\beta _{l_{n}}=2$ by (5.5), hence $\varPsi _{\eta }(r)/\varPsi _{\boldsymbol{l}}(r)=r^{2}/r^{\beta _{\boldsymbol{l},0}}=1$ for any $r\in [1,\infty )$, and also see for any $n\in \mathbb {N}$ that

$$\begin{aligned} 2^{-n-n_{0}}\le \eta \biggl (\frac{\eta ^{-1}(2^{-n-n_{0}})}{\eta ^{-1}(2^{-n_{0}})}\biggr ) \le \eta (1/L^{\boldsymbol{l}}_{n}) \le \eta \biggl (\frac{6}{5}\frac{\eta ^{-1}(2^{-n-n_{0}})}{\eta ^{-1}(2^{-n_{0}})}\biggr ) \le c2^{-n} \end{aligned}$$

(5.6)

by (5.4) and (5.2), where $c:=2^{2-n_{0}}\bigl (\frac{6}{5}/\eta ^{-1}(2^{-n_{0}})\bigr )^{\beta _{\eta }}$, and thus that

$$\begin{aligned} \frac{\varPsi _{\eta }(1/L^{\boldsymbol{l}}_{n})}{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})} =\frac{T^{\boldsymbol{l}}_{n}\eta (1/L^{\boldsymbol{l}}_{n})}{(L^{\boldsymbol{l}}_{n})^{2}} =2^{n}\eta (1/L^{\boldsymbol{l}}_{n})\prod _{k=1}^{n}\biggl (1-\frac{5}{6}l_{k}^{-1}-\frac{1}{6}l_{k}^{-2}\biggr ) \in [c',c], \end{aligned}$$

(5.7)

where $c':=2^{-n_{0}}\prod _{k=1}^{\infty }(1-\frac{5}{6}l_{k}^{-1}-\frac{1}{6}l_{k}^{-2})\in (0,1)$ by (5.5). Now for any $n\in \mathbb {N}$ and any $s\in [1,l_{n}]$, by (3.1) we have

$$\begin{aligned} \frac{\varPsi _{\boldsymbol{l}}(s/L^{\boldsymbol{l}}_{n})}{s^{2}\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})} =\frac{1}{s^{2}}\biggl (1+\frac{3l_{n}-4}{l_{n}-1}(s-1)\biggr ) \biggl (1+\frac{\frac{2}{3}l_{n}-\frac{8}{9}}{l_{n}-1}(s-1)\biggr ) \in [1,2), \end{aligned}$$

(5.8)

and it follows from $\eta (s/L^{\boldsymbol{l}}_{n})\in \bigl [\eta (1/L^{\boldsymbol{l}}_{n}),\eta (1/L^{\boldsymbol{l}}_{n-1})\bigr ]$, (5.6), (5.8) and (5.7) that

$$\begin{aligned} \frac{\varPsi _{\eta }(s/L^{\boldsymbol{l}}_{n})}{\varPsi _{\boldsymbol{l}}(s/L^{\boldsymbol{l}}_{n})}&=\frac{\varPsi _{\eta }(s/L^{\boldsymbol{l}}_{n})}{\varPsi _{\eta }(1/L^{\boldsymbol{l}}_{n})} \frac{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})}{\varPsi _{\boldsymbol{l}}(s/L^{\boldsymbol{l}}_{n})} \frac{\varPsi _{\eta }(1/L^{\boldsymbol{l}}_{n})}{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})} \\&=\frac{\eta (s/L^{\boldsymbol{l}}_{n})}{\eta (1/L^{\boldsymbol{l}}_{n})} \frac{s^{2}\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})}{\varPsi _{\boldsymbol{l}}(s/L^{\boldsymbol{l}}_{n})} \frac{\varPsi _{\eta }(1/L^{\boldsymbol{l}}_{n})}{\varPsi _{\boldsymbol{l}}(1/L^{\boldsymbol{l}}_{n})} \in [c'/2,(c^{2}\vee c)2^{n_{0}+1}], \end{aligned}$$

proving that $\varPsi _{\eta }(r)/\varPsi _{\boldsymbol{l}}(r)\in [c'/2,(c^{2}\vee c)2^{n_{0}+1}]$ for any $r\in (0,\infty )$. Lastly, combining this result with Lemma 3.2, Theorem 3.3 and Remark 1.2-(2) shows that $(K^{\boldsymbol{l}},d_{\boldsymbol{l}},m_{\boldsymbol{l}},\mathcal {E}^{\boldsymbol{l}},\mathcal {F}_{\boldsymbol{l}})$ satisfies $fHKE (\varPsi _{\eta })$.    $\square$

The decay rate of $\varPsi _{\eta }(r)/r^{2}=\eta (r\wedge 1)$ for $\eta$ as in Theorem 5.1 can be arbitrarily slow in the sense stated in the following proposition.

Proposition 5.2

Let $\varPsi :[0,\infty )\rightarrow [0,\infty )$ be a homeomorphism satisfying (1.6). Then there exists a homeomorphism $\eta :[0,1]\rightarrow [0,1]$ with the properties $\eta (0)=0$ and (5.1) such that $\eta (r)\ge c\varPsi (r)/r^{2}$ for any $r\in (0,1]$ for some $c\in (0,\infty )$.

Proof

Noting (1.6), define $\eta _{0}:[0,1]\rightarrow [0,\infty )$ by $\eta _{0}(r):=\sup _{s\in (0,r]}\varPsi (s)/s^{2}$ ($\eta _{0}(0):=0$), so that $\eta _{0}$ is continuous and non-decreasing and $\eta _{0}((0,1])\subset (0,\infty )$, and set $s_{n}:=\max \eta _{0}^{-1}(2^{-n}\eta _{0}(1))$ for $n\in \mathbb {N}\cup \{0\}$, so that $s_{0}=1$, $0<s_{n}<s_{n-1}$ for any $n\in \mathbb {N}$ and $\lim _{n\rightarrow \infty }s_{n}=0$. Define a homeomorphism $\eta :[0,1]\rightarrow [0,1]$ by

$$\begin{aligned} \eta (r):=\biggl (1+\frac{r-2^{-n^{2}}s_{n}}{2^{-(n-1)^{2}}s_{n-1}-2^{-n^{2}}s_{n}}\biggr )2^{-n} \end{aligned}$$

(5.9)

for $n\in \mathbb {N}$ and $r\in [2^{-n^{2}}s_{n},2^{-(n-1)^{2}}s_{n-1}]$ and $\eta (0):=0$. Then since $\eta ^{-1}(2^{1-n})=2^{-(n-1)^{2}}s_{n-1}$ and $0<s_{n}<s_{n-1}$ for any $n\in \mathbb {N}$,

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{\eta ^{-1}(2^{-n})}{\eta ^{-1}(2^{1-n})} =\sum _{n=1}^{\infty }\frac{2^{-n^{2}}s_{n}}{2^{-(n-1)^{2}}s_{n-1}} \le \sum _{n=1}^{\infty }2^{1-2n}=\frac{2}{3}<\infty , \end{aligned}$$

namely $\eta$ satisfies (5.1), and for any $n\in \mathbb {N}$ and any $r\in [s_{n},s_{n-1}]$ we have

$$\begin{aligned} \eta (r)\ge \eta (s_{n})\ge \eta (2^{-n^{2}}s_{n})=2^{-n}=\frac{\eta _{0}(s_{n-1})}{2\eta _{0}(1)} \ge \frac{\eta _{0}(r)}{2\eta _{0}(1)}\ge \frac{\varPsi (r)/r^{2}}{2\eta _{0}(1)}, \end{aligned}$$

i.e., $\eta (r)\ge c\varPsi (r)/r^{2}$ with $c:=(2\eta _{0}(1))^{-1}\in (0,\infty )$ for any $r\in (0,1]$.    $\square$

We conclude this paper with the following proposition, which gives criteria for verifying (5.1) for concrete homeomorphisms $\eta :[0,1]\rightarrow [0,1]$ with $\eta (0)=0$, and some applications of it to $\eta (r)=1/\log (e-1+r^{-1})$ in Example 5.4 below.

Proposition 5.3

Let $\eta :[0,1]\rightarrow [0,1]$ be a homeomorphism with $\eta (0)=0$, let $\delta \in [0,\infty )$, $\alpha ,\beta \in (0,\infty )$ and assume that there exists $c\in (0,\infty )$ such that

$$\begin{aligned} \frac{\eta (R)}{\eta (r)}\le 1+\delta +\frac{c(R/r)^{\beta }}{(\log (e-1+R^{-1}))^{\alpha }} {for\,any}\,\, r,R\in (0,1]\,{with}\, r\le R. \end{aligned}$$

(5.10)

• (1) If $\delta <1$ and $\beta <\alpha$, then $\eta$ satisfies (5.1).

• (2) Let $\widetilde{\eta }:[0,1]\rightarrow [0,1]$ be a homeomorphism with $\widetilde{\eta }(0)=0$, let $\widetilde{\delta }\in [0,1)$ and assume that there exist $\widetilde{\alpha },\widetilde{c}\in (0,\infty )$ such that $\widetilde{\eta }$ satisfies (5.10) with $\widetilde{\delta },\widetilde{\alpha },1,\widetilde{c}$ in place of $\delta ,\alpha ,\beta ,c$. Then $\widetilde{\eta }\circ \eta$ satisfies (5.10) with $\frac{1}{2}(1+\widetilde{\delta }),c'$ in place of $\delta ,c$ for some $c'\in (0,\infty )$. In particular, if $\beta <\alpha$, then $\widetilde{\eta }\circ \eta$ satisfies (5.1).

Proof

• (1) Set $s_{n}:=\eta ^{-1}(2^{-n})$ for $n\in \mathbb {N}\cup \{0\}$. For each $n\in \mathbb {N}$, we see from $\eta (s_{n-1})/\eta (s_{n})=2$, (5.10) with $(r,R)=(s_{n},s_{n-1})$ and $\delta <1$ that

  $$\begin{aligned} \frac{s_{n}}{s_{n-1}}\le \frac{c^{1/\beta }(1-\delta )^{-1/\beta }}{(\log (e-1+s_{n-1}^{-1}))^{\alpha /\beta }}, \end{aligned}$$

  (5.11)

  and from $\eta (1)/\eta (s_{n-1})=2^{n-1}$, (5.10) with $(r,R)=(s_{n-1},1)$ and $\delta <1$ that $2^{n-1}\le 1+\delta +cs_{n-1}^{-\beta }\le (2+c)s_{n-1}^{-\beta }$, whence, provided $n\ge 2+2\log _{2}(2+c)$,

  $$\begin{aligned} \log (e-1+s_{n-1}^{-1})\ge \log (s_{n-1}^{-1}) \ge \frac{\log 2}{\beta }(n-1-\log _{2}(2+c)) \ge \frac{\log 2}{2\beta }n. \end{aligned}$$

  (5.12)

  It follows from (5.11), (5.12) and $\alpha /\beta >1$ that

  $$\begin{aligned} \sum _{n=1}^{\infty }\frac{\eta ^{-1}(2^{-n})}{\eta ^{-1}(2^{1-n})} =\sum _{n=1}^{\infty }\frac{s_{n}}{s_{n-1}} \le \sum _{n=1}^{n_{c}-1}\frac{s_{n}}{s_{n-1}} +\sum _{n=n_{c}}^{\infty }\frac{c^{1/\beta }(2\beta /\log 2)^{\alpha /\beta }}{(1-\delta )^{1/\beta }n^{\alpha /\beta }} <\infty , \end{aligned}$$

  where $n_{c}:=3+\lfloor 2\log _{2}(2+c)\rfloor$, proving (5.1).

• (2) Set $\widetilde{r}:=\eta ^{-1}\bigl (\exp \bigl (-(2\widetilde{c}(1+\delta )/(1-\widetilde{\delta }))^{1/\widetilde{\alpha }}\bigr )\bigr )$ and let $r,R\in (0,1]$ satisfy $r\le R$. By (5.10) for $\widetilde{\eta }$ and $\eta$ and $(\log (e-1+\eta (R)^{-1}))^{-\widetilde{\alpha }}\le 1$ we have

  $$\begin{aligned} \begin{aligned} \frac{\widetilde{\eta }\circ \eta (R)}{\widetilde{\eta }\circ \eta (r)}&\le 1+\widetilde{\delta }+\frac{\widetilde{c}}{(\log (e-1+\eta (R)^{-1}))^{\widetilde{\alpha }}}\frac{\eta (R)}{\eta (r)} \\&\le 1+\widetilde{\delta }+\frac{\widetilde{c}(1+\delta )}{(\log (e-1+\eta (R)^{-1}))^{\widetilde{\alpha }}} +\frac{\widetilde{c}c(R/r)^{\beta }}{(\log (e-1+R^{-1}))^{\alpha }}. \end{aligned} \end{aligned}$$

  (5.13)

  If $R\le \widetilde{r}$, then $\widetilde{c}(1+\delta )/(\log (e-1+\eta (R)^{-1}))^{\widetilde{\alpha }}\le \frac{1}{2}(1-\widetilde{\delta })$ by the definition of $\widetilde{r}$ and hence (5.13) yields (5.10) with $\widetilde{\eta }\circ \eta ,\frac{1}{2}(1+\widetilde{\delta }),\widetilde{c}c$ in place of $\eta ,\delta ,c$, whereas if $R>\widetilde{r}$, then we see from (5.13), $\widetilde{\delta }<1$ and $(\log (e-1+\eta (R)^{-1}))^{-\widetilde{\alpha }}\le 1 \le (\log (e-1+\widetilde{r}^{-1})/\log (e-1+R^{-1}))^{\alpha }\wedge (R/r)^{\beta }$ that (5.10) with $\widetilde{\eta }\circ \eta ,\frac{1}{2}(1+\widetilde{\delta }),c'$ in place of $\eta ,\delta ,c$ holds, where $c':=\widetilde{c}(1+\delta )(\log (e-1+\widetilde{r}^{-1}))^{\alpha }+\widetilde{c}c$. In particular, if $\beta <\alpha$, then $\widetilde{\eta }\circ \eta$ satisfies (5.1) by $\frac{1}{2}(1+\widetilde{\delta })<1$ and (1).    $\square$

Example 5.4

Define homeomorphisms $\eta _{k}:[0,1]\rightarrow [0,1]$, $k\in \mathbb {N}$, inductively by

$$\begin{aligned} \eta _{1}(r):=\frac{1}{\log (e-1+r^{-1})} {(\eta _{1}(0):=0)} \quad \text {and}\quad \eta _{k+1}:=\eta _{1}\circ \eta _{k}, k\in \mathbb {N}. \end{aligned}$$

(5.14)

Then $\eta _{k}$ satisfies (5.10) with $\delta =\frac{1}{2}$ and $\alpha =1$ for some $c\in (0,\infty )$ for any $\beta \in (0,\infty )$ and any $k\in \mathbb {N}$. Indeed, this follows by a straightforward induction on k based on Proposition 5.3-(2), which is applicable with $\eta =\eta _{k}$ and $\widetilde{\eta }=\eta _{1}$ since $\eta _{1}$ is easily seen to satisfy (5.10) with $\delta =0$, $\alpha =1$ and $c=(e\beta )^{-1}$ for any $\beta \in (0,\infty )$ as follows: for any $r,R\in (0,1]$ with $r\le R$,

$$\begin{aligned} \begin{aligned} \frac{\eta _{1}(R)}{\eta _{1}(r)}&=1+\frac{\log \dfrac{e-1+r^{-1}}{e-1+R^{-1}}}{\log (e-1+R^{-1})} =1+\frac{\log \dfrac{R}{r}+\log \dfrac{1+(e-1)r}{1+(e-1)R}}{\log (e-1+R^{-1})} \\&\le 1+\frac{\beta ^{-1}\log ((R/r)^{\beta })}{\log (e-1+R^{-1})} \le 1+\frac{(e\beta )^{-1}(R/r)^{\beta }}{\log (e-1+R^{-1})}. \end{aligned} \end{aligned}$$

(5.15)

As a consequence, for each $k\in \mathbb {N}$, recalling that $\varPsi _{\eta _{k}}:[0,\infty )\rightarrow [0,\infty )$ is defined by $\varPsi _{\eta _{k}}(r):=r^{2}\eta _{k}(r\wedge 1)$, we conclude from Proposition 5.3-(1) that $\eta _{k}$ satisfies (5.1), thus from Theorem 5.1 that there exists $\boldsymbol{l}_{k}=(l_{k,n})_{n=1}^{\infty }\in (\mathbb {N}\setminus \{1,2,3,4\})^{\mathbb {N}}$ with $\sum _{n=1}^{\infty }l_{k,n}^{-1}<\infty$ such that $\varPsi _{\eta _{k}}(r)/\varPsi _{\boldsymbol{l}_{k}}(r)\in [c_{k}^{-1},c_{k}]$ for any $r\in (0,\infty )$ for some $c_{k}\in [1,\infty )$, and thereby that $(K^{\boldsymbol{l}_{k}},d_{\boldsymbol{l}_{k}},m_{\boldsymbol{l}_{k}},\mathcal {E}^{\boldsymbol{l}_{k}},\mathcal {F}_{\boldsymbol{l}_{k}})$ satisfies $fHKE (\varPsi _{\eta _{k}})$.