**The Laplacian on some self-conformal fractals** **and Weyl’s asymptotics for its eigenvalues:**

**A survey of the analytic aspects**

**Naotaka Kajino**

**Abstract.**

This article surveys the analytic aspects of the author’s recent
studies on the construction and analysis of a*“geometrically canonical”*

*Laplacian* on circle packing fractals invariant with respect to certain
Kleinian groups (i.e., discrete groups of M¨obius transformations on
the Riemann sphereCb =C*∪ {∞}), including the classicalApollonian*
*gasket* and some*round Sierpi´nski carpets. The main result on Weyl’s*
asymptotics for its eigenvalues is of the same form as that by Oh and
Shah [Invent. Math.**187**(2012), 1–35, Theorem 1.4] on the asymptotic
distribution of the circles in a very large class of such fractals.

*§***1.** **Introduction**

This article, which is a considerable expansion of [12], concerns the author’s recent studies in [11, 14, 15, 16] on Weyl’s eigenvalue asymp- totics for a “geometrically canonical” Laplacian defined by the author on circle packing fractals which are invariant with respect to certain Kleinian groups (i.e., discrete groups of M¨obius transformations onCb:=

C*∪ {∞}*), including the classical*Apollonian gasket*(Figure 1) and some
*round Sierpi´nski carpets* (Figure 5). Here we focus on sketching the con-
struction of the Laplacian, the proof of its uniqueness and basic proper-
ties, and the analytic aspects of the proof of the eigenvalue asymptotics;

2010 *Mathematics Subject Classification.* Primary 28A80, 35P20, 53C23;

Secondary 31C25, 37B10, 60J35.

*Key words and phrases.* Apollonian gasket, Kleinian groups, round
Sierpi´nski carpets, Dirichlet forms, Laplacian, Weyl’s eigenvalue asymptotics.

This work was supported by JSPS KAKENHI Grant Numbers JP25887038, JP15K17554, JP18K18720 and by the Research Institute for Mathemati- cal Sciences, an International Joint Usage/Research Center located in Kyoto University.

the reader is referred to [13] for a survey of the ergodic-theoretic aspects of the proof of the eigenvalue asymptotics.

This article is organized as follows. First in *§*2 we introduce the
Apollonian gasket*K(*D) and recall its basic geometric properties. In*§*3,
after a brief summary of how the Laplacian on*K(*D) was discovered by
Teplyaev in [34], we give its definition and sketch the proof of the result
in [14] that it is the infinitesimal generator of the*unique* strongly local,
regular symmetric Dirichlet form over*K(*D) with respect to which the
inclusion map*K(*D)*,→*Cis*harmonic* on the complement of the three
outmost vertices. In *§*4, we state the principal result in [14] that the
Laplacian on*K(*D) satisfies Weyl’s eigenvalue asymptotics of the same
form as the asymptotic distribution of the circles in *K(*D) by Oh and
Shah in [30, Corollary 1.8], and sketch the proof of certain estimates
on the eigenvalues required to conclude Weyl’s asymptotics by applying
the ergodic-theoretic result explained in [13]. Finally, in*§*5 we present a
partial extension of these results to the case of round Sierpi´nski carpets
which are invariant with respect to certain concrete Kleinian groups.

*Notation.* We use the following notation throughout this article.

(0) The symbols*⊂*and*⊃*for set inclusion*allow*the case of the equality.

(1) N:=*{n∈*Z*|n >*0*}*, i.e., 0*̸∈*N.

(2) Cb:=C*∪ {∞}* denotes the Riemann sphere.

(3) *i*:=*√*

*−*1 denotes the imaginary unit. The real and imaginary parts
of*z∈*Care denoted by Re*z* and Im*z, respectively.*

(4) The cardinality (number of elements) of a set*A*is denoted by #A.

(5) Let*E* be a non-empty set. We define id*E*:*E→E* by id*E*(x) :=*x.*

For*x∈E, we define***1*** _{x}*=

**1**

^{E}

_{x}*∈*R

*by*

^{E}**1**

*(y) :=*

_{x}**1**

^{E}*(y) :={*

_{x}_{1 if}

_{y}_{=}

_{x,}0 if*y**̸*=*x.*

For*u*:*E→*[*−∞,*+*∞*] we set *∥u∥*sup:=*∥u∥*sup,E := sup_{x}_{∈}_{E}*|u(x)|*.
(6) Let *E* be a topological space. The Borel *σ-field of* *E* is denoted
by B(E). For *A* *⊂* *E, its interior, closure and boundary inE* are
denoted by int_{E}*A,A** ^{E}*and

*∂*

_{E}*A, respectively, and whenE*=Cthey are simply denoted by int

*A,A*and

*∂A, respectively. We setC*(E) :=

*{u* *|* *u*:*E→*R,*u*is continuous*}*, supp* _{E}*[u] :=

*u*

^{−}^{1}(R

*\ {*0

*}*)

*for*

^{E}*u∈ C*(E), and

*C*c(E) :=

*{u∈ C*(E)

*|*supp

*[u] is compact*

_{E}*}*.

(7) Let *n* *∈* N. The Lebesgue measure on (R^{n}*,*B(R* ^{n}*)) is denoted by
vol

*n*. The Euclidean inner product and norm on R

*are denoted by*

^{n}*⟨·,·⟩*and

*| · |*, respectively. For

*A*

*⊂*R

*and*

^{n}*f*:

*A*

*→*C we set

**Lip**

_{A}*f*:= sup

_{x,y}

_{∈}

_{A, x}

_{̸}_{=y}

^{|}

^{f(x)}

_{|}

_{x}

^{−}

_{−}

^{f(y)}

_{y}

_{|}*(sup*

^{|}*∅*:= 0). For a non-empty open subset

*U*of R

*and*

^{n}*u*:

*U*

*→*Rwith

**Lip**

_{U}*u <*+

*∞*, the first- order partial derivatives of

*u, which exist vol*

*n*-a.e. on

*U*, are denoted by

*∂*1

*u, . . . , ∂*

*n*

*u, and we set∇u*:= (∂1

*u, . . . , ∂*

*n*

*u).*

*§***2.** **The Apollonian gasket and its fractal geometry**

In this section, we introduce the Apollonian gasket and state its geometric properties needed for our purpose. The same framework is presented also in [13, Section 2], but we repeat it here for the reader’s convenience. The following definition and proposition form the basis of the construction and further detailed studies of the Apollonian gasket.

**Definition 2.1** (tangential disk triple). (0) We set*S* :=*{*1,2,3*}*.
(1) Let *D*_{1}*, D*_{2}*, D*_{3} *⊂* C be either three open disks or two open disks

and an open half-plane. The triple D := (D1*, D*2*, D*3) of such sets
is called a*tangential disk triple* if and only if #(D*j**∩D**k*) = 1 (i.e.,
*D** _{j}* and

*D*

*are*

_{k}*externally*tangent) for any

*j, k∈S*with

*j*

*̸*=

*k. If*D is such a triple consisting of three disks, then the open triangle in Cwith vertices the centers of

*D*

_{1}

*, D*

_{2}

*, D*

_{3}is denoted by

*△*(D).

(2) Let D= (D_{1}*, D*_{2}*, D*_{3}) be a tangential disk triple. The open subset
C*\*∪

*j**∈**S**D**j* of C is then easily seen to have a unique bounded
connected component, which is denoted by*T*(D) and called the*ideal*
*triangle* associated withD. We also set*{q** _{j}*(D)

*}*:=

*D*

_{k}*∩D*

*for each (j, k, l)*

_{l}*∈ {*(1,2,3),(2,3,1),(3,1,2)

*}*and

*V*

_{0}(D) :=

*{q*

*(D)*

_{j}*|j*

*∈S}*. (3) A tangential disk tripleD= (D

_{1}

*, D*

_{2}

*, D*

_{3}) is called

*positively oriented*

if and only if its associated ideal triangle*T*(D) is to the left of*∂T*(D)
when*∂T*(D) is oriented so as to have*{q**j*(D)*}*^{3}*j=1* in this order.

Finally, we define

TDT^{+}:=*{D|*D is a positively oriented tangential disk triple*},*
TDT* ^{⊕}*:=

*{D|*D= (D

_{1}

*, D*

_{2}

*, D*

_{3})

*∈*TDT

^{+},

*D*

_{1}

*, D*

_{2}

*, D*

_{3}are disks

*}.*

The following proposition is classical and can be shown by some
elementary (though lengthy) Euclidean-geometric arguments. We set
rad(D) := *r* and curv(D) := *r*^{−}^{1} for each open disk *D* *⊂*C of radius
*r∈*(0,+*∞*) and curv(D) := 0 for each open half-plane*D⊂*C.

**Proposition 2.2.** *Let* D = (D1*, D*2*, D*3)*∈* TDT^{+}*, set* (α, β, γ) :=

(curv(D1),curv(D2),curv(D3))

*and setκ*:=*κ(*D) :=*√*

*βγ*+*γα*+*αβ.*

(1) *Let* *D*_{cir}(D) *⊂* C *denote the* circumscribed disk *of* *T(*D), i.e., the
*unique open disk with* *{q*_{1}(D), q_{2}(D), q_{3}(D)*} ⊂* *∂D*_{cir}(D). *Then*
*T*(D)*\ {q*1(D), q2(D), q3(D)*} ⊂D*cir(D), *∂D*cir(D) *is orthogonal to*

*∂D**j* *for any* *j∈S, and*curv(Dcir(D)) =*κ.*

(2) *There exists a unique* inscribed disk*D*in(D)*of* *T*(D), i.e., a unique
*open disk* *D*in(D)*⊂*C *such that* *D*in(D)*⊂T*(D) *and* #(Din(D)*∩*
*D**j*) = 1 *for anyj∈S. Moreover,* curv(Din(D)) =*α*+*β*+*γ*+ 2κ.

The following notation is standard in studying self-similar sets.

(a)Examples without a half-plane (b)Example with a half-plane
Figure 1. The Apollonian gaskets*K*(D) associated withD*∈*TDT^{+}

**Definition 2.3.** (1) We set*W*0:=*{∅}*, where*∅*is an element called
the *empty word,* *W** _{m}*:=

*S*

*for*

^{m}*m∈*Nand

*W*

*:=∪*

_{∗}*m**∈N∪{*0*}**W** _{m}*.
For

*w∈W*

*, the unique*

_{∗}*m∈*N

*∪ {*0

*}*satisfying

*w∈W*

*is denoted by*

_{m}*|w|*and called the

*length*of

*w.*

(2) Let *w, v* *∈* *W** _{∗}*,

*w*=

*w*1

*. . . w*

*m*,

*v*=

*v*1

*. . . v*

*n*. We define

*wv*

*∈W*

*by*

_{∗}*wv*:=

*w*1

*. . . w*

*m*

*v*1

*. . . v*

*n*(w

*∅*:=

*w,*

*∅v*:=

*v).*We also de- fine

*w*

^{(1)}

*. . . w*

^{(k)}for

*k*

*≥*3 and

*w*

^{(1)}

*, . . . , w*

^{(k)}

*∈*

*W*

*inductively by*

_{∗}*w*

^{(1)}

*. . . w*

^{(k)}:= (w

^{(1)}

*. . . w*

^{(k}

^{−}^{1)})w

^{(k)}. For

*w∈W*

*and*

_{∗}*n∈*N

*∪ {*0

*}*we set

*w*

*:=*

^{n}*w . . . w∈W*

_{n}

_{|}

_{w}*. We write*

_{|}*w≤v*if and only if

*w*=

*vτ*for some

*τ*

*∈W*

*, and write*

_{∗}*w̸≍v*if and only if neither

*w≤v*nor

*v≤w*holds.

Proposition 2.2-(2) enables us to define natural “contraction maps”

Φ* _{w}*:TDT

^{+}

*→*TDT

^{+}for each

*w∈W*

*, which in turn is used to define the Apollonian gasket*

_{∗}*K(*D) associated withD

*∈*TDT

^{+}, as follows.

**Definition 2.4.** We define maps Φ_{1}*,*Φ_{2}*,*Φ_{3}:TDT^{+}*→*TDT^{+} by

(2.1)

Φ_{1}(D) := (D_{in}(D), D_{2}*, D*_{3}),

Φ2(D) := (D1*, D*in(D), D3), D= (D1*, D*2*, D*3)*∈*TDT^{+}*.*
Φ_{3}(D) := (D_{1}*, D*_{2}*, D*_{in}(D)),

We also set Φ*w* := Φ*w*_{m}*◦ · · · ◦*Φ*w*_{1} (Φ* _{∅}* := id

_{TDT}+) andD

*w*:= Φ

*w*(D) for

*w*=

*w*

_{1}

*. . . w*

_{m}*∈W*

*andD*

_{∗}*∈*TDT

^{+}.

**Definition 2.5** (Apollonian gasket). Let D *∈* TDT^{+}. We define
the*Apollonian gasket* *K(*D) associated withD (see Figure 1) by
(2.2) *K(*D) :=*T*(D)*\*∪

*w**∈**W*_{∗}*D*in(D*w*) =∩

*m**∈N*

∪

*w**∈**W*_{m}*T*(D*w*).

The curvatures of the disks involved in (2.2) admit the following simple expression.

**Definition 2.6.** We define 4*×*4 real matrices*M*_{1}*, M*_{2}*, M*_{3} by

(2.3) *M*_{1}:=

1 0 0 0 1 1 0 1 1 0 1 1 2 0 0 1

*, M*_{2}:=

1 1 0 1 0 1 0 0 0 1 1 1 0 2 0 1

*, M*_{3}:=

1 0 1 1 0 1 1 1 0 0 1 0 0 0 2 1

and set *M**w* := *M**w*1*· · ·M**w**m* for *w* =*w*1*. . . w**m* *∈* *W** _{∗}* (M

*:= id4*

_{∅}*×*4).

Note that then for any*n∈*N*∪ {*0*}* we easily obtain
(2.4)

*M*_{1}*n* =

1 0 0 0
*n*^{2} 1 0 *n*
*n*^{2} 0 1 *n*
2n 0 0 1

*, M*_{2}*n*=

1 *n*^{2} 0 *n*
0 1 0 0
0 *n*^{2} 1 *n*
0 2n 0 1

*, M*_{3}*n*=

1 0 *n*^{2} *n*
0 1 *n*^{2} *n*
0 0 1 0
0 0 2n 1

*.*
**Proposition 2.7.** *Let* D = (D_{1}*, D*_{2}*, D*_{3})*∈*TDT^{+}*, let* *α, β, γ, κ* *be*
*as in Proposition* 2.2, let*w∈W*_{∗}*and*(D_{w,1}*, D*_{w,2}*, D** _{w,3}*) :=D

*w*

*. Then*(2.5) (

curv(D*w,1*),curv(D*w,2*),curv(D*w,3*), κ(D*w*))

= (α, β, γ, κ)M*w**.*
*Proof.* This follows by an induction in*|w|*using Proposition 2.2-(2)

and Definition 2.4. Q.E.D.

We next collect basic facts regarding the Hausdorﬀ dimension and
measure of*K(*D). For each*s∈*(0,+*∞*) letH* ^{s}* : 2

^{C}

*→*[0,+

*∞*] denote the

*s-dimensional Hausdorﬀ (outer) measure on*Cwith respect to the Euclidean metric, and for each

*A⊂*Clet dimH

*A*denote its Hausdorﬀ dimension; see, e.g., [25, Chapters 4–7] for details. As is well known, it easily follows from the definition ofH

*that the image*

^{s}*f*(A) of

*A⊂*C by

*f*:

*A→*Cwith

**Lip**

_{A}*f <*+

*∞*satisfiesH

*(f(A))*

^{s}*≤*(Lip

_{A}*f*)

*H*

^{s}*(A) for any*

^{s}*s∈*(0,+

*∞*) and hence in particular dimH

*f*(A)

*≤*dimH

*A. On*the basis of this observation, we easily get the following lemma.

**Lemma 2.8.** *Let* D*,*D^{′}*∈* TDT^{+}*. Then there exists* *c* *∈* (0,+*∞*)
*such that*H* ^{s}*(K(D))

*≤c*

*H*

^{s}*(K(D*

^{s}*))*

^{′}*for anys∈*(0,+

*∞*). In particular, dimH

*K(*D) = dimH

*K(*D

*).*

^{′}*Proof.* Let*f*_{D}*′**,*D denote the unique orientation-preserving M¨obius
transformation on Cb such that *f*_{D}*′**,*D(q* _{j}*(D

*)) =*

^{′}*q*

*(D) for any*

_{j}*j*

*∈*

*S.*

Then*f*_{D}*′**,*D(K(D* ^{′}*)) =

*K(*D), since a M¨obius transformation onCb maps any open disk in Cb onto another. Now the assertion follows from the observation in the last paragraph and

**Lip**

_{D}cir(D* ^{′}*)

*f*

_{D}

*′*

*,*D

*<*+

*∞*. Q.E.D.

**Definition 2.9.** Noting Lemma 2.8, we define

(2.6) *d*_{AG}:= dim_{H}*K(*D), whereD*∈*TDT^{+} is arbitrary.

**Theorem 2.10**(Boyd [2]; see also [7, 26, 27]).

(2.7) 1.300197*< d*_{AG}*<*1.314534.

Moreover, for the*d*_{AG}-dimensional Hausdorﬀ measureH^{d}^{AG}(K(D))
of *K(*D) we have the following theorem, which was proved first by
Sullivan [33] through considerations on the isometric action of M¨obius
transformations on the three-dimensional hyperbolic space, and later by
Mauldin and Urba´nski [26] through purely two-dimensional arguments.

**Theorem 2.11**([33, Theorem 2], [26, Theorem 2.6]).

(2.8) 0*<*H^{d}^{AG}(K(D))*<*+*∞* *for any* D*∈*TDT^{+}*.*

*Remark* 2.12. The self-conformality of*K(*D) is required most cru-
cially in the proof of Theorem 2.11, and is heavily used further to obtain
certain equicontinuity properties of*{H*^{d}^{AG}(K(D*w*))*}**w**∈**W** _{∗}* as a family of
functions of(

curv(D_{1}),curv(D_{2}),curv(D_{3}))

, where (D_{1}*, D*_{2}*, D*_{3}) :=D.
This equicontinuity is the key to verifying the ergodic-theoretic assump-
tions of Kesten’s renewal theorem [19, Theorem 2], which is then applied
to conclude Theorem 4.4 below.

*§***3.** **The canonical Dirichlet form on the Apollonian gasket**
In this section, we introduce the canonical Dirichlet form on the
Apollonian gasket*K(*D), whose infinitesimal generator is our Laplacian
on*K(*D), and state its properties established by the author in [14]; see
[6, 4] for the basics of the theory of regular symmetric Dirichlet forms.

Before giving its actual definition, we briefly summarize how it has
been discovered. The initial idea for its construction was suggested by
the theory of analysis on the*harmonic Sierpi´nski gasket* *K** _{H}* (Figure 2,
right) due to Kigami [20, 22]. This is a compact subset ofCdefined as
the image of a

*harmonic map*Φ :

*K→*Cfrom the

*Sierpi´nski gasket*

*K*(Figure 2, left) toC. More precisely, let

*V*0 =

*{q*1

*, q*2

*, q*3

*}*be the set of the three outmost vertices of

*K, let (E,F*) be the (self-similar)

*standard*

*Dirichlet form*on

*K*(so that

*F*is known to be a dense subalgebra of (

*C*(K),

*∥ · ∥*sup)), and let

*h*

^{K}_{1}

*, h*

^{K}_{2}

*∈ F*be

*E-harmonic*on

*K*

*\V*0 and satisfy

*E*(h

^{K}

_{j}*, h*

^{K}*) =*

_{k}*δ*

*jk*for any

*j, k*

*∈ {*1,2

*}*(see [10, Sections 2 and 3] and the references therein for details). Then we can define a con- tinuous map Φ :

*K*

*→*C by Φ(x) := (

*h*^{K}_{1}(x), h^{K}_{2}(x))

, and its image
*K** _{H}* := Φ(K) is called the harmonic Sierpi´nski gasket. In fact, Kigami
has proved in [20, Theorem 3.6] that Φ :

*K*

*→*

*K*

*is injective and hence a homeomorphism, and further in [20, Theorem 4.1] that a one- dimensional, measure-theoretic “Riemannian structure” can be defined*

_{H}**-**
Φ :=

(*h*^{K}_{1}
*h*^{K}_{2}

)

:*K→K*_{H}*,→*C
*h*^{K}* _{j}* :

*E*-harmonic on

*K\V*0

*E*(h^{K}_{j}*, h*^{K}* _{k}* ) =

*δ*

*jk*

Figure 2. Sierpi´nski gasket*K* and harmonic Sierpi´nski gasket*K** _{H}*
on

*K*through the embedding Φ and the

*E-energy measureµ*

^{1}of Φ, which plays the role of the “Riemannian volume measure” and is given by (3.1)

*µ*:=

*µ*

_{⟨}

_{h}*K*

1*⟩*+*µ*_{⟨}_{h}*K*

2*⟩*= “*|∇*Φ*|*^{2}*d*vol ”;

here *µ*_{⟨}_{u}* _{⟩}* denotes the

*E*-energy measure of

*u*

*∈ F*playing the role of

“*|∇u|*^{2}*d*vol” and defined as the unique Borel measure on *K*such that
(3.2)

∫

*K*

*f dµ*_{⟨}_{u}* _{⟩}*=

*E*(f u, u)

*−*1

2*E*(f, u^{2}) for any*f* *∈ F*.
Kigami has also proved in [22, Theorem 6.3] that the heat kernel of
(K, µ,*E,F*) satisfies the two-sided*Gaussian* estimate of the same form
as for Riemannian manifolds, and further detailed studies of (K, µ,*E,F*)
have been done in [9, 23, 10]; see [10] and the references therein for
details.

As observed from Figures 1 and 2, the overall geometric structure of
the Apollonian gasket*K(*D) resembles that of the harmonic Sierpi´nski
gasket*K** _{H}*, and then it is natural to expect that the above-mentioned
framework of the measurable Riemannian structure on

*K*induced by the embedding Φ :

*K→K*

*can be adapted to the setting of*

_{H}*K(*D) for D

*∈*TDT

*to construct a “geometrically canonical” Dirichlet form on*

^{⊕}*K(*D). Namely, it is expected that there exists a non-zero strongly local regular symmetric Dirichlet form (

*E*

^{D}

*,F*

_{D}) over

*K(*D)

*with respect to*

*which the coordinate functions*Re(

*·*)

*|*

*K(*D)

*,*Im(

*·*)

*|*

*K(*D)

*are harmonic on*

*K(*D)

*\V*0(D). The possibility of such a construction was first noted by Teplyaev in [34, Theorem 5.17], and in [14] the author has completed the construction of (

*E*

^{D}

*,F*

_{D}) and further proved its uniqueness and concrete identification, summarized as follows. We start with some definitions.

**Definition 3.1.** (1) A subset*C* ofCis called a*circular arc*if and
only if*C*=*{z*0+re^{iθ}*|θ∈*[α, β]*}*for some*z*0*∈*C,*r∈*(0,+*∞*) and

1*µ*was first introduced in [24] and is called the*Kusuoka measure* on*K.*

*α, β∈*Rwith*α < β. In this case we set cent(C) :=z*_{0}, rad(C) :=*r*
and*D**C*:= int*{*(1*−t) cent(C) +tz|z∈C,t∈*[0,1]*}*.

(2) For a circular arc*C, the length measure on (C,*B(C)) is denoted by
H^{1}*C*, the gradient vector along*C* at *x∈C* of a function *u*:*C→*R
is denoted by *∇**C**u(x) providedu*is diﬀerentiable at*x, and we set*
*W*^{1,2}(C) := *{u∈* R^{C}*|* *u*is a.c. on*C,|∇**C**u| ∈L*^{2}(C,H^{1}*C*)*}*, where

“a.c.” is an abbreviation of “absolutely continuous”.

(3) We define*h*1*, h*2:C*→*Rby*h*1(z) := Re*z* and*h*2(z) := Im*z.*

**Definition 3.2.** LetD= (D_{1}*, D*_{2}*, D*_{3})*∈*TDT* ^{⊕}*. We define
(3.3) AD:=

*{T*(D)

*∩∂D*

*j*

*|j∈S} ∪ {∂D*in(D

*w*)

*|w∈W*

_{∗}*}*and set

*K*

^{0}(D) := ∪

*C**∈A*D*C, so that each* *C* *∈* AD is a circular arc,

∪

*C**∈A*D*D**C*=*△*(D)*\K(*D),∪

*C,A**∈A*D*, C**̸*=A(C*∩A) =*∪

*w**∈**W*_{∗}*V*0(D*w*),
and an induction in*|w|*easily shows that for any*w∈W** _{∗}*,

(3.4) A_{D}*w*=*{C∩K(*D*w*)*|C∈*A_{D}*} \ {∅}.*

The canonical Dirichlet form (*E*^{D}*,F*_{D}) on*K(*D) and the associated

“Riemannian volume measure” similar to (3.1) turn out to be expressed
explicitly in terms of the circle packing structure of*K(*D), as follows.

**Definition 3.3**(cf. [14, Theorems 5.11 and 5.13]). LetD*∈*TDT* ^{⊕}*.
(1) We define a Borel measure

*µ*

^{D}on

*K(*D) by

(3.5) *µ*^{D}:=∑

*C**∈A*D

rad(C)H^{1}*C*(*· ∩C),*

so that for any *w* *∈W** _{∗}* we have

*µ*

^{D}(K(D

*w*)) = 2 vol

_{2}(

*△*(D

*w*)) by (3.4),∪

*C**∈A*Dw*D** _{C}*=

*△*(D

*w*)

*\K(*D

*w*) and vol

_{2}(K(D

*w*)) = 0.

(2) For each *u* *∈* R^{K}^{0}^{(}^{D}^{)} with *u|**C* a.c. on *C* for any *C* *∈* A_{D}, we
define a *µ*^{D}-a.e. defined,R^{2}-valued Borel measurable map*∇*_{D}*u*by
(*∇*_{D}*u)|**C* := *∇**C*(u*|**C*) for each *C* *∈* A_{D}, so that *|∇*_{D}*u|*^{2}*dµ*^{D} =

∑

*C**∈A*D*|∇**C*(u*|**C*)*|*^{2}rad(C)*d*H^{1}*C*. Then we further define
(3.6) *F*D:=*W*_{D}^{1,2}:=

{

*u∈*R^{K}^{0}^{(}^{D}^{)}

*u|**C**∈W*^{1,2}(C) for any*C∈*A_{D},

*|∇*_{D}*u| ∈L*^{2}(K(D), µ^{D})

}

and set *C*_{D} := *{u* *∈ C*(K(D)) *|* *u|**K*^{0}(D) *∈ F*_{D}*}* and *C*_{D}^{lip} := *{u* *∈*
*C*(K(D))*|* **Lip**_{K(}_{D}_{)}*u <*+*∞}*, which are considered as linear sub-
spaces of*F*_{D}through the linear injection*C*(K(D))*∋u7→u|**K*^{0}(D)*∈*
R^{K}^{0}^{(}^{D}^{)}. Noting that *⟨∇*D*u,∇*D*v⟩ ∈L*^{1}(K(D), µ^{D}) for any*u, v* *∈*

*F*_{D}, we also define a bilinear form*E*^{D}:*F*_{D}*× F*_{D} *→*Ron*F*_{D} by
*E*^{D}(u, v) :=

∫

*K(*D)

*⟨∇*_{D}*u,∇*_{D}*v⟩dµ*^{D}

=∑

*C**∈A*D

∫

*C*

*⟨∇**C*(u*|**C*),*∇**C*(v*|**C*)*⟩*rad(C)*d*H^{1}*C**.*
(3.7)

In particular, setting*dµ*^{D}_{⟨}_{u}* _{⟩}*:=

*|∇*D

*u|*

^{2}

*dµ*

^{D}for each

*u∈ F*D, we have

*µ*

^{D}=

*µ*

^{D}

_{⟨}

_{h}1*|**K(D)**⟩*+*µ*^{D}_{⟨}_{h}

2*|**K(D)**⟩* as the counterpart of (3.1) for*K(*D).

**Theorem 3.4**([14, Theorem 5.18]).*Let*D*∈*TDT^{⊕}*and setF*_{D}^{0}*,0*:=

*{u∈ F*_{D}*|u|**V*0(D)= 0*}. Then*(*E*^{D}*,F*_{D})*is an irreducible, strongly local,*
*regular symmetric Dirichlet form onL*^{2}(K(D), µ^{D})*with a coreC*_{D}^{lip}*, and*
(3.8)

∫

*K(*D)

*u*^{2}*dµ*^{D}*≤*40κ(D)^{−}^{2}*E*^{D}(u, u) *for anyu∈ F*_{D}^{0}*,0**.*
*Moreover, the inclusion map* *F*D *,→L*^{2}(K(D), µ^{D})*is a compact linear*
*operator under the norm∥u∥*_{F}_{D}:= (*E*^{D}(u, u)+∫

*K(*D)*u*^{2}*dµ*^{D})^{1/2}*onF*_{D}*.*
**Theorem 3.5** ([14, Theorem 5.23]). *Let* D *∈* TDT^{⊕}*, let* *µ*^{′}*be a*
*finite Borel measure on* *K(*D)*with* *µ** ^{′}*(U)

*>*0

*for any non-empty open*

*subsetU*

*ofK(*D), and let(

*E*

^{′}*,F*

*)*

^{′}*be a strongly local, regular symmetric*

*Dirichlet form onL*

^{2}(K(D), µ

*)*

^{′}*withE*

*(u, u)*

^{′}*>*0

*for someu∈ F*

^{′}*. Then*

*the following two conditions are equivalent:*

(1) *Any* *h∈ {h*1*|**K(*D)*, h*2*|**K(*D)*}* *is inF*^{′}*and is* *E** ^{′}*-harmonic

*onK(*D)

*\*

*V*0(D), i.e.,

*E*

*(h, v) = 0*

^{′}*for anyv∈ F*

^{′}*∩ C*(K(D))

*withv|*

*V*0(D)= 0.

(2) *F*^{′}*∩ C*(K(D)) =*C*D *andE*^{′}*|**C*D*×C*D =*cE*^{D}*|**C*D*×C*D *for some* *c∈*R*.*
*Remark* 3.6. In contrast to the case of*K(*D) described in Definition
3.3, Theorems 3.4 and 3.5, the standard Dirichlet form (*E,F*) on the
Sierpi´nski gasket*K*satisfies*µ*_{⟨}_{u}* _{⟩}*(K

^{0}) = 0 for any

*u∈ F*by [10, Lemma 8.26] and [8, Lemma 5.7], where

*K*

^{0}denotes the union of the boundaries of the equilateral triangles constituting

*K. In particular, (E,F*) can- not be expressed as the sum of any weighted one-dimensional Dirichlet forms on Φ(K

^{0})

*⊂*

*K*

*similar to (3.7). The author does not have a good explanation of the reason for this diﬀerence, and it would be very nice to give one. A naive guess could be that some suﬃcient smoothness of the relevant curves might be required for the validity of an expression like (3.7) of a non-zero strongly local regular symmetric Dirichlet form satisfying the analog of Theorem 3.5-(1); indeed, the curves constituting Φ(K*

_{H}^{0}) are

*C*

^{1}but not

*C*

^{2}by [22, Theorem 5.4-(2)], whereas the corre- sponding curves

*C*

*∈*AD in

*K(*D) are circular arcs and therefore real

analytic. While this guess itself might well be correct, it would be still unclear how smooth the relevant curves should need to be.

The rest of this section is devoted to a brief sketch of the proof of
Theorems 3.4 and 3.5, which is rather long and occupies the whole of [14,
Sections 4 and 5]. It starts with identifying what the*trace* *E*^{D}*|**V** _{m}*(D),
(3.9)

*E*

^{D}

*|*

*V*

*m*(D)(u, u) := inf

*v**∈F*D*, v**|**Vm(D)*=u*E*^{D}(v, v), *u∈*R^{V}^{m}^{(}^{D}^{)}*,*
of (*E*^{D}*,F*_{D}) to*V**m*(D) :=∪

*w**∈**W*_{m}*V*0(D*w*)*should*be for any*m∈*N∪{0*}*.
In view of the desired properties of (*E*^{D}*,F*_{D}) in Theorem 3.5, the forms
*{E*^{D}*|**V**m*(D)*}**m**∈N∪{*0*}*should have the properties in the following theorem.

**Theorem 3.7** ([34, Theorem 5.17]). *Let* D *∈* TDT^{⊕}*. Then there*
*exists{E**m*^{D}*}**m**∈N∪{*0*}* *such that the following hold for anym∈*N*∪ {*0*}:*
(1) *E**m*^{D} *is a symmetric Dirichlet form onℓ*^{2}(V*m*(D)). *E**m*^{D}(1*x**,***1***y*) = 0 =

*E**m*^{D}(1*x**,***1)***for anyx, y* *∈V**m*(D)*with{τ∈W**m**|x, y* *∈V*0(D*τ*)*}*=*∅.*
(2) *Both* *h*1*|**V** _{m}*(D)

*andh*2

*|*

*V*

*(D)*

_{m}*are*

*E*

*m*

^{D}

*-harmonic onV*

*m*(D)

*\V*0(D).

(3) *E**m*^{D}(u, u) = min_{v}_{∈R}*Vm+1 (*D)*, v**|**Vm(D)*=u*E**m+1*^{D} (v, v)*for anyu∈*R^{V}^{m}^{(}^{D}^{)}*.*
(4) *E**m*^{D}(h1*|**V**m*(D)*, h*1*|**V**m*(D)) +*E**m*^{D}(h2*|**V**m*(D)*, h*2*|**V**m*(D)) = 2 vol2(*△*(D)).

Teplyaev’s proof of Theorem 3.7 in [34] is purely Euclidean-geometric
and provides no further information on *{E**m*^{D}*}**m**∈N∪{*0*}*. The author has
identified it as follows, by applying a refinement of [28, Corollary 4.2].

**Theorem 3.8** ([14, Theorem 4.18]). *For each*D= (D1*, D*2*, D*3)*∈*
TDT^{⊕}*, a sequence* *{E**m*^{D}*}**m**∈N∪{*0*}* *as in Theorem* 3.7 *is unique, and*
(3.10) *E*0^{D}(u, u) =∑

*j**∈**S*

*κ(*D)^{2}+ curv(D*j*)^{2}
2κ(D) curv(D* _{j}*)

(*u(q** _{j+1}*(D))

*−u(q*

*(D)))2*

_{j+2}*for any* *u∈*R^{V}^{0}^{(}^{D}^{)}*, whereq** _{j+3}*(D) :=

*q*

*(D)*

_{j}*for*

*j*

*∈S. Moreover, for*

*any*D

*∈*TDT

^{⊕}*, any*

*m∈*N

*∪ {*0

*}*

*and anyu∈*R

^{V}

^{m}^{(}

^{D}

^{)}

*,*

(3.11) *E**m*^{D}(u, u) =∑

*w**∈**W*_{m}*E*0^{D}* ^{w}*(u

*|*

*V*0(D

*w*)

*, u|*

*V*0(D

*w*)).

LetD*∈*TDT* ^{⊕}*. Theorem 3.7-(3) allows us to apply to

*{E*

*m*

^{D}

*}*

*m*

*∈N∪{*0

*}*

the general theory from [21, Chapter 2] of constructing a Dirichlet form
by taking the*“inductive limit”*of Dirichlet forms on finite sets. Namely,
setting*V** _{∗}*(D) :=∪

*m**∈N∪{*0*}**V**m*(D), we can define a linear subspace*F*_{D}* ^{′}*
ofR

^{V}

^{∗}^{(}

^{D}

^{)}and a bilinear form

*E*

*:*

^{′D}*F*

_{D}

^{′}*× F*

_{D}

^{′}*→*Ron

*F*

_{D}

*by*

^{′}*F*_{D}* ^{′}* :={

*u∈*R^{V}^{∗}^{(}^{D}^{)}lim_{m}_{→∞}*E**m*^{D}(u*|**V** _{m}*(D)

*, u|*

*V*

*(D))*

_{m}*<*+

*∞*}

*,*(3.12)

*E** ^{′D}*(u, v) := lim

_{m}

_{→∞}*E*

*m*

^{D}(u

*|*

*V*

*(D)*

_{m}*, v|*

*V*

*(D))*

_{m}*∈*R

*,*

*u, v*

*∈ F*

_{D}

^{′}*.*(3.13)

The next step of the proof of Theorems 3.4 and 3.5 is the following
identification of (*E*^{′D}*,F*_{D}* ^{′}* ) as (

*E*

^{D}

*,F*D), i.e., as given by (3.6) and (3.7).

**Theorem 3.9** ([14, Theorem 5.13]). *Let* D*∈*TDT^{⊕}*. Then* *F*_{D}* ^{′}* =

*{u|*

*V*

*(D)*

_{∗}*|*

*u∈ F*D

*}, the mapping*

*F*D

*∋*

*u7→u|*

*V*

*(D)*

_{∗}*∈ F*

_{D}

^{′}*is a linear*

*isomorphism, andE*

*(u*

^{′D}*|*

*V*

*(D)*

_{∗}*, v|*

*V*

*(D)) =*

_{∗}*E*

^{D}(u, v)

*for any*

*u, v∈ F*

_{D}

*.*

*Sketch of the proof.* By Theorem 3.7-(2),(3) and (3.12) we have
*h*1*|**V** _{∗}*(D)

*, h*2

*|*

*V*

*(D)*

_{∗}*∈ F*

_{D}

*, which together with (3.12) implies that*

^{′}*C*

^{′}_{D}:=

*{u∈ C*(K(D))*|u|**V** _{∗}*(D)

*∈ F*

_{D}

^{′}*}*is a dense subalgebra of (

*C*(K(D)),

*∥·∥*sup) with

*h*1

*|*

*K(*D)

*, h*2

*|*

*K(*D)

*∈ C*

_{D}

^{lip}

*⊂ C*

_{D}

*. Hence at this stage we can al- ready define the*

^{′}*E*

*-energy measure*

^{′D}*µ*

^{′D}

_{⟨}

_{u}*of*

_{⟩}*u*

*∈ C*

_{D}

*by (3.2) with*

^{′}*K(*D),

*E*

^{′D}*,C*

_{D}

*in place of*

^{′}*K,E,F*, and the analog of (3.1) by

*µ*

*:=*

^{′D}*µ*^{′D}_{⟨}_{h}

1*|**K(D)**⟩*+*µ*^{′D}_{⟨}_{h}

2*|**K(D)**⟩*. Then it follows from Theorem 3.7-(4) and (3.11)
that *µ** ^{′D}*(K(D

*w*)) = 2 vol2(

*△*(D

*w*)) =

*µ*

^{D}(K(D

*w*)) for any

*w*

*∈*

*W*

*, whence*

_{∗}*µ*

*=*

^{′D}*µ*

^{D}.

Now that*µ** ^{′D}*has been identified as

*µ*

^{D}given by (3.5), it is natural to guess

^{2}that

*F*

_{D}

^{′}*⊂ {u|*

*V*

*(D)*

_{∗}*|u∈ F*D

*}*and that

*E*

*(u*

^{′D}*|*

*V*

*(D)*

_{∗}*, u|*

*V*

*(D)) =*

_{∗}*E*

^{D}(u, u) for any

*u∈ F*

_{D}with

*u|*

*V*

*(D)*

_{∗}*∈ F*

_{D}

*. This guess is not diﬃcult to verify, first for any piecewise linear*

^{′}*u*

*∈ F*

_{D}by direct calculations based on Theorem 3.7-(2), (3.10), (3.11) and (3.13), and then for any

*u∈ F*

_{D}with

*u|*

*V*

*(D)*

_{∗}*∈ F*

_{D}

*by using the canonical approximation of*

^{′}*u*by piecewise linear functions; here

*u∈ F*

_{D}is called

*m-piecewise linear,*where

*m*

*∈*N

*∪ {*0

*}*, if and only if

*u|*

*K*

^{0}(D

*w*) is a linear combination of

*h*1

*|*

*K*

^{0}(D

*w*)

*, h*2

*|*

*K*

^{0}(D

*w*)

*,*

**1**

*0(D*

_{K}*w*) for any

*w∈W*

*m*, and

*piecewise linear*if and only if

*u*is

*m-piecewise linear for somem∈*N

*∪ {*0

*}*.

Finally, for any*u∈ F*_{D}, some direct calculations using (3.10), (3.7)
and (3.4) show that*E*0^{D}* ^{w}*(u

*|*

*V*

_{0}(D

*w*)

*, u|*

*V*

_{0}(D

*w*))

*≤*7∫

*K(*D*w*)*|∇*_{D}*u|*^{2}*dµ*^{D}for
any*w∈W** _{∗}*, which together with (3.11) yields

*E*

*m*

^{D}(u

*|*

*V*

*(D)*

_{m}*, u|*

*V*

*(D))*

_{m}*≤*

*E*

^{D}(u, u) for any

*m∈*N

*∪ {*0

*}*, whence

*u|*

*V*

*(D)*

_{∗}*∈ F*

_{D}

*by (3.12). Q.E.D.*

^{′}The last main step of the proof of Theorem 3.4 is to prove (3.8), which is based mainly on (3.5), (3.7) and the following lemma.

**Lemma 3.10**([14, Lemma 5.19]). *Let* *C⊂*C*be a circular arc, let*
*u∈*R^{C}*satisfy***Lip**_{C}*u <*+*∞, and for* *a∈*R*defineI*_{C}^{a}*u*:*D*_{C}*→*R*by*
(3.14) *I**C*^{a}*u((1−t) cent(C) +tz) := (1−t)a*+tu(z), (t, z)*∈*[0,1]*×C.*

2This is how the author first came up with the expressions (3.6) and (3.7).

*Then for anya∈*[min_{C}*u,*max_{C}*u],***Lip**_{D}

*C**I**C*^{a}*u≤√*

5**Lip**_{C}*uand*
(3.15)

2 21

∫

*D**C*

*|∇I**C*^{a}*u|*^{2}*d*vol2*≤*

∫

*C*

*|∇**C**u|*^{2}rad(C)*d*H^{1}*C**≤*2

∫

*D**C*

*|∇I**C*^{a}*u|*^{2}*d*vol2*.*
*Further, withu** ^{C}*:=H

^{1}

*C*(C)

^{−}^{1}∫

*C**u d*H*C*^{1}*, for anya∈ {*0, u^{C}*},*
(3.16) 2

∫

*D**C*

*|I**C*^{a}*u|*^{2}*d*vol2*≤*

∫

*C*

*u*^{2}rad(C)*d*H^{1}*C**≤*4

∫

*D**C*

*|I**C*^{a}*u|*^{2}*d*vol2*.*
Combining Lemma 3.10 with (3.5) and (3.7), we obtain the follow-
ing.

**Lemma 3.11** ([14, Lemma 5.21]). *Let* D *∈* TDT^{⊕}*and* *u* *∈ C*^{lip}_{D}*.*
*Noting△*(D)*\*(K(D)*\K*^{0}(D)) =∪

*C**∈A*D*D**C**, defineI*_{D}^{0}*u∈*R^{△}^{(}^{D}^{)} *by*
(3.17)

*I*_{D}^{0}*u|**K(*D):=*u,* *I*_{D}^{0}*u|**D** _{C}* :=

{*I**C*^{0}(u*|**C*) *ifC⊂∂T*(D),

*I**C*^{u}* ^{C}*(u

*|*

*C*)

*ifC̸⊂∂T*(D),

*C∈*A

_{D}

*.*

*If alsou|*

*V*0(D)= 0, then

*I*

_{D}

^{0}

*u|*

*∂*

*△*(D)= 0,

**Lip**

_{△}_{(}

_{D}

_{)}

*I*

_{D}

^{0}

*u≤√*

5**Lip**_{K(}_{D}_{)}*u,*
2

21

∫

*△*(D)

*|∇I*_{D}^{0}*u|*^{2}*d*vol_{2}*≤ E*^{D}(u, u)*≤*2

∫

*△*(D)

*|∇I*_{D}^{0}*u|*^{2}*d*vol_{2}*,*
(3.18)

2

∫

*△*(D)

*|I*_{D}^{0}*u|*^{2}*d*vol2*≤*

∫

*K(*D)

*u*^{2}*dµ*^{D} *≤*4

∫

*△*(D)

*|I*_{D}^{0}*u|*^{2}*d*vol2*.*
(3.19)

*Sketch of the proof of Theorem* 3.4. Recall the following classical
fact implied by [5, Lemma 6.2.1, Theorems 4.5.1, 4.5.3 and 6.1.6]: if*Q*
is an open rectangle inCwhose smaller side length is*δ∈*(0,+*∞*), then
(3.20)

∫

*Q*

*u*^{2}*d*vol2*≤* *δ*^{2}
*π*^{2}

∫

*Q*

*|∇u|*^{2}*d*vol2

for any *u* *∈* R* ^{Q}* with

**Lip**

_{Q}*u <*+

*∞*and

*u|*

*∂Q*= 0. Since

*△*(D)

*⊂Q*for some such

*Q*with

*δ*= 3κ(D)

^{−}^{1}and then each

*u*

*∈*R

^{△}^{(}

^{D}

^{)}with

**Lip**

_{△}_{(}

_{D}

_{)}

*u <*+

*∞*and

*u|*

*∂*

*△*(D) = 0 can be extended to

*Q*by setting

*u|*

*Q*

*\△*(D):= 0 so as to satisfy

**Lip**

_{Q}*u <*+

*∞*and

*u|*

*∂Q*= 0, we easily see from Lemma 3.11 and (3.20) that (3.8) holds for any

*u∈ F*

_{D}

^{0}

*,0*

*∩ C*

_{D}

^{lip}.

Now, by utilizing the canonical approximation of each *u∈ F*_{D} by
piecewise linear functions as in the sketch of the proof of Theorem 3.9
above, we can show that (3.8) extends to any*u∈ F*_{D}^{0}*,0*, which implies
*F*D*⊂L*^{2}(K(D), µ^{D}), and that the inclusion map*F*D*,→L*^{2}(K(D), µ^{D})

is the limit in operator norm of finite-rank linear operators and hence compact. The rest of the proof is straightforward. Q.E.D.

*Sketch of the proof of Theorem* 3.5. The implication from (2) to
(1) is immediate from Theorem 3.9 and Theorem 3.7-(2),(3). That
from (1) to (2) can be shown by defining the trace*E|**V** _{m}*(D)of (

*E,F*) to

*V*

*m*(D) for

*m∈*N

*∪{*0

*}*in essentially the same way as (3.9), proving that

*{E|*

*V*

*(D)*

_{m}*}*

*m*

*∈N∪{*0

*}*satisfies Theorem 3.7-(1),(2),(3) by the assumption of (1) and then applying Theorem 3.8 to conclude that

*{E|*

*V*

*(D)*

_{m}*}*

*m*

*∈N∪{*0

*}*=

*{cE*

*m*

^{D}

*}*

*m*

*∈N∪{*0

*}*for some

*c∈*R, which is easily seen to imply (2). Q.E.D.

*§***4.** **Weyl’s eigenvalue asymptotics for the Apollonian gasket**
The following proposition is an easy consequence of Theorem 3.4;

see also [5, Exercise 4.2, Corollary 4.2.3, Theorems 4.5.1 and 4.5.3].

**Proposition 4.1.** *Let*D*∈*TDT^{⊕}*, letV* *be a finite subset ofV** _{∗}*(D)

*and setF*

_{D}

^{0}

*,V*:=

*{u∈ F*D

*|u|*

*V*= 0

*}. Then*(

*E*

^{D}

*|*

_{F}_{D,V}

^{0}

_{×F}_{D,V}

^{0}

*,F*

_{D}

^{0}

*,V*)

*is*

*a strongly local, regular symmetric Dirichlet form onL*

^{2}(K(D)

*\V, µ*

^{D}),

*and there exists a unique non-decreasing sequence{λ*

^{D}

_{n}

^{,V}*}*

*n*

*∈N*

*⊂*[0,+

*∞*)

*such that*

*−L*

_{D}

*,V*

*φ*

^{D}

_{n}*=*

^{,V}*λ*

^{D}

_{n}

^{,V}*φ*

^{D}

_{n}

^{,V}*for any*

*n*

*∈*N

*for some complete*

*orthonormal system*

*{φ*

^{D}

_{n}

^{,V}*}*

*n*

*∈N*

*⊂ D*(

*L*D

*,V*)

*of*

*L*

^{2}(K(D)

*\V, µ*

^{D}); here

*L*

_{D}

*,V*:

*D*(

*L*

_{D}

*,V*)

*→*

*L*

^{2}(K(D)

*\V, µ*

^{D})

*denotes the*Laplacian, i.e., the

*non-positive self-adjoint operator on*

*L*

^{2}(K(D)

*\V, µ*

^{D}), associated with (

*E*

^{D}

*|*

_{F}_{D,V}

^{0}

_{×F}_{D,V}

^{0}

*,F*

_{D}

^{0}

*,V*). Also,lim

*n*

*→∞*

*λ*

^{D}

_{n}*= +*

^{,V}*∞, and for anyn∈*N

*,*(4.1)

*λ*

^{D}

_{n}*= min*

^{,V}{ max

*u**∈**L**\{*0*}*

*E*^{D}(u, u)

∫

*K(*D)*u*^{2}*dµ*^{D}

*Lis a linear subspace*
*of* *F*_{D}^{0}_{,V}*,*dim*L*=*n*

}
*.*

The proof of the following theorem is the principal aim of [14].

**Theorem 4.2**([14, Theorem 7.1]). *There existsc*_{AG}*∈*(0,+*∞*)*such*
*that for any*D*∈*TDT^{⊕}*and any finite subset* *V* *of* *V** _{∗}*(D),

(4.2) lim

*λ**→*+*∞*

#*{n∈*N*|λ*^{D}_{n}^{,V}*≤λ}*

*λ*^{d}^{AG}* ^{/2}* =

*c*AGH

^{d}^{AG}(K(D)).

The rest of this section outlines the analytic aspects of the proof of Theorem 4.2. It can be deduced from the following theorem applicable to more general counting functions, including the classical one given by

#*{w∈W*_{∗}*|*curv(D_{in}(D*w*))*≤λ}*, whose asymptotic behavior analogous
to (4.2) has been obtained first by Oh and Shah in [30, Corollary 1.8].