Volume growth of quasihyperbolic balls

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VOLUME GROWTH OF QUASIHYPERBOLIC

BALLS

RIKU KLÉN, VILLE SUOMALA, MATTI VUORINEN,AND XIAOHUI ZHANG

Abstract. The purpose of this paper is to discuss the no-tion of the quasihyperbolic volume and to find growth esti-mates for the quasihyperbolic volume of balls in a domainG ⊂ R

n , in terms of the radius. It turns out that in thecase of domains with Ahlfors regular boundaries, the rateof growth depends not merely on the radius but also on themetric structure of the boundary.

Keywords. Quasihyperbolic volume, uniform porosity, Q-regularity

2010 Mathematics Subject Classification. 51M10, 51M25,

28A80

1. Introduction

Since its introduction three decades ago, the quasihyperbolicmetric has become a popular tool in many subfields of geometricfunction theory. For instance, in the study of quasiconformalmaps of Rn and of Banach spaces [Va1], analysis of metric spaces[H] and hyperbolic type metric [HIMPS]. A natural question iswhether and to what extent, the results of hyperbolic geometryhave counterparts for the quasihyperbolic geometry. For instancein [K] it was noticed that some facts from hyperbolic trigonometryof the plane have counterparts in the quasihyperbolic setup whilesome have not.

The purpose of this paper is to discuss the notion of the quasi-hyperbolic volume and to find growth estimates for the quasihy-perbolic volume of balls in a domain G ⊂ R

n , in terms of theradius. It turns out that the rate of growth depends not merelyon the radius but also on the metric structure of the boundary.Below we give an explicit growth estimate for the case of domainswith Ahlfors regular boundary.

For a compact set E ⊂ Rn , and 0 < s < t , we consider the

layer sets E(s, t) = z : s ≤ d(z, E) ≤ t and relate its volume tothe metric size of E via its (Hausdorff) dimension under the ad-ditional assumption that the boundary of E be Ahlfors Q-regularfor some 0 < Q < n. It is practically equivalent to formulate

File: QVol120824.tex, printed: 2012-8-28, 0.26

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http://arxiv.org/abs/1208.5355v1

2 R. KLÉN, V. SUOMALA, M. VUORINEN, AND X. ZHANG

this idea in terms of the number of those Whitney cubes of theWhitney decomposition of Rn \ E that meet E(s, t) . This ideagoes back to [MV]—in Lemmas 3.1 and 3.3 we will refine someresults of [MV]. One of the key ideas of [MV] was to estimatefrom above the metric size of E in terms of the size of the layersets.

In the last section, we estimate from below the growth of thequasihyperbolic volume of balls in the case when G = R

n \ Eis ψ-uniform and E is Ahlfors regular. The main result of thepaper is Theorem 4.19 which gives a lower bound for the volumegrowth.

2. Notation

The quasihyperbolic distance between two points x and y in aproper subdomain G of the Euclidean space R

n, n ≥ 2, is definedby

kG(x, y) = infα∈Γxy

∫

α

|dz|d(z)

,

where d(z) = d(z, ∂G) is the (Euclidean) distance between thepoint z ∈ G and the boundary of G (denoted ∂G), and Γxy is thecollection of all rectifiable curves in G joining x and y. Some basicproperties of the quasihyperbolic metric can be found in [V1].

In particular, G.J. Martin and B.G. Osgood [MO, page 38]showed that for x, y ∈ R

n \ 0 and n ≥ 2

(2.1) kRn\0(x, y) =

√θ2 + log2

|x||y| ,

where θ = ∡(x, 0, y) ∈ [0, π].For the volume of the unit ball in R

n we use the notation Ωn =m(B(0, 1)) = πn/2/Γ((n/2) + 1) where m is the n-dimensionalLebesgue measure and Γ stands for the usual Γ-function, see[AS, Ch 6]. The surface area of the unit sphere is ωn−1 =mn−1(S

n−1(0, 1)) = nΩn .The quasihyperbolic volume of a Lebesgue measurable set A ⊂

G is defined by

volkG(A) =

∫

A

dm(z)

d(z)n.

We also use volk(A) for volkG(A) if the domain G is clear from

the context.We assume n ≥ 2 and use the notation Bn(x, r) = z ∈ R

n :|x − z| < r for the Euclidean ball, and its boundary is thesphere Sn−1(x, r) = ∂Bn(x, r), where the center x can be omittedif x = 0.

VOLUME GROWTH OF QUASIHYPERBOLIC BALLS 3

Below we use the Whitney decomposition of the complementof a closed set in R

n. If E ⊂ Rn is non-empty and closed, then

Rn \ E can be presented [S, p. 16] as a union of closed dyadic

cubes Qkj

(2.2) Rn \ E =

⋃

k∈Z

Nk⋃

j=1

Qkj ,

where the cubes Qkj have the following properties:

(i) each Qkj has sides parallel to the coordinate axes and

edges of length 2−k,(ii) the interiors of the cubes Qk

j and Qki are mutually dis-

joint provided i 6= j,(iii) the distance between the cube Qk

j and E satisfies thefollowing inequalities

(2.3) 2−k√n ≤ d(Qk

j , E) ≤ 22−k√n.

The decomposition (2.2) is called the Whitney decomposition, thecubes Qk

j are called the Whitney cubes and the set Qkj : j =

1, . . . , Nk is called the kth generation of the cubes.For 0 < s < t <∞ and E ⊂ R

n, we define

E(s) = x ∈ Rn : d(E, x) ≤ s

and

(2.4) E(s, t) = x ∈ Rn : s ≤ d(E, x) ≤ t.

Because diam(Qkj ) =

√n2−k we have by (2.3)

(2.5) Qkj ⊂ E(

√n2−k, 5

√n2−k).

We say that E ⊂ Rn is (α, r0)-uniformly porous [Va2] if for all

x ∈ E and 0 < r < r0 there is y ∈ B(x, r) with d(y, E) ≥ αr.A set E is uniformly porous, if it is (α, r0)-uniformly porous forsome α, r0 > 0. We refer to α, r0 and n as the uniform porositydata.

A set E ⊂ Rn is Q-regular for 0 < Q < n, if there is a (Borel

regular, outer-) measure µ with spt(µ) = E and constants 0 <α ≤ β <∞ such that

αrQ ≤ µ(B(x, r)) ≤ βrQ, for all x ∈ E and 0 < r < diam(E) .

Here spt(µ) denotes the smallest closed set with full µ-measure.We refer to Q,α, β and diam(E) as the Q-regularity data.

There is a close connection between the notions of uniformporosity and Q-regularity. Indeed, it is well known and easy tosee (e.g. [BHR, Lemma 3.12]) that if E is Q-regular for some0 < Q < n, then it is uniformly porous. Conversely, if E isuniformly porous, then it is a subset of some Q-regular set for


some 0 < Q < n. See [JJKRRS, Theorem 5.3] for a more precisequantitative statement.

3. Number of Whitney cubes

Recall that Nk denotes the number of the kth generation Whit-ney cubes of Rn \ E.

Lemma 3.1. If E ⊂ Rn is compact and uniformly porous, then

there are constants 0 < c < C < ∞ and k0 ∈ N (depending onlyon the uniform porosity data) such that for k ≥ k0, it holds

c2knm(E(2−k)) ≤ Nk ≤ C2knm(E(2−k)) ,

where Nk = Nk+n0+ . . .+Nk+n1

, n0 is the smallest integer satisfy-ing 8

√n2−n0 ≤ 1 and n1 is the largest integer with 80

√n2−n1 ≥ 1.

Proof. Let 0 < λ < 1 < Λ < ∞. Choose 0 < α < (1 + λ)/(2Λ)and r0 > 0 such that E is (2α, r0)-uniformly porous. Given r > 0such that R = 1

2α(1 + λ)r < r0, let B1, . . . , BN be a maximal

collection of disjoint balls of radius R centered at E. Then byelementary geometry

(3.2) ΩnNRn ≤ m(E(R)) ≤ Ωn3

nNRn .

For each Bi, we can find yi ∈ 12Bi with d(yi, E) ≥ αR = 1

2(1 +

λ)r. Moreover (since we live in the Euclidean space), we can infact find such yi with d(yi, E) =

12(1+λ)r. Then B(yi,

12(1−λ)r) ⊂

E(r) \ E(λr) and thus, combined with (3.2),

m(E(r) \ E(λr)) ≥ NΩn

(1− λ

2

)n

rn ≥(α(1− λ)

3(1 + λ)

)n

3nΩnNRn

≥(α(1− λ)

3(1 + λ)

)n

m(E(Λr)) ,

recall that R ≥ Λr.Applying the above estimate with r = 2−k−3, λ = 1

2and Λ = 8

and using (2.5) implies

m(E(2−k)) ≤ cm(E(2−k−3) \E(2−k−4))

≤ c

n1∑

j=n0

Nk+j2−(k+j)n ≤ cm(E(2−k))

where c < +∞ depends only on n and α. This completes theproof.

We also need the following simple fact

Lemma 3.3. If E ⊂ Rn is compact and Q-regular for some 0 <

Q < n, then there are constants 0 < c < C <∞ (depending onlyon the Q-regularity data) such that crn−Q ≤ m(E(r)) ≤ Crn−Q

for 0 < r < diam(E).


Proof. Let B1, . . . , BN be a maximal collection of balls of radiusr centered at E. Then, as in (3.2),

N⋃

i=1

Bi ⊂ E(r) ⊂N⋃

i=1

3Bi ,

and in particular,

(3.4) NΩnrn ≤ m(E(r)) ≤ NΩn3

nrn .

Using the Q-regularity, we have

(3.5) NαrQ ≤∑

i

µ(Bi) ≤ µ(E) ≤∑

i

µ(3Bi) ≤ Nβ3QrQ .

The claim follows by combining (3.4) and (3.5).

Putting the two previous Lemmas together, we obtain:

Corollary 3.6. If E ⊂ Rn is compact and Q-regular for some

0 < Q < n, then there are constants 0 < c < C <∞ and k0 ∈ N

(depending only on the Q-regularity data) such that for all k ≥ k0,it holds

c2kQ ≤ Nk ≤ C2kQ .

Remark 3.7. The porosity and regularity assumptions in theabove results are essential because without them, we cannot guar-antee that m(E(2r) \ E(r)) is comparable to m(E(r)) for smallr > 0.

4. Growth of the quasihyperbolic volume

As mentioned in the Introduction, our goal is to find estimatesfor the asymptotic behaviour of volk(Bk(x, r)) as r → ∞ for x ∈G when the domain G is fixed. As an example, we first considervery regular domains for which the correct asymptotics can beobtained by direct calculation.

4.1. Unit ball. Let us find the quasihyperbolic volume of Bk(r)in the unit ball G = Bn. We need the following integral repre-sentation for the hypergeometric function: for c > b > 0,

F (a, b; c; x) =1

B(b, c− b)

∫ 1

0

tb−1(1− t)c−b−1(1− xt)−adt


where B(b, c− b) = Γ(b)Γ(c− b)/Γ(c). By definition

volk(Bn(r)) =

∫

Bn(r)

dm(x)

(1− |x|)n = ωn−1

∫ r

0

tn−1

(1− t)ndt

= ωn−1rn

∫ 1

0

tn−1(1− rt)−ndt

=ωn−1r

n

nF (n, n;n+ 1; r)

=ωn−1r

n

n(1− r)n−1F (1, 1;n+ 1; r),(4.2)

where the last equality follows from [AS, 15.3.3]

F (a, b; c; x) = (1− x)c−a−bF (c− a, c− b; c; x).

Hence for 0 < r <∞

volk(Bk(r)) =ωn−1 (tanh(r/2))

n

n (1− tanh(r/2))n−1F (1, 1;n+ 1; tanh(r/2))

∼ 21−nωn−1

n− 1e(n−1)r (r → ∞),

since by [AS, 15.1.20]

F (1, 1;n+ 1; 1) =Γ(n+ 1)Γ(n− 1)

Γ(n)2=

n

n− 1.

Corollary 4.3. For s > 0, λ > 1, G = Bn and E = ∂G

volk(E(s, λs))

volk(E(s,∞))→ 1− λ1−n

as s→ 0.

Proof. Let 0 < s < 1/λ. Then the claim is equivalent to

volk(Bn(1− s) \Bn(1− λs)

volk(Bn(1− s))→ 1− λ1−n

as s→ 0.By (4.2)

volk(Bn(1− s) \Bn(1− λs)

volk(Bn(1− s))

= 1− λ1−n(1− λs)nF (1, 1;n+ 1; 1− λs)

(1− s)nF (1, 1, n+ 1, 1− s)

→ 1− λ1−n (s→ 0).


Remark 4.4. For G = Bn, let mh(Bn(r)) be the hyperbolic

volume of the Euclidean ball Bn(r). Then we have(

2

1 + r

)n

volk(Bn(r)) = ωn−1

∫ r

0

2ntn−1

(1 + r)n(1− t)ndt

≤ ωn−1

∫ r

0

2ntn−1

(1− t2)ndt = mh(B

n(r))

and

mh(Bn(r)) ≤ ωn−1

∫ r

0

2ntn−1

(1− t)ndt = 2nvolk(B

n(r)).

It is easy to see thatmh(Bn(r)) = ωn−12

nrnF (n/2, n; 1+n/2; r2)/n.Corollary 4.3 is true for the hyperbolic metric with the same lim-iting value 1− λn−1.

4.5. Punctured space. We will consider the growth of quasihy-perbolic volume in multiply connected domains. It can be shownthat in R

2 \ 0 the quasihyperbolic area of Bk(x, r) is equal toπr2, if r ≤ π, and 2π

√r2 − π2+2π2 arctan(π/

√r2 − π2), if r > π.

Therefore

2π√r2 − π2 ≤ volk(Bk(x, r)) ≤ 4πr

if r > π. We will now find similar lower and upper bounds forthe quasihyperbolic volume of Bk(x, r) in R

n \ 0 for n > 2.

Proposition 4.6. For x ∈ Rn \ 0, n > 2 and r > π

2ωn−1

√r2 − π2 ≤ volk(Bk(x, r)) ≤ 2ωn−1r .

Proof. Since Bk(x, r) is invariant in the inversion in Sn−1(|x|) wehave volk(Bk(x, r)) = 2volk(Bk(x, r) ∩ Bn(|x|)). By [V1, (3.9)]

Bk(x, r) ⊂ Rn \ Bn(|x|e−r). Let S = Sn−1(|x|e−

√r2−π2

), then by

(2.1) maxy∈S

kRn\0(x, y) = r implies Bn(|x|) \ Bn(|x|e−√r2−π2

) ⊂Bk(x, r). Hence we have

(4.7) volk(Bn(|x|) \Bn(|x|e−

√r2−π2

)) ≤ volk(Bk(x, r)∩Bn(|x|))and

(4.8) volk(Bk(x, r) ∩ Bn(|x|)) ≤ volk(Bn(|x|) \Bn(|x|e−r)).

Next let us find the quasihyperbolic volume of annulus E(a, b) =x ∈ R

n | a < |x| < b, 0 < a < b <∞. By definition(4.9)

volk(E(a, b)) =

∫

E(a,b)

dm(x)

|x|n = ωn−1

∫ b

a

tn−1

tndt = ωn−1 log

b

a.

Now the assertion follows from the equations (4.7) and (4.8).


Corollary 4.10. For s > 0, λ > 1, G = Bn \ 0 and E = ∂G

volk(E(s, λs))

volk(E(s,∞))→ 1− λ1−n

as s→ 0.

Proof. Let s < 1/λ. Then E(s, λs) = E1∪E2 where E1 = Bn(1−s)\Bn(1−λs) and E2 = Bn(λs)\Bn(s). Then by (4.2) and (4.9)

volkG(E(s, λs)) = volkBn (E1) + volk

Rn\0(E2)

=ωn−1(1− s)n

nsn−1F (1, 1;n+ 1, 1− s)

−ωn−1(1− λs)n

n(λs)n−1F (1, 1;n+ 1, 1− λs) + ωn−1 log λ

∼ ωn−1

n− 1(1− λ1−n)s1−n (s→ 0).

Similarly E(s,∞) = Bn(1 − s) \ Bn(s) = E ′1 ∪ E ′

2 where E ′1 =

Bn(1− s) \Bn(1/2) and E ′2 = Bn(1/2) \Bn(s), and

volkG(E(s,∞)) = volkBn (E

′1) + volk

Rn\0(E ′

2)

=ωn−1(1− s)n

nsn−1F (1, 1;n+ 1, 1− s)

−ωn−1

2nF (1, 1;n+ 1, 1/2) + ωn−1 log(1/2s)

∼ ωn−1

n− 1s1−n + ωn−1 log(1/s) (s→ 0).

Now we have

lims→0

volk(E(s, λs))

volk(E(s,∞))= lim

s→0

ωn−1

n− 1(1− λ1−n)s1−n

ωn−1

n− 1s1−n + ωn−1 log(1/s)

= 1− λ1−n.

4.11. Half space. The next two propositions concern the case ofthe half space, in which case the quasihyperbolic volume coincideswith the classical hyperbolic volume. The hyperbolic volume ofhyperbolic simplexes in the upper half space has been consideredin [Mi].

Proposition 4.12. For x ∈ H2 and r > 0 we have

volk(Bk(x, r)) = 2π(cosh r − 1).

Proof. Let us choose x = te2 implying that

Bk(x, r) = B2(e2 cosh r, sinh r).


Now

volk(Bk(x, r)) =

∫ t cosh r+t sinh r

t cosh r−t sinh r

√t2 sinh2 r − (cosh r − h)2

h2dh

= 2π(cosh r − 1).

Proposition 4.13. For x ∈ H3 and r > 0 we have

volk(Bk(x, r)) = π(sinh(2r)− 2r).

Proof. Let us choose x = te2 implying that

Bk(x, r) = B2(e2 cosh r, sinh r).

Now

volk(Bk(x, r)) =

∫ t cosh r+t sinh r

t cosh r−t sinh r

π(t2 sinh2 r − (cosh r − h)2)

h3dh

= π(sinh(2r)− 2r).

4.14. More general domains. Our next goal is to prove growthestimates of the above type for larger class of domains, so calledϕ-uniform domains. To that end, we first observe that the quasi-hyperbolic volume of the Whitney cubes is essentially constant.

Lemma 4.15. Let E be a closed subset of Rn and let G be a

component of Rn \ E. Then for the Whitney cubes of R

n \ Econtained in G we have

2−2nn−n/2 ≤ volk(Qkj ) ≤ n−n/2.

Proof. Since m(Qkj ) = 2−kn we have by (2.3)

volk(Qkj ) ≥

2(k−2)n

√nn 2−kn = 2−2nn−n/2,

and

volk(Qkj ) ≤

2kn√nn2

−kn = n−n/2.

Theorem 4.16. Let G be a proper subdomain of Rn with compact

and Q−regular (0<Q<n) boundary. There exists a constant C <∞ such that for each x ∈ G and sufficiently large r > 0, we have

volk(Bk(x, r)) ≤ CeQr.


Proof. Let E = ∂G. We have

Bk(x, r) ⊂ z ∈ G : e−rd(x) < d(z) < erd(x) =: D

and

Qkj ⊂ E(2−k

√n, 5 · 2−k

√n).

If 5 · 2−k√n < e−rd(x) or 2−k

√n > erd(x), i.e.

k >r + log(5

√n/d(x))

log 2=: K, or k < −r − log(

√n/d(x))

log 2=: K ′

then Qkj ∩D = ∅, and thus

Bk(x, r) ⊂ ∪[K]k=[K ′] ∪

Nk

j=1 Qkj .

Note that Nk ≤M(diam(E))2nk always holds for all k ∈ N [MV]and therefore Nk ≤ C(diam(E), k0) for k ≤ k0, where k0 is as inLemma 3.1. Hence for all sufficiently large r, we may estimate

volk(Bk(x, r)) ≤[K]∑

k=[K ′]

Nk∑

j=1

volk(Qkj )

≤ n−n/2

[K]∑

k=[K ′]

Nk

≤ n−n/2(k0 − [K ′])C(diam(E), k0)

+n−n/2

[K]∑

k=k0

c2knm(E(2−k))

≤ n−n/2(k0 − [K ′])C(diam(E), k0)

+n−n/2

[K]∑

k=k0

c2knc′(2−k)n−Q

≤ n−n/2(k0 − [K ′])C(diam(E), k0)

+cc′n−n/2

1− 2−Q2KQ

≤ CeQr,

where after the first inequality we only sum over those Whitneycubes that are contained in G, the third and fourth inequalitiesfollow from Lemmas 3.1 and 3.3, respectively, and the last in-equality holds for sufficiently large r by the definition of K andK ′. Note that by adapting the constants, we may assume thatLemma 3.3 holds for all 0 < r < 2−k0.

Let ϕ : [0,∞) → [0,∞) be a continuous strictly increasing func-tion with ϕ(0) = 0. A domain G ⊂ R

n is said to be ϕ−uniform


[V2] if

(4.17) kG(x, y) ≤ ϕ(|x− y|/mind(x, ∂G), d(y, ∂G))for all x, y ∈ G. In particular, the domain G is C−uniform ifϕ(t) = C log(1 + t) with C > 1. A domain is called uniform ifit is a C−uniform domain for some C > 1. These domains havebeen studied by many authors, see [GO, KLSV, M, Va3, V2].

Lemma 4.18. Let ∅ 6= E ⊂ Rn be an (α, r0)-uniformly porous

compact set such that G = Rn \E is a ψ-uniform domain and let

x ∈ G. Then there exists r1 = r1(x) > 0 such that

d(w,Bk(x, r)) ≤diam(∂G) + d(x) + r0

αψ−1(r)

for all w ∈ ∂G and r ≥ r1.

Proof. Let w ∈ ∂G. Since ∂G is uniformly (α, r0)-porous, foreach s ∈ (0, r0] there exists y ∈ G∩Bn(w, s) with d(y) > αs. Byψ-uniformity of G,

kG(x, y) ≤ ψ

( |x− y|mind(x), d(y)

)≤ ψ

(diam(∂G) + d(x) + r0

mind(x), αs

),

which implies

d

(w,Bk

(x, ψ

(diam(∂G) + d(x, ∂G) + r0

mind(x), αs

)))≤ s .

Substituting s = (diam(∂G) + d(x) + r0)/(αψ−1(r)) this reads

d(w,Bk(x, r)) ≤diam(∂G) + d(x) + r0

αψ−1(r)

for r ≥ ψ ((diam(∂G) + d(x) + r0) /mind(x), αr0) =: r1.

Theorem 4.19. Let E ⊂ Rn be a closed Q−regular set with

0 < Q < n such that G = Rn \ E is a ψ−uniform domain. Then

for each x ∈ D there is c > 0 and r1 > 0 such that

volk(Bk(x, r)) ≥ c(ψ−1(r))Q ,

for all r > r1.

Proof. Let α, r0 > 0 be such that ∂G is (α, r0)-uniformly porousand set C = (diam(∂G) + d(x) + r0)/(α

√n). Furthermore, let K

be the largest integer such that

2−K ≥ C

ψ−1(r).

Then, by combining Lemma 4.18 and (2.5), we see that for m ≤K, all the mth-generation Whitney cubes belong to Bk(x, r). To-gether with Corollary 3.6 and 4.15 this yields for large values of


r the required estimate

volk(Bk(x, r)) ≥ c0NK ≥ c12KQ ≥ c(ψ−1(r))Q ,

where c > 0 only depends on the Q-regularity data and d(x).

By combining Theorem 4.16, Theorem 4.19 and the definitionof uniform domain we get the following corollary.

Corollary 4.20. Let E ⊂ Rn be a compact Q−regular set with

0 < Q < n such that G = Rn \ E is a uniform domain with the

uniformity constant L > 1. Then for each x ∈ G and sufficientlylarge r > 0,

ceQr/L ≤ volk(Bk(x, r)) ≤ CeQr,

where C < ∞ only depends on the Q-regularity data and L andc > 0 depends only on the Q-regularity data, L, and d(x).

Example 4.21. Let E ⊂ Rn be a self-similar set whose comple-

ment is a uniform domain. For instance, E can be a Cantor seton a hyperplane, the 1

4-Cantor set in the plane, or more generally

any self-similar set satisfying the strong separation condition (Seee.g. [F] for the definitions). Then E is Q-regular in its dimensionand Corollary 4.20 can be applied.

Remark 4.22. Although we stated the Theorem 4.19 for un-bounded domains, essentially the same proof works for all ψ-uniform domains G ⊂ R

n whose boundary is Q-regular and uni-formly porous in G. The last assumption means that that theporosity holes in the definition of the uniform porosity lie com-pletely inside G. For instance, in R

2, ∂G can be the Von-Kochsnowflake curve. More generally, the boundary could be the unionof a finite number of such Qi-regular pieces, and in this case wecould apply the lower bound with the exponent rminQi/L andthe upper bound with the exponent rmaxQi.

It is not known what is the best possible lower bound in Corol-lary 4.20. It is an interesting open problem if the exponent Qr/Lcould be replaced by cr for some uniform constant c > 0 inde-pendent of L.

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(Riku Klén, Matti Vuorinen, Xiaohui Zhang) Department of Mathe-

matics and Statistics, University of Turku, 20014 Turku, Finland

E-mail address: [email protected], [email protected], [email protected]

http://arxiv.org/abs/0812.4369


(Ville Suomala) Department of Mathematical sciences, P.O Box

3000, FI-90014 University of Oulu, Finland

E-mail address: [email protected]

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Volume growth of quasihyperbolic balls

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