Koskela, P. and K. Wildrick (2008) “Exceptional Sets for Quasiconformal Mappings in General Metric Spaces,”
International Mathematics Research Notices, Vol. 2008, Article ID rnn020, 32 pages.
doi:10.1093/imrn/rnn020
Exceptional Sets for Quasiconformal Mappings in General
Metric Spaces
P. Koskela and K. Wildrick
Correspondence to be sent to: kewildri@jyu.fi
A theorem of Balogh, Koskela, and Rogovin states that in Ahlfors Q-regular metric spaces
which support a p-Poincaré inequality, 1 ≤ p ≤ Q, an exceptional set of σ -finite (Q − p)dimensional Hausdorff measure can be taken in the definition of a quasiconformal
mapping while retaining Sobolev regularity analogous to that of the Euclidean setting.
Through examples, we show that the assumption of a Poincaré inequality cannot be
removed.
In memoriam: Juha Heinonen (1960–2007)
1 Introduction
Classically, a homeomorphism f : → of domains in Rn , n ≥ 2, is said to be quasi1,n
(, ), and there is a constant K ≥ 1 such that D f(x)n ≤ K J f (x)
conformal if f ∈ Wloc
for almost every x ∈ . In particular, f is absolutely continuous on n-almost every path
in .
For a homeomorphism f : X → Y of metric spaces, we define for all x ∈ X and
r>0
L f (x, r) := sup{dY ( f(x), f(y)) : d X (x, y) ≤ r},
l f (x, r) := inf{dY ( f(x), f(y)) : d X (x, y) ≥ r},
L f (x, r)
L f (x, r)
H f (x) := lim sup
and h f (x, r) := lim inf
.
r→0
l f (x, r)
l f (x, r)
r→0
Received November 29, 2007; Revised February 4, 2008; Accepted February 7, 2008
Communicated by Prof. Stanislav Smirnov
C The Author 2008. Published by Oxford University Press. All rights reserved. For permissions,
please e-mail: journals.permissions@oxfordjournals.org.
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University of Jyväskylä
2 P. Koskela and K. Wildrick
In the 1960’s, Gehring [3, 4] and Väisälä [16] gave the following metric characterization of quasiconformality in Euclidean space.
Theorem 1.1 ([3]). An orientation preserving homeomorphism f : → of domains in
Rn , n ≥ 2, is quasiconformal if and only if there is a constant H ≥ 1 and a set E ⊆ of
σ -finite (n − 1)-dimensional Hausdorff measure such that H f (x) < ∞ for all x ∈ \E and
H f (x) ≤ H for almost every point x ∈ .
Theorem 1.1 led to the definition of a quasiconformal mapping on Carnot groups
spaces is quasiconformal if there is a constant H ≥ 1 such that H f (x) ≤ H for all x ∈ X.
One fact that makes the study of such mappings a viable and rich field is that in sufficiently nice metric spaces, they possess regularity similar to that of quasiconformal
mappings in Euclidean space. The following theorem, which is now well known, can be
deduced from [7, 9].
Theorem 1.2 ([6, 9]). A quasiconformal homeomorphism between bounded, Ahlfors Q1,Q
and hence
regular metric spaces which support a Q-Poincaré inequality, Q > 1, is in Wloc
absolutely continuous on Q-almost every curve.
For related results in the group setting, see [12–14], and [10]. The severity of the
assumptions on the mapping in Theorem 1.2 can be reduced, assuming the ambient space
is Euclidean [6, 8]. Balogh, Koskela, and Rogovin generalized these results to a metric
setting where no Poincaré inequality is assumed [1].
Theorem 1.3 ([1]). Let (X, d X , μ) be a proper, locally Ahlfors Q-regular metric measure
space, Q > 1, and suppose that E ⊆ X has σ -finite (Q − p)-dimensional Hausdorff measure for some 1 ≤ p < Q. Let f : X → Y be a homeomorphism to a metric measure space
(Y, dY , ν) such that Y is proper and locally Ahlfors Q-regular off f(E). If there is a constant
H < ∞ with h f (x) < ∞ for all x ∈ X\E and h f (x) ≤ H for μ-almost every point x ∈ X, then
1, p
f ∈ Wloc (X; Y). In particular, f is absolutely continuous on p-almost every rectifiable
path in X.
Theorem 1.3 generalizes Theorem 1.2 in three major ways. First, it demands a
bound only on h f , rather than on H f . Second, it allows for an exceptional set. Third, it
does not require that the metric spaces support a Poincaré inequality. The price to pay
for this is a reduction in the regularity achieved. Specifically, the regularity provided by
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and more general metric spaces. According to [7], a homeomorphism f : X → Y of metric
Exceptional Sets for Quasiconformal Mappings 3
1, p
1,Q
Theorem 1.3 is Wloc (X; Y) rather than Wloc
(X; Y). This is due to combination of the second
and third generalizations listed above, as Theorem 1.2 and the following theorem from
[1] show.
Theorem 1.4 ([1]). Assume the hypotheses of Theorem 1.3, and further assume that X
1,Q
(X; Y).
supports a p-Poincaré inequality. Then f ∈ Wloc
In the statements of Theorems 1.3 and 1.4 in [1], the target space Y was assumed
to be locally Ahlfors Q-regular rather than locally Ahlfors Q-regular off f(E), but the
the precise definitions to Section 2.
Our main result shows that the loss of regularity in Theorem 1.3 is unavoidable
in the general metric setting, and is not an artifact of the proof of Theorem 1.3.
Theorem 1.5. For all α > 0 and > 0, there is a homeomorphism f : X → Y of metric
measure spaces and a set E ⊆ X such that
(i)
X is compact, quasiconvex, and Ahlfors 2-regular,
(ii)
Y is compact and locally Ahlfors 2-regular off f(E),
(iii) dim H (E) ≤ α, and 0 < Hdim H (E) (E) < ∞,
(iv)
(v)
H f (x) = 1 for all x ∈ X\E,
1,q
f∈
/ Wloc (X; Y) for some q < 2 − dim H (E) + .
The conclusions (i)–(iv) above fulfill the hypotheses of Theorem 1.3, so we see that
1,2−dim H (E)
(X; Y). By Theorem 1.4, if X supported a 2 − dim H (E)f as above is in the space Wloc
1,2
(X; Y). Our result shows
Poincaré inequality, then f would have to be in the space Wloc
that this cannot be the case. We suspect that X supports a p-Poincaré inequality for
some p < 2 − dim H (E) + , but due to the technical nature of the construction, we are
unable to provide a concise proof of this. It would also be interesting to have an example
fulfilling the requirements of Theorem 1.5 where Y is globally Ahlfors 2-regular. In this
paper, we focus on dimension 2 only for simplicity—similar constructions can be made
in any integral dimension greater than 1.
Our construction is quite concrete and is in the spirit of the following classical example, which shows that the size of the exceptional set in Theorem 1.1 cannot
be increased. Let = (0, 1) × R, let C be a regular Cantor set in [0, 1] of dimension 0 < < 1, and let c: [0, 1] → [0, 1] be the corresponding Cantor function. Define
f : → by f(x, y) = (x, y + c(x)). Then H f = 1 except on C × R, which is a set of σ -finite
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proofs given therein provide the slightly more general versions stated above. We defer
4 P. Koskela and K. Wildrick
(1 + )-dimensional Hausdorff measure. However, f is not absolutely continuous on any
horizontal line traversing . The family of such lines has positive 2-modulus, and so f
1,2
(, ) and therefore does not satisfy the analytic definition of quasiconforis not in Wloc
mality.
2 Notation, Definitions, and Basic Facts
Throughout this section, let (X, d X , μ) and (Y, dY , ν) be metric measure spaces. The concepts we will introduce are fairly standard. A more complete discussion can be found in
Given a point x ∈ X and a radius r > 0, we employ the following notation for
balls:
B(X,d) (x, r) = {y ∈ X : d(x, y) < r} and
B (X,d) (x, r) = {y ∈ X : d(x, y) ≤ r}.
A metric space is said to be proper if every closed and bounded set is compact.
Where it will not cause confusion, we will replace B(X,d) (x, r) by B X (x, r), Bd (x, r),
or B(x, r). A similar convention will be made for any other objects, which depend on the
ambient metric space. If τ > 0, and B = B(x, r) is a ball, then we set τ B = B(x, τr). For
> 0 and E ⊆ X, we denote
N (E) =
B(x, )
and
N (E) =
x∈E
B(x, ).
x∈E
A homeomorphism f : X → Y is called an s-similarity, s > 0, if for all x, y ∈ X,
dY ( f(x), f(y)) = sd X (x, y).
If there is a constant L ≥ 1 such that for all x, y ∈ X,
d X (x, y)
≤ dY ( f(x), f(y)) ≤ Ld X (x, y),
L
(2.1)
then f is called bi-Lipschitz. If only the second inequality in (2.1) is assumed to hold,
then f is called Lipschitz.
We denote the length of an interval I ⊆ R by |I |. We define a path in X to be
a continuous, nonconstant map γ : I → X where I ⊆ R is a compact interval. A path
γ : I → X is called rectifiable if it is of finite length. The space X is called quasiconvex if
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[5, 7].
Exceptional Sets for Quasiconformal Mappings 5
there is a constant L ≥ 1 such that for each pair of points x, y ∈ X, there is a path in X
connecting x to y of length no greater than Ld(x, y).
Any rectifiable path γ : [a, b] → X has a unique parameterization γs : [0, length(γ )]
→ X such that for all t ∈ [a, b],
γ (t) = γs (length(γ |[a,t] )).
The path γs is called the arc length parameterization of γ , and it is 1-Lipschitz.
integral of ρ over γ by
γ
ρ ds :=
ρ ◦ γs (t) dt.
[0,length(γ )]
Given a collection of paths ⊆ X, we define the p-modulus of , p > 1, by
mod p(
) = inf
ρ p dμ,
X
where the infimum is taken over all Borel functions ρ : X → [0, ∞] such that for all locally
rectifiable paths γ ∈ ,
γ
ρ ds ≥ 1.
Such a function ρ is said to be admissible for the path family . A condition is said to be
true on p-almost every path in X if the collection of paths in X where the condition does
not hold has p-modulus 0.
Given an open set U ⊆ X and a mapping f : U → Y, we say that a Borel function
ρ : U → [0, ∞] is an upper gradient of f in U if, for each rectifiable path γ : [0, 1] → U , we
have
dY ( f(γ (0)), f(γ (1))) ≤
γ
ρ ds.
1, p
Let f : X → Y be a continuous map. Then f is in the Sobolev space Wloc (X; Y),
1 ≤ p ≤ ∞, if for each relatively compact open subset U ⊆ X, the map f has an upper
gradient g ∈ L p(U ) in U , and there is a point x0 ∈ U such that u(x) := dY ( f(x0 ), f(x)) ∈ L p(U ).
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Given a Borel function ρ : X → [0, ∞] and a rectifiable path γ in X, we define the
6 P. Koskela and K. Wildrick
A continuous mapping f : X → Y is said to be absolutely continuous on a rectifiable path γ in X if the map f ◦ γs : [0, length(γ )] → Y is absolutely continuous in the usual
sense. As in the Euclidean setting, Sobolev maps of metric spaces (which are defined to
be continuous) have absolute continuity properties [15, Proposition 3.1].
1, p
Theorem 2.1 ([15]). If X is proper, then each f ∈ Wloc (X; Y) is absolutely continuous on
p-almost every rectifiable path in X.
A concept closely related to modulus is capacity. Given disjoint, closed subsets
paths in U , which connect E to F . The p-capacity, 1 ≤ p < ∞, of the condenser (E, F ; U )
is defined by
cap p(E, F ; U ) = inf
ρ p dμ,
U
where the infimum is taken over all upper gradients ρ of functions u : U → R such that
u| E ≤ 0 and u| F ≥ 1. If we also require that u is Lipschitz, the resulting quantity is called
the Lipschitz capacity and denoted by cap Lp(E, F ; U ).
Remark 2.2. Let u be a real-valued function on a metric space X, and set u = min{u, 1}.
Then for all x, y ∈ X,
|
u(x) − u(y)| ≤ |u(x) − u(y)|.
(2.2)
Hence any upper gradient ρ of u is also an upper gradient of u. Moreover, inequality
(2.2) implies that u is Lipschitz if u is Lipschitz. Thus, in the definitions of capacity
and Lipschitz capacity of a condenser (E, F ; U ), it suffices to only consider functions
u : U → R such that u| E ≤ 0 and u| F = 1.
Finding a nontrivial lower bound for the modulus of a path family is frequently
difficult. The following theorem [7, Proposition 2.17] provides a tool for doing so.
Theorem 2.3 ([7]). Let (X, d, μ) be a compact and quasiconvex metric measure space,
and let E, F be disjoint continua in X. Then for all q > 0,
capqL (E, F ; X) ≤ modq (E, F ; X).
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E and F of an open set U in X, we define the condenser (E, F ; U ) to be the collection of all
Exceptional Sets for Quasiconformal Mappings 7
Any metric space (X, d) carries a natural family of measures. For any Q ≥ 0, we
define the Q-dimensional Hausdorff measure of a subset E ⊆ X by
Q
Q,
(E) := lim H(X,d)
(E),
H(X,d)
→0
Q,
where H(X,d)
(E) is the Carathéodory premeasure defined as follows. Let B be the collec-
tion of all covers of E by closed sets in X of diameter no greater than . Then
C∈B
(diam B) Q .
B∈C
The Hausdorff dimension of a metric space (X, d) is defined by
dim H (X) := sup{Q ≥ 0 : H Q (X) > 0}.
For a full description of Hausdorff measure and the Carathéodory construction, see [2,
Chapter 2.10]. Note that our definition differs from that in literature as we do not include
a dimensional normalization constant.
The Q-dimensional Hausdorff content of a subset E ⊆ X is given by
Q
(E) = inf
H∞
(diam B) Q ,
B∈C
where the infimum is now taken over all covers C of E by closed sets in X.
Remark 2.4. It is an easy exercise to show that if [a, b] ⊆ R is a nondegenerate interval,
Q
([b − a]) = (b − a) Q .
then for all Q ≥ 0 we have H∞
Remark 2.5. If (Y, d) is a metric space, and E ⊆ X ⊆ Y, then
Q
Q
Q
H(Y,d)
(E) ≤ H(X,d)
(E) ≤ 2 Q H(Y,d)
(E),
and a similar statement holds for Hausdorff content.
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Q,
(E) := inf
H(X,d)
8 P. Koskela and K. Wildrick
The metric measure space (X, d, μ) is called Ahlfors Q-regular, Q ≥ 0, if there
exists a constant K ≥ 1 such that for all a ∈ X and 0 < r ≤ diam X, we have
rQ
≤ μ(B d (a, r)) ≤ Kr Q .
K
(2.3)
Remark 2.6. If (X, d, μ) is Ahlfors Q-regular with constant K, and diam X ≤ r ≤
C diam X for some C ≥ 1, then (2.3) remains valid if K is replaced by C Q K.
say that a metric space (X, d) is Ahlfors Q-regular if the metric measure space (X, d, H Q )
is Ahlfors Q-regular. In the sequel, we will deal only with metric measure spaces where
the measure is H Q for some Q ≥ 0, eliminating this confusion.
Remark 2.8. If a metric space (X, d) is Ahlfors Q-regular, then its completion
is as well, with a constant depending only on the original constant and Q. See
[17, Proposition 2.10].
The metric measure space (X, d, μ) is locally Ahlfors Q-regular if for every compact subset V ⊆ X, there is a constant K ≥ 1 and a radius r0 > 0 such that for each point
a ∈ V and radius 0 < r ≤ r0 , the inequalities in (2.3) are satisfied.
The following definition is perhaps nonstandard. Let E ⊆ X. We say that (X, d, μ)
is locally Ahlfors Q-regular off of E if there is a constant K ≥ 1 such that for each point
a ∈ X\E, there is a radius ra > 0 such that for each 0 < r ≤ ra , the inequalities in (2.3) are
satisfied.
The space (X, d, μ) is said to support a p-Poincaré inequality, 1 ≤ p < ∞ if there
are constants C , τ ≥ 1 such that if B is a ball in X, u : τ B → R is a bounded continuous
function, and ρ is an upper gradient of u, then
1/q
q
− |u − u B | dμ ≤ C diam(B) − ρ dμ
.
B
τB
Note that if (X, d, μ) supports a p-Poincaré inequality, 1 ≤ p < ∞, then it also supports
a q-Poincaré inequality for all q ≥ p. The Poincaré inequality can be thought of as a
requirement that a space contains “many” curves. See [5, 7] for more information.
Many of the properties we have described above are invariant under similarities
and bi-Lipschitz maps. We leave the proof of the following proposition to the reader.
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Remark 2.7. If (X, d, μ) is Ahlfors Q-regular, then (X, d, H Q ) is so as well. Thus we may
Exceptional Sets for Quasiconformal Mappings 9
Proposition 2.9. Let (X, d X , H Q ) and (Y, dY , H Q ) be metric measure spaces, and suppose
that f : X → Y is either a similarity or a bi-Lipschitz homeomorphism. Then
(i) if X is quasiconvex, then so is Y,
(ii) if X is Ahlfors Q-regular, then so is Y,
(iii) if X supports a p-Poincaré inequality, then so does Y.
If f is a similarity, then the constant associated with each condition on Y is the same as
the constant associated with that condition on X. If f is bi-Lipschitz, then the constant
associated with each condition on Y depends only on the constant associated with that
Metric measure spaces, which are Ahlfors Q-regular and support a p-Poincaré inequality, p ≤ Q, enjoy several important geometric properties [11]. For example, they are
quasiconvex with constant depending only on the constants associated with the Ahlfors
regularity condition and the Poincaré inequality. Such spaces are also nice analytically.
The following theorem, adapted to our needs from the more general [7, Theorem 5.9],
shows that capacity type estimates are available.
Theorem 2.10 ([7]). Let (X, d, H Q ) be a bounded Ahlfors Q-regular metric measure space,
which supports a p-Poincaré inequality for some 1 ≤ p ≤ Q. Let E and F be compact
subsets of X, and suppose that there are constants Q − p < s ≤ Q and λ > 0 such that
s
s
(E), H∞
(F ) ≥ λ(diam X)s .
min H∞
Then there is a constant C ≥ 1, depending only on s, λ, and the data associated with X,
such that
ρ p dH Q ≥ C −1 (diam X) Q− p,
X
whenever u is a continuous function on X with u| E ≤ a and u| F ≥ b, where b − a ≥ 1/4,
and ρ is an upper gradient of u.
3 Cantor Sets
Fix an integer n ≥ 1. We now discuss the construction of a regular Cantor set C n of
Hausdorff dimension (log3 2)/n. Let U1,1 be an open interval of length 3−n (3n − 2) removed
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condition on X and the bi-Lipschitz constant.
10 P. Koskela and K. Wildrick
Fig. 1..
The first two steps in the construction of a Cantor set.
from the center of [0, 1], and U11 and U12 the remaining closed intervals of length 3−n . Set
For i ≥ 1 an integer, define the index sets
Ji := {1, . . . , 2i−1 } and
Ki := {1, . . . , 2i }.
Assume that Cni , the intervals {Ui, j } j∈Ji , and the intervals {Uik }k∈Ki have been defined.
j
For j ∈ Ji+1 = Ki , define Ui+1, j to be the central open subinterval of Ui of length 3−(i+1)n
k
(3n − 2), and let {Ui+1
}k∈Ki+1 be the collection of remaining closed subintervals of length
3−(i+1)n . See Figure 1.
We may now set
Cni+1 = Cni \
Ui+1, j =
j∈Ji+1
k
Ui+1
.
k∈Ki+1
This inductively defines Cni for all positive integers i. Finally, we set
Cn =
∞
Cni .
i=1
We may assume that for each positive integer i, the intervals {Uik }k∈Ki are ordered so
that the right endpoint of Uik is less than the left endpoint of Uik+1 , and similarly for the
intervals {Ui, j } j∈Ji .
For ease of notation, for each positive integer i, we define the positive real number
wi by
wi =
3−in (3n − 2)
,
2
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Cn1 = [0, 1]\U1,1 = U11 ∪ U12 .
Exceptional Sets for Quasiconformal Mappings 11
which is half the length of any Ui, j where j ∈ Ji . Furthermore, let ui, j be the center of the
open interval Ui, j ; thus
Ui, j = (ui, j − wi , ui, j + wi ).
Similarly, for k ∈ Ki , we set uik to be the center of Uik .
The Cantor set Cn is self-similar in the following sense. For each positive integer
i and each k ∈ Ki , there is a 3−in -similarity φik : Cn → Cn ∩ Uik . Using this, it is not hard to
show that for any positive integer i and k ∈ Ki , we have
(3.1)
This equation also holds for i = 0 under the convention U01 = [0, 1].
Note that the sets Ui, j and Uik and the quantities wi depend implicitly on n. Setting
n = 1 recovers the standard “one-third Cantor set”. We will often refer simultaneously
to Cn and Cm where m < n. In this situation we will denote the intervals removed in the
construction of Cm by
{Vi, j : i ∈ Z+ , j ∈ Ji }
and the remaining intervals by
Vik : i ∈ Z+ , k ∈ Ki .
We will not have need for alternate versions of the quantities wi defined above; they will
always refer to the Cantor set labeled Cn .
Each Cantor set Cn gives rise to a Cantor function cn : [0, 1] → [0, 1], which is the
unique continuous function satisfying
cn (t) =
2j − 1
2i
for t ∈ Ui j .
Remark 3.1. The function cn is not absolutely continuous. To see this, let i be a positive
integer and k ∈ Ki , and let {Iα } ⊆ [0, 1] be a finite collection of (open or closed) intervals
covering Cn ∩ Uik . Denoting the initial and terminal points of Iα by aα and bα , respectively,
we have
α
|cn (bα ) − cn (aα )| ≥ 2−i .
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(log 2)/n H(log3 2)/n Cn ∩ Uik = 2−i = H∞ 3
Cn ∩ Uik .
12 P. Koskela and K. Wildrick
This shows that H1 (cn (Cn ∩ Uik )) ≥ 2−i , but it follows from the construction that H1 (Cn ∩
Uik ) = 0.
We will need a lemma regarding the Hausdorff content of certain sets involving
the Cantor sets described above. Because of how it will be used, we employ the notation
established for Cm .
Lemma 3.2. Let i ≥ 1 be an integer. For each k ∈ Ki , suppose we are given sets E k , E k ⊆ R
Vik ⊆ E k ∪ E k .
Further suppose that there is a subset K ⊆ Ki with card K ≤ (card Ki )/2, such that for all
k∈
/K
(log3 2)/m
(E k ) <
H∞
2−i
.
4
(3.2)
Then
(log 2)/m
H∞ 3
E k
k ∈K
/
≥
1
.
4
Proof. Towards a contradiction, suppose there is a cover {E α }α of
1
(diam E α )(log3 2)/m < .
4
α
By (3.2), for each k ∈
/ K there is a cover {Fβk }β of E k with
(log3 2)/m 2−i
diam Fβk
≤
.
4
β
Then the collection
{E α }α ∪
k ∈K
/
Fβk
β
∪ Vik k∈K
k ∈K
/
E k such that
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such that
Exceptional Sets for Quasiconformal Mappings 13
covers Cm and
(log3 2)/m (log3 2)/m
diam Fβk
diam Vik
(diam E α )(log3 2)/m +
+
α
k ∈K
/
<
β
k∈K
−i
1
2
+ (card Ki )
+ (card K)2−i ≤ 1.
4
4
(log3 2)/m
However, by (3.1) we have H∞
(Cm ) = 1, yielding a contradiction.
In this section we construct the spaces to be used in Theorem 1.5. Fix integers 1 ≤ m < n
such that n/m ∈ Z. We will consider Cn and Cm as defined and notated in the previous
section. All subsets of R2 are endowed with the metric inherited from the standard
2-norm on R2 , which is denoted by · .
For the remainder of the paper, the notation A B means that there is a positive
constant C depending only on n and m such that A ≤ C B. The notation A ≈ B means that
A B and B A.
For each integer i ≥ 1, define new index sets Ji and Ki by
Ji := { j ∈ Ji : Ui, j ∩ [u1,1 , u2,2 ] = ∅},
Ki := k ∈ Ki : Uik ∩ [u1,1 , u2,2 ] = ∅ .
We also define
X1,1 := {(x, y) ∈ [0, 1] × R : 0 ≤ x − u1,1 ≤ w1 − dist(y, Cm )},
X2,2 := {(x, y) ∈ [0, 1] × R : 0 ≤ u2,2 − x ≤ w2 − dist(y, Cm )}.
If i ≥ 3 is an integer and j ∈ Ji , define
Xi, j := {(x, y) ∈ [0, 1] × R : |x − ui, j | ≤ wi − dist(y, Cm )}.
(4.1)
Note that the function y → dist(y, Cm ) is the maximal 1-Lipschitz function on R, which
takes the value 0 at each point of Cm .
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4 The Construction and its Basic Properties
14 P. Koskela and K. Wildrick
At left, X when n = 2 and m = 1. At right, a magnified view of a portion of X
1
showing a single component W2,2
of X2,2 and several components of X3,3 .
We now describe the construction to be used in Theorem 1.5. Set
⎛
X=⎝
⎞
Xi, j ⎠ ∪ ((Cn ∩ [u1,1 , u2,2 ]) × Cm ).
i≥1, j∈Ji
See Figure 2. Note that for each y ∈ Cm , the line segment [u1,1 , u2,2 ] × {y} is contained in
X.
Define f : [0, 1] × R → [0, 1] × R by
f(x, y) = (x, y + cn (x)).
(4.2)
Let Y = f(X), and set E := X ∩ (Cn × Cm ). We make X and Y into metric measure spaces
by equipping them with the ambient metric from R2 and the measure H2 . By Remark 2.5,
H2X is comparable to the restriction of HR2 2 to X, and thus they are equivalent for our
purposes. A similar statement applies to Y.
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Fig. 2..
Exceptional Sets for Quasiconformal Mappings 15
Proposition 4.1. The mapping f : X → Y is a homeomorphism and satisfies H f (x, y) = 1
for all points (x, y) ∈ X\E.
Proof. It is clear that f is a homeomorphism. If (x, y) ∈ X\E, then there are integers
i ≥ 1 and j ∈ Ji such that x is an interior point of Ui, j . It follows that there is an open
ball B containing (x, y) such that f| B is an isometry. The result follows.
The remainder of this section is devoted to establishing geometric and analytic
properties of X and Y. To do so, we first examine the sets Xi, j . For a given y ∈ R, the
the following lemma will enable us to describe the components of Xi, j .
Lemma 4.2. Let i ≥ 1 be an integer. Then
N wi (Cm ) =
k
,
N wi V(i−1)n/m
(4.3)
k∈K(i−1)n/m
and the union is disjoint.
Proof. It follows from the definitions that
N wi (Cm ) ⊆
k
.
N wi V(i−1)n/m
(4.4)
k∈K(i−1)n/m
k
k
On the other hand, suppose y ∈ N wi (V(i−1)n/m
) for some k ∈ K(i−1)n/m . If y ∈
/ V(i−1)n/m
, then
k
, which are in Cm . If
y is a distance at most wi from one of the endpoints of V(i−1)n/m
k
, we note that either y ∈ Cm or there exists an integer i0 > (i − 1)n/m and an
y ∈ V(i−1)n/m
integer j0 ∈ Ji0 such that y ∈ Vi0 , j0 , the endpoints of which are contained in Cm . We have
(i−1)n
|Vi0 , j0 | ≤ 3−
m
+1 m
(3m − 2) = 3−in 3n−m (3m − 2) ≤ 2wi .
Thus dist(y, C) ≤ wi . This, along with (4.4), shows that (4.3) holds. To see that the union
k
k
and V(i−1)n/m
with k = k are separated by
is disjoint, consider that any two sets V(i−1)n/m
the interval V(i−1)n/m, j for some j ∈ J(i−1)n/m . Disjointness now follows from
|V(i−1)n/m, j | = 3−in 3n (3m − 2) > 2wi ,
completing the proof.
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inequality defining Xi, j has a solution x ∈ [0, 1] if and only if y ∈ N wi (Cm ). Consequently,
16 P. Koskela and K. Wildrick
A component Wi,k j of Xi,k j when i ≥ 3
If points (x1 , y) and (x2 , y) are in some Xi, j , then the horizontal line segment
[x1 , x2 ] × y is also contained in Xi, j . Thus Lemma 4.2 implies that the components of Xi, j
may be indexed by k ∈ K(i−1)n/m ; we denote them by
k
Wi,k j = {(x, y) ∈ [0, 1] × N wi V(i−1)n/m
: |x − ui, j | ≤ wi − dist(y, Cm )},
(4.5)
with the obvious modifications for the components of X1,1 and X2,2 . See Figures 2 and 3.
The following two remarks follow immediately from this description. Let i ≥ 1
be an integer, j ∈ Ji , and k, k ∈ K(i−1)n/m .
Remark 4.3. If (x, y) ∈ Wi,k j and (x , y ) ∈ Wi,k j , where k = k , then
k
k
|y − y| ≥ dist V(i−1)n/m
− 2wi ≥ 3−in (3n+m − 3n+1 + 2) ≥ 2(3−in ).
, V(i−1)n/m
Remark 4.4. If (x, y) ∈ Wi,k j , then there is a point y ∈ Cm such that (x, y ) ∈ Wi,k j and
|y − y | ≤ dist(x, ∂Ui, j ).
Moreover, y may be chosen so that the line segment connecting (x, y) to (x, y ) is contained
in Wi,k j .
The following elementary but useful lemma follows from the fact that in the
construction of Cn , each removed interval is comparable in length to each interval not
removed.
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Fig. 3..
Exceptional Sets for Quasiconformal Mappings 17
Lemma 4.5. Let i > 1 be an integer, j ∈ Ji , and i0 < i. Then there is an index j0 ∈ Ji0 such
that for any x ∈ Ui, j ,
|x − ui0 , j0 | ≤ 2wi0 .
Proof. Since i0 < i and j ∈ Ji , the construction of Cn implies that there is some k0 ∈ Ki0
such that Ui, j ⊆ Uik00 . We may find j0 ∈ Ji0 such that Uik00 is adjacent to Ui0 , j0 . Now
The next proposition describes the self-similarity of the space X. Let
Z = {(x, y) ∈ [0, 1] × R : |x − u1,1 | ≤ w1 − dist(y, Cm )}.
Note that
X1,1 = Z ∩ ([u1,1 , u1,1 + w1 ] × R).
1
. For each k ∈
Proposition 4.6. The set X1,1 is connected and hence equal to W1,1
k
k
: X1,1 → W2,2
. For all integers i ≥ 3, j ∈ Ji , and
Kn/m , there is a 3−n -similarity s2,2
k ∈ K(i−1)n/m , there is a 3−(i−1)n -similarity si,k j : Z → Wi,k j . In this case, si,k j maps X1,1 onto
Wi,k j ∩ ([ui, j , ui, j + wi ] × R).
Proof. The first assertion is clear. The second is proven similarly to the third, which we
now establish. Fix integers i ≥ 3, j ∈ Ji and k ∈ K(i−1)n/m . Define sik : R → R by
sik (y) =
(y − 1/2)
k
+ v(i−1)n/m
.
3(i−1)n
k
. Furthermore,
Then sik maps C bijectively onto C ∩ V(i−1)n/m
dist(y, Cm )
.
dist sik (y), Cm =
3(i−1)n
Set
si,k j (x, y) =
wi
(x − u1,1 ) + ui, j , sik (y) .
w1
Noting that wi /w1 = 3−(i−1)n , (4.1) and (4.5) show that si,k j is a homeomorphism from Z to
Wi,k j ; a simple calculation shows that it is a 3−(i−1)n -similarity.
The final assertion is easily verified.
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|x − ui0 , j0 | ≤ Uik00 + wi0 ≤ 3−i0 n + wi0 ≤ 2wi0 .
18 P. Koskela and K. Wildrick
The following proposition is tedious but elementary to verify. We omit the proof.
Proposition 4.7. Let i ≥ 2 be an integer. There is a constant L ≥ 1, independent of n,
m, and i, such that each of X1,1 , Z , and X1,1 ∩ ([u1,1 , u1,1 + (w1 − wi )]) are L-bi-Lipschitz
equivalent to the unit square [0, 1]2 .
Propositions 2.9, 4.6, and 4.7, along with the fact that the unit square is quasiconvex, Ahlfors 2-regular, and supports a 1-Poincaré inequality, show the following
corollary.
X1,1 ∩ ([u1,1 , u1,1 + (w1 − wi )]),
and
Wi,k j ∩ ([ui, j , ui, j + wi ] × R)
are quasiconvex, Ahlfors 2-regular, and support a 1-Poincaré inequality. Moreover, the
constants associated with each condition are independent of i, j, k, n, and m.
We now establish some global properties of X and Y. It is clear that both spaces
are compact.
Remark 4.9. As the collection {∂Ui, j }i∈Z+ , j∈Ji is dense in Cm ∩ [u1,1 , u2,2 ], we have
X=
Xi, j .
i∈Z+ , j∈Ji
Proposition 4.10. The space X is -quasiconvex where depends only on n and m. Proof. Let (x1 , y1 ) and (x2 , y2 ) be points in X. We wish to show that there is a path γ in X
connecting these points such that
length(γ ) (x1 , y1 ) − (x2 , y2 ).
Remark 4.9 shows that we may assume
(x1 , y1 ), (x2 , y2 ) ∈
i∈Z+ , j∈Ji
Xi, j .
(4.6)
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Corollary 4.8. Let i ≥ 2, j ∈ Ji , and k ∈ K(i−1)n . The sets X1,1 , Wi,k j ,
Exceptional Sets for Quasiconformal Mappings 19
Case 1. Assume that there is some integer i ≥ 1 and j ∈ Ji such that both (x1 , y1 )
and (x2 , y2 ) are in Xi, j . Let (x1 , y1 ) ∈ Wi,k1j and (x2 , y2 ) ∈ Wi,k2j where k1 , k2 ∈ K(i−1)n/m .
If |y1 − y2 | < 2(3−in ), then Remark 4.3 implies that k1 = k2 , and the desired path
connecting (x1 , y1 ) to (x2 , y2 ) exists by Corollary 4.8. Thus we may assume that there is
some integer i0 < i such that
2 3−(i0 +1)n ≤ |y1 − y2 | < 2(3−i0 n ),
(4.7)
an index j0 ∈ Ji0 such that
max{|x1 − ui0 , j0 |, |x2 − ui0 , j0 |} ≤ 2wi0 .
Let β1 be a path parameterizing the horizontal line segment from (x1 , y1 ) to (ui0 , j0 , y1 ), and
let β2 be a path parameterizing the horizontal line segment from (ui0 , j0 , y2 ) to (x2 , y2 ). Since
y1 , y2 ∈ Cm , these paths are in X. The second inequality in (4.7) along with Remark 4.3
implies that there is some k0 ∈ K(i0 −1)n/m such that (ui0 , j0 , y1 ) and (ui0 , j0 , y2 ) are in Wik00, j0 . By
Corollary 4.8, there is a path γ0 connecting these points with
length(γ0 ) |y1 − y2 |.
Thus the concantenation γ1 = β1 · γ0 · β2 is a path in X connecting (x1 , y1 ) to (x2 , y2 ) with
length(γ1 ) |y1 − y2 | + 4wi0 |y1 − y2 | (x1 , y1 ) − (x2 , y2 ).
(4.8)
We now remove the assumption that y1 , y2 ∈ Cm . By Remark 4.4, there is a point
y1
∈ Cm and a path α1 in Wi,k1j parameterizing the line segment from (x1 , y1 ) to (x1 , y1 ) with
length(α1 ) ≤ wi . Similarly, there is a point y2 ∈ Cm and a path α2 in Wi,k2j parameterizing
the line segment from (x2 , y2 ) to (x2 , y2 ) with length(α2 ) ≤ wi . From the first inequality in
(4.7), we see that wi |y1 − y2 |. Thus
(x1 , y1 ) − (x2 , y2 ) ≤ (x1 , y1 ) − (x2 , y2 ) + 2wi (x1 , y1 ) − (x2 , y2 ).
By the discussion leading to (4.8), there is a path γ1 connecting (x1 , y1 ) to (x2 , y2 ) with
length(γ1 ) (x1 , y1 ) − (x2 , y2 ) (x1 , y1 ) − (x2 , y2 ).
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Assume for the moment that y1 and y2 are points in Cm . By Lemma 4.5, there is
20 P. Koskela and K. Wildrick
Thus γ = α1 · γ1 · α2 is a path in X connecting (x1 , y1 ) to (x2 , y2 ) with
length(γ ) wi + (x1 , y1 ) − (x2 , y2 ) + wi (x1 , y1 ) − (x2 , y2 ).
Case 2. Now assume that there are integers i1 , i2 ≥ 1, j1 ∈ Ji1 , j2 ∈ Ji2 , k1 ∈
K(i1 −1)n/m , and k2 ∈ K(i2 −1)n/m , such that (x1 , y1 ) ∈ Wik11, j1 and (x2 , y2 ) ∈ Wik22, j2 , where Xi1 , j1 =
Xi2 , j2 . This implies that
Thus by Remark 4.4, we may find a point y1 ∈ Cm such that (x1 , y1 ) ∈ Wik11, j1 , and there is a
path α in X parameterizing the line segment connecting (x1 , y1 ) to (x1 , y1 ) with
length(α) = |y1 − y1 | ≤ dist(x1 , ∂Ui1 , j1 ) ≤ |x1 − x2 |.
(4.9)
Since y1 ∈ Cm , there is a path β in X parameterizing the line segment connecting (x1 , y1 )
to (x2 , y1 ) with
length(β) = |x1 − x2 |.
Consider that by (4.9),
(x2 , y1 ) − (x2 , y2 ) ≤ |y1 − y1 | + |y1 − y2 | (x1 , y1 ) − (x2 , y2 ).
Noting that (x2 , y1 ) and (x2 , y2 ) are both in Xi2 , j2 , Case 1 above provides a path γ connecting
them with
length(γ ) (x1 , y1 ) − (x2 , y2 ).
Now α · β · γ is a path in X connecting (x1 , y1 ) to (x2 , y2 ) with
length(α · β · γ ) (x1 , y1 ) − (x2 , y2 ),
as desired.
These cases exhaust all possibilities, completing the proof.
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dist(x1 , ∂Ui1 , j1 ) ≤ |x1 − x2 |.
Exceptional Sets for Quasiconformal Mappings 21
Proposition 4.11. The space X is Ahlfors 2-regular, with constant depending only on n
and m.
Proof. Let (x, y) ∈ X, and r ≤ diam(X) 1. By Remarks 2.8 and 4.9, we may assume that
there are indices i ∈ Z, j ∈ Ji , and k ∈ K(i−1)n/m such that (x, y) ∈ Wi,k j .
Since X is endowed with the ambient metric from R2 , it follows from Remark
2.5 and the Ahlfors 2-regularity of the plane that there is a constant κ ≥ 1, not even
depending on n, such that
Thus it suffices to show the corresponding lower bound.
If r ≤ 6(3n )wi , then by Proposition 4.6 we see that r diam Wi,k j . Hence by Corollary 4.8 and Remark 2.6, there is a constant K depending only on n such that
H2X (B X ((x, y), r)) r 2
(4.10)
If 6(3n )wi ≤ r ≤ diam X, then we may find an integer 1 ≤ i0 < i such that
6wi0 ≤ r ≤ 6(3n )wi0 .
Note that 6(3n )w1 ≥ 6 ≥ diam X, so this case is not possible for i = 1. Combining Remark
4.4 and Lemma 4.5, we may find indices j0 ∈ Ji0 and k0 ∈ K(i0 −1)n/m and a point (x , y ) ∈
Wik00, j0 and
(x, y) − (x , y ) ≤ 3wi0 ≤ r/2.
Thus
B X ((x, y), r) ⊇ B X ((x , y ), r/2).
and hence the discussion leading to (4.10) implies
H2X (B X ((x, y), r)) ≥ H2X (B X ((x , y ), r/2)) r 2 .
Thus X is Ahlfors 2-regular with constant depending only on n and m.
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H2X (B X ((x, y), r)) ≤ κr 2 .
22 P. Koskela and K. Wildrick
Proposition 4.12. The space Y is locally Ahlfors 2-regular off f(E).
Proof. This follows from Corollary 4.8 and the fact that f is an isometry when restricted
to the interior of each Xi, j .
5 The Proof of Theorem 1.5
For integers 1 ≤ m < n with n/m ∈ Z, let X, E, Y, and f be as defined in Section 4. For
R2 and (x, y) ∈ X, denote by (x, y) L the reflection of the point (x, y) across the line L. Given
vertical lines L and L that intersect the x-axis in the points l and l , respectively, set
[L 1 , L 2 ] = X ∩ ([l, l ] × R).
Let L 0 be the vertical line {u1,1 } × R and R0 be the vertical line {u2,2 } × R. Define
the path families
= {γ : [0, 1] → X : γ (0) ∈ L 0 , γ (1) ∈ R0 },
0 = {γ ∈ : f ◦ γs is absolutely continuous}.
We now show that the curve family 0 may be ignored in computing the modulus of .
If γ : I → R2 is any path, we denote the components of γ by γ x and γ y, so that for
any t ∈ I ,
γ (t) = (γ x (t), γ y(t)).
Lemma 5.1. If γ ∈ 0 , then H1 (((γs )x )−1 (Cn )) > 0.
Proof. It suffices to show that if H1 (((γs )x )−1 (Cn )) = 0, then f ◦ γs is not absolutely continuous. Set F = ((γs )x )−1 (Cn ), and let 0 < δ < 1/8. By assumption we may find finitely many
intervals {(aα , bα )} ⊆ [0, length(γ )] which cover F and satisfy
α
|bα − aα | < δ.
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the remainder of the paper we will use the following notation. If L is a vertical line in
Exceptional Sets for Quasiconformal Mappings 23
Since γs is continuous and connects L 0 to R0 , we have that
γsx (aα , bα ) ⊇ Cn ∩ U23 .
α
The fact that the 1-norm is equivalent to the 2-norm on R2 , the definition of f, and the
triangle inequality show that there is a universal constant c > 0 such that
f ◦ γs (bα ) − f ◦ γs (aα ) ≥ c
s
s
s
s
α
y
From Remark 3.1 and the fact that γs is 1-Lipschitz, we may conclude that
f ◦ γs (bα ) − f ◦ γs (aα ) ≥ c (1/4 − δ) ≥ c/8.
α
Since δ can be made arbitrarily small, this shows that f ◦ γs is not absolutely
continuous.
Lemma 5.2. Let q > 0. Then modq (
) = modq (
\
0 ).
Proof. It follows from the definitions that modq (
\
0 ) ≤ modq (
). Thus it suffices to
show that modq (
\
0 ) ≥ modq (
). Let ρ : X → [0, ∞] be admissible for \
0 , and define
ρ : X → [0, ∞] by
ρ (x, y ) =
∞
x ∈ Cn
ρ(x, y )
x∈
/ Cn .
Then ρ is a Borel function, and it follows from Lemma 5.1 that ρ is admissible for .
Since H2 ((Cn × R) ∩ X) = 0, we have
q
ρ dH =
X
ρ q dH2 .
2
X
Taking the infimum over all functions ρ that are admissible for \
0 yields the desired
result.
Proof of Theorem 1.5. We begin by determining the correct parameters n and m to use
in the construction, and setting the value of q in condition (v).
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α
cn ◦ γ x (bα ) − cn ◦ γ x (aα ) − γ y(bα ) − γ y(aα ).
24 P. Koskela and K. Wildrick
Let α > 0 and > 0 be given. Fix a positive integer m such that (2 log3 2)/m ≤
min{α, }. For any positive integer n, we define
1
1
+
q(n) = 2 − (log3 2)
n m
+ n−1/2
and
δ(n) =
q(n)(q(n) − 1) log3 2
.
n + (q(n) − 1 − n) log3 2
(5.1)
Note that if n−1/2 ≤ log3 2/m, then
1 < q(n) < 2 and
0 < δ(n) <
2 log3 2
.
n(1 − log3 2)
(5.2)
log3 2/m and the following inequalities are satisfied:
(log3 2)
1
1
+
n m
((n/m) − 1) log3 2
≤ n−1/2 ,
n − log3 2
−
δ(n) +
(5.3)
log3 2
≤ n−1/2 .
n
(5.4)
This is possible because for sufficiently large n, the quantities on the left-hand sides
of (5.3) and (5.4) are bounded above by a linear function of 1/n. Set q = q(n), δ = δ(n).
Inequalities (5.3) and (5.4) now imply the following:
1<
2−
log3 2
m
n − log3 2
n − log3 2
0<δ≤
≤q
(5.5)
log3 2
− (2 − q).
m
(5.6)
Let X, Y, E, and f be as defined in Section 4 with parameters n and m as above.
Then, as (2 log3 2)/m ≤ α and n > m, we have
dim H (E) = (log3 2)
1
1
+
n m
≤α
and
0 < Hdim H (E) (E) < ∞.
These facts, along with Propositions 4.1, 4.10, 4.11, and 4.12, provide the information
needed to show that X, Y, f, and E satisfy conditions (i)–(iv) of Theorem 1.5. As n−1/2 ≤
1,q
/ Wloc (X; Y ).
log3 2/m < , we see that q < 2 − dim H (E) + . It remains to show that f ∈
Since f is not absolutely continuous on any path in the family \
0 defined above, by
Theorem 2.1 we need only show that modq (
\
0 ) > 0.
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We now fix an integer n > m that is a multiple of m, and so large that n−1/2 ≤
Exceptional Sets for Quasiconformal Mappings 25
The first few lines.
By Theorem 2.3 and Lemma 5.2, we have
capqL (L 0 , R0 ; X) ≤ modq (
) = modq (
\
0 ).
Thus it suffices to show that capqL (L 0 , R0 ; X) > 0. As per Remark 2.2, let v : X → R be
a c-Lipschitz function, c ≥ 1, with v| L 0 ≤ 0 and v| R0 = 1. Let ρ : X → [0, ∞] be an upper
gradient of v in X. We will show that
ρ q dH2 κ > 0,
(5.7)
X
where κ > 0 depends only on n and m. The basic idea is to use the symmetry and selfsimilarity of the space X to keep track of where the function v grows.
We now inductively define some sequences of lines and functions. Set C 0 = R0 ,
v0 = v, and ρ0 = ρ. Let L 1 be the vertical line through the point (u1,1 + (w1 − w2 ), 0). Notice
that
u3,3 =
u1,1 + (w1 − w2 ) + u2,2
.
2
Let C 1 be the vertical line through (u3,3 , 0). Then the reflection of the line L 1 through the
line C 1 is C 0 . See Figure 4.
Define v1 : [L 1 , C 1 ] → R by
v1 (x, y ) = v0 (x, y ) + 1 − v0 ((x, y )C 1 ).
Then v1 is 2c-Lipschitz,
v1 | L 1 = v0 | L 1 + 1 − v0 | R0 = v0 | L 1 ,
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Fig. 4..
26 P. Koskela and K. Wildrick
and v1 |C 1 = 1. Furthermore, the function ρ1 : [L 1 , C 1 ] → [0, ∞] defined by
ρ1 (x, y ) = ρ0 (x, y ) + ρ0 ((x, y )C 1 )
is an upper gradient of v1 . We have
q
X
ρ0 dH2 ≥
q
[L 0 ,L 1 ]
ρ0 dH2 + 2(1−q)
q
[L 1 ,C 1 ]
ρ1 dH2 .
vertical line through the point (u1,1 + (w1 − wi+1 ), 0), and let C i be the vertical line such
that the reflection of L i in C i is C i−1 . Define vi : [L i , C i ] → R by
vi (x, y ) = vi−1 (x, y ) + 1 − vi−1 ((x, y )C i ).
We see inductively that vi is 2i c Lipschitz and that vi | L i = vi−1 | L i and vi |C i = 1. Furthermore, the function ρi : [L i , C i ] → [0, ∞] defined by
ρi (x, y ) = ρi−1 (x, y ) + ρi−1 ((x, y )C i )
is an upper gradient of vi . Finally, we see by induction that for any positive integer i0 ,
the following inequality holds:
ρ dH ≥
q
X
2
i
0 −1
(1−q)i
2
i=0
[L i ,L i+1 ]
q
ρi
q
dH + 2
2
(1−q)i0
[L i0 ,C i0 ]
ρi0 dH2 .
(5.8)
We will use the first term on the right-hand side of inequality (5.8) to keep track of the
change of v on large pieces of X, and the second term for small pieces.
Let M be the vertical line {u1,1 + w1 } × R. If (x, y ) ∈ X ∩ M, then y ∈ Cm and
dist((x, y ), X ∩ C i ) = 3−(i+2)n + wi+2 =
3−(i+1)n
2
for each integer i ≥ 1. Since vi |C i = 1 and vi is 2i c-Lipschitz, we may find an integer i0 ≥ 2
such that vi0 | M ≥ 3/4. Note that i0 depends on c, and so our final estimates should be
independent of i0 . See Figure 5.
The line C i0 −1 intersects the x-axis at the point ui0 +1, j0 for where j0 = 2i0 −1 + 1 ∈
Ji0 +1 . By the symmetry of [L i0 , C i0 −1 ] about the vertical line C i0 , and the symmetry of
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Now, let i ≥ 2 and assume that L i−1 , C i−1 , vi−1 , and ρi−1 are defined. Let L i be the
Exceptional Sets for Quasiconformal Mappings 27
After choosing i0 .
Xi0 +1, j0 about the line C i0 −1 , we see that there is an isometry from [L i0 , M] to
Xi0 +1, j0 ∩ ([ui0 +1, j0 , ui0 +1, j0 + wi0 +1 ] × R).
Thus we may index the components of [L i0 , M] by Ki0 n/m . For each k ∈ Ki0 n/m , set
Wk to be the component of [L i0 , M] that corresponds to
Wik0 +1, j0 ∩ ([ui0 +1, j0 , ui0 +1, j0 + wi0 +1 ] × R).
Note that by Corollary 4.8, the components Wk of [L i0 , M] are Ahlfors 2-regular and
support a 1-Poincaré inequality (and hence a q-Poincaré inequality).
For each k ∈ Ki0 n/m , set
E k = {(x, y ) ∈ Wk ∩ L i0 : vi0 (x, y ) < 1/2}.
Note that Wk ∩ L i0 is equal to the vertical line segment
{u1,1 + (w1 − wi0 +1 )} × N wi0 +1 Vik0 n/m .
Thus by Remarks 2.4 and 2.5, we see that
(log3 2)/m
H∞
(Wk ∩ L i0 ) ≈ 2−i0 n/m .
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Fig. 5..
28 P. Koskela and K. Wildrick
We now define a collection of indices that selects the components Wk on which the value
of vi0 on L i0 is “often” less than 1/2, in the sense of the appropriate Hausdorff content:
2−i0 n/m
(log 2)/m
.
(E k ) ≥
K S := k ∈ Ki0 n/m : H∞ 3
4
(5.9)
Informally, an index k ∈ Ki0 n/m is in K S if v changes substantially on small pieces of X
that are roughly at the height of Vik0 n/m .
It follows from Proposition 4.6 that
Thus for each k ∈ K S ,
(log3 2)/m
H∞
(E k ) (diam Wk )(log3 2)/m .
For k ∈ K S , set
Fk = Wk ∩ M = {u1,1 + w1 } × Cm ∩ Vik0 n/m .
Then vi0 ≥ 3/4 on Fk . Moreover, Remark 2.5 and Equation (3.1) imply that
(log3 2)/m
H∞
(log3 2)/m Cm ∩ Vik0 n/m = 2−i0 n/m ≈ (diam Wk )(log3 2)/m .
(Fk ) ≈ H∞
Note that (5.6) implies 2 − q < (log3 2)/m ≤ 2. Furthermore, we have 1 ≤ q ≤ 2. Thus for
k ∈ K S , we may apply Theorem 2.10 to the space Wk , using the sets E k and Fk , and the
function vi0 . The conclusion is that for each k ∈ K S ,
ρi0 dH2 (diam Wk )2−q 3−i0 n(2−q) .
q
Wk
We now proceed in two cases. First suppose that card(K S ) ≥ (card Ki0 n/m )/2 =
i0 n/m−1
2
. As {Wk }k∈KS is a disjointed family of measurable subsets of [L i0 , C i0 ], we have
⎛
ρi0 ≥ 2(1−q)i0 ⎝
q
2(1−q)i0
[L i0 ,C i0 ]
⎞
k∈K S
ρi0 dH2 ⎠ 2(1−q)i0 2i0 n/m 3−i0 n(2−q) .
q
Wk
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(diam Wk )(log3 2)/m = (3−i0 n diam X1,1 )(log3 2)/m ≈ 2−i0 n/m .
Exceptional Sets for Quasiconformal Mappings 29
The final inequality in (5.5) is equivalent to
n
n(2 − q)
−
≥ 0,
i0 (1 − q) +
m
log3 2
and so
q
2(1−q)i0
[L i0 ,C i0 ]
ρi0 1.
Now suppose that card K S ≤ (card Ki0 n/m )/2. In this case, we work with the first
term on the right-hand side of (5.8). Since vi | L i = vi−1 | L i for each positive integer i, we
may define a continuous function V : [L 0 , L i0 ] → R such that V = vi on [L i , L i+1 ] for 0 ≤
i ≤ i0 − 1. Similarly, P : [L 0 , L i0 ] → [0, ∞] defined by P = ρi on [L i , L i+1 ], 0 ≤ i ≤ i0 − 1, is
an upper gradient of V. Using V and P , we may interpret the first term on the right-hand
side of (5.8) as a “weighted capacity” in the following manner.
Recall the definition of δ > 0 from (5.1). The definition of P and Hölder’s inequality
show that
P
q−δ
dH ≤
2
[L 0 ,L i0 ]
i
0 −1 i=0
[L i ,L i+1 ]
q
ρi
(q−δ)/q
dH
2
2
δ/q
H ([L i , L i+1 ])
.
(5.10)
We may cover [L i , L i+1 ] by a rectangle of height 2 and width wi+1 − wi+2 , showing that
H2 ([L i , L i+1 ]) wi+1 − wi+2 ≤ 3−in .
From these estimates and application of Hölder’s inequality to the sum, we see that
P q−δ dH2 [L 0 ,L i0 ]
i
0 −1 [L i ,L i+1 ]
i=0
i −1
0
i=0
(q−δ)/q
q
2(1−q)i
(1−q)i
2
[L i ,L i+1 ]
ρi dH2
q
ρi
dH
2
2i(q−1)(q−δ)/q 3−inδ/q
(q−δ)/q i −1
0
i=0
The number δ is defined so that
(q − 1)(q − δ)
n
−
δ
log3 2
= −n < 0.
2
(q−1)(q−δ)
− logn 2
δ
3
i
δ/q
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By inequality (5.8), this yields the desired conclusion (5.7).
30 P. Koskela and K. Wildrick
From this, we see that
q/(q−δ)
P
q−δ
dH
2
[L 0 ,L i0 ]
i
0 −1
q
(1−q) j
2
[L i ,L i+1 ]
i=0
ρi dH2 .
(5.11)
This estimate shows how the right-hand term above may be thought of as a capacity.
For each k ∈ Ki0 n/m , set
Since V| L i0 = vi0 , we see that
E k ∪ E k = Wk ∩ L i0 .
As we have assumed that card K S ≤ (card Ki0 n/m )/2, it follows from Lemma 3.2 that
⎛
(log3 2)/m
H∞
⎝
⎞
E k ⎠ 1.
k ∈K
/ S
Moreover, since i0 ≥ 2, we see that
(log3 2)/m
H∞
(L 0 ∩ X) ≈ 1 ≈ (diam[L 0 , L i0 ])(log3 2)/m ,
and V| L 0 = v| L 0 ≤ 0. The inequalities in (5.6) show that 1 ≤ (q − δ) ≤ 2 and that 2 − (q −
δ) < (log3 2)/m ≤ 2. By Proposition 4.8, the space [L 0 , L i0 ] is Ahlfors 2-regular and supports a (q − δ)-Poincaré inequality. Thus we may apply Theorem 2.10 to the space [L 0 , L i0 ],
using the sets L 0 ∩ X and k∈K
/ S E k , and the function V. We conclude that
P q−δ dH2 diam([L 0 , L i0 ])2−(q−δ) 1.
[L 0 ,L i0 ]
Noting that the exponent q/(q − δ) depends only on n and m, inequalities (5.8) and (5.11)
now yield the desired conclusion.
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E k = {(x, y ) ∈ Wk ∩ L i0 : V(x, y ) ≥ 1/2}.
Exceptional Sets for Quasiconformal Mappings 31
Acknowledgments
This paper is dedicated to the memory of Juha Heinonen, a friend, mentor, and leader.
His knowledge and friendship have been invaluable throughout our careers and several
conversations with him contributed to this paper. He is sorely missed. The second author
would like to thank the University of Jyväskylä Department of Mathematics for its
hospitality during the winter of 2006, when much of this research was conducted. Thanks
also to Mario Bonk for conversations regarding Lemma 5.1.
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