CONFORMAL DIMENSION DOES NOT ASSUME
VALUES BETWEEN ZERO AND ONE
LEONID V. KOVALEV
Abstract
We prove that the conformal dimension of any metric space is at least one unless it is
zero. This confirms a conjecture of J. T. Tyson [23, Conj. 1.2].
1. Introduction
Let (X, dX ) be a metric space. Given a ∈ X and r ≥ 0, we write B(a, r) = {x ∈
X : dX (x, a) < r}. The Hausdorff dimension H dim X is defined as the infimum of
all numbers s > 0 such that for every ε > 0, the entire space X can be covered by a
collection of balls B(ai , ri ) such that i ris < ε. Although the Hausdorff dimension
is not preserved by homeomorphisms, one can use it to define a topological invariant,
T dim X := inf{H dim Y : Y is a metric space homeomorphic to X},
called the topological dimension of X (see [13]).
An injective mapping f : X → Y is called quasi-symmetric, or η-quasisymmetric, if there is a homeomorphism η : [0, ∞) → [0, ∞) such that
d (x , x ) dY (f (x1 ), f (x3 ))
X 1 3
≤η
dY (f (x2 ), f (x3 ))
dX (x2 , x3 )
for any distinct points x1 , x2 , x3 ∈ X (see [11], [22]). The conformal dimension
C dim X is defined in the same way as T dim X, only with the additional requirement
that X and Y be homeomorphic via a quasi-symmetric mapping:
C dim X := inf{H dim Y : Y is quasi-symmetrically homeomorphic to X}.
Conformal dimension, which was introduced by Pansu [18], arises naturally in studies
of Gromov hyperbolic spaces (see [5] – [8]).
For any metric space X, we have T dim X ≤ C dim X ≤ H dim X. All three
dimensions have different ranges of possible values. Using Cantor-type sets, one can
show that the Hausdorff dimension can assume any value in the interval [0, ∞] (see
[17]). The topological dimension is always integer when it is finite (see [13]). Given
DUKE MATHEMATICAL JOURNAL
c 2006
Vol. 134, No. 1, Received 6 April 2005. Revision received 22 November 2005.
2000 Mathematics Subject Classification. Primary 51F99; Secondary 47H06, 46B20.
1
2
LEONID V. KOVALEV
any s ∈ {0} ∪ [1, ∞], one can construct a metric space X such that C dim X = s (see
[4], [6], [18], [23]); moreover, X can be chosen compact and totally disconnected.
Tyson conjectured in [3], [23], and [24] (see also [11, p. 122]) that C dim X ∈
/ (0, 1)
for any metric space X. The following theorem confirms Tyson’s conjecture, thus
completing the description of the possible values of C dim X.
THEOREM 1.1
The conformal dimension of any metric space is either zero or at least one.
Since nonseparable spaces have infinite conformal dimension, we consider only separable metric spaces. By the Banach-Fréchet-Mazur theorem (see [1, Chap. XI, Sec. 8]),
every such space can be isometrically embedded into C[0, 1], the space of continuous
real-valued functions on [0, 1] with the uniform norm. Therefore, Theorem 1.1 follows
from the next theorem.
THEOREM 1.2
Let V be a real separable Banach space, and let E ⊂ V be a set such that H dim
E < 1. For any ε > 0, there exists a quasi-symmetric homeomorphism f : V → V
such that H dim f (E) ≤ ε.
This statement remains true if the Hausdorff dimension H dim is replaced by the
upper Minkowski dimension, lower Minkowski dimension, upper packing dimension,
or lower packing dimension.
See [17, pp. 76, 81] for definitions of the Minkowski and packing dimensions.
Tyson [24] proved a version of Theorem 1.1 for the conformal Assouad dimension
(defined in [15, p. 1284], [24, p. 643]). For a set E ⊂ Rn , Tyson [24] defined the global
conformal dimension GC dim E as inf{H dim f (E)}, where f ranges over all quasisymmetric mappings of Rn onto itself. Obviously, GC dim E ≥ C dim E for any E.
Strict inequality holds when E ⊂ R3 is Antoine’s necklace, which has GC dim E = 1
and C dim E = 0 (see [4]). However, Theorem 1.2 implies that any set E ⊂ Rn with
GC dim E < 1 has GC dim E = C dim E = 0.
Theorem 1.2 follows from a more general result on the existence of contractive
quasi-symmetric mappings in Banach spaces (see Theorem 4.1). Sections 2 and 3
contain preliminaries concerning nonlinear accretive operators and Lebesgue-Bochner
spaces. Some auxiliary results are reminiscent of [16]. A part of Lemma 2.6 was
inspired by [14].
2. Quasi-symmetric accretive mappings
Throughout the article, V is a real separable Banach space, V ∗ is its dual, and F :
∗
V → 2V is the duality map defined by
F(x) = ϕ ∈ V ∗ : ϕ2 = x2 = ϕ(x) .
CONFORMAL DIMENSION
3
The semi–inner products on V are
x, y+ = max ϕ(x) : ϕ ∈ F(y)
and
x, y− = min ϕ(x) : ϕ ∈ F(y) .
It is easy to see that |
x, y± | ≤ xy, x, y+ is convex in x, and x, y− is concave
in x. Furthermore,
x + βy, y± = x, y± + βy2 ,
β ∈ R.
(2.1)
See [10] for more on the duality map and semi–inner products.
Definition 2.1
Let T : V → V be an injective mapping, and let η : [0, ∞) → [0, ∞) be a homeomorphism. We say that T is a QSA mapping (or η-QSA, to be more precise) if the
following inequality holds for any distinct points x, y, z ∈ V :
T (x) − T (z) ≤ η
x − z T (y) − T (z), y − z
−
y − z
y − z
.
(2.2)
Quasi-symmetric mappings were introduced by Tukia and Väisälä [22] in the
context of general metric spaces. Later, Väisälä [25] studied quasi-symmetric and related mappings in Banach spaces (see also [12]). Our abbreviation QSA is meant
to indicate that QSA mappings are not only quasi-symmetric but also accretive
(see [10, p. 124]); the latter means that T (x) − T (y), x − y+ ≥ 0 for all x, y ∈ V .
Of course, the QSA condition is stronger than being quasi-symmetric and accretive.
2.2
Let T : V → V be an injective mapping. Then T is QSA if and only if T is quasisymmetric and there is c > 0 such that
LEMMA
T (x) − T (y), x − y− ≥ cT (x) − T (y)x − y,
x, y ∈ V .
(2.3)
Proof
If T is η-QSA, then T is η-quasi-symmetric, and (2.3) holds with c = 1/η(1).
Conversely, if T is η-quasi-symmetric and (2.3) holds, then T is c−1 η-QSA.
䊐
Väisälä [25] proved that the mapping Sα (x) = xα−1 x, α > 0, is quasi-symmetric in
every Banach space. This is in contrast to Proposition 2.3, which incidentally shows
that the inverse of a QSA mapping need not be QSA itself. Two notational remarks are
in order. First, the expression xα−1 x is interpreted as 0 when x = 0. Second, C(·) is
a strictly positive constant that depends only on the parameters listed in parentheses.
Its value may change from one appearance to another.
4
LEONID V. KOVALEV
PROPOSITION 2.3
The mapping Sα : V → V is QSA for 0 < α < 2. It is not necessarily QSA for
α ≥ 2.
Proof
Since Sα is quasi-symmetric, it suffices to prove that (2.3) holds when 0 < α < 2 and
fails when α ≥ 2.
Case 1: 0 < α ≤ 1. Given x, y ∈ V , let a = x, let b = y, and let d = x − y.
We assume that a, b, d > 0, for otherwise, (2.3) immediately follows from (2.1).
Interchanging and normalizing x and y, we can make sure that b ≤ a = 1. The
triangle inequality implies |d − 1| ≤ b.
First, we estimate Sα (x) − Sα (y) from above:
Sα (x) − Sα (y) = x − bα−1 y ≤ x − y + (bα−1 − 1)y = d − b + bα .
(2.4)
Next, we use (2.1) to find a lower bound for Sα (x) − Sα (y), x − y− :
x − bα−1 y, x − y− = bα−1 x − y2 + (1 − bα−1 )
x, x − y− ≥ bα−1 d 2 + d − bα−1 d.
(2.5)
Comparing (2.4) and (2.5), we see that (2.3) follows once we prove
1 − bα−1 + bα−1 d ≥ C(α)(d − b + bα )
(2.6)
for all 0 < b < 1 such that |d − 1| ≤ b. Since (2.6) is linear in d, it suffices to consider
the values d = 1 ± b. When d = 1 + b, inequality (2.6) holds with C(α) = 1. When
d = 1 − b, (2.6) takes the form
1 − bα ≥ C(α)(1 − 2b + bα ).
(2.7)
The concavity of the function b → bα implies bα ≤ 1 + α(b − 1); hence
1 − bα ≥ α(1 − b)
and
1 − 2b + bα ≤ (2 − α)(1 − b).
Therefore, inequalities (2.7) and (2.6) hold with C(α) = α/(2 − α).
Case 2: 1 < α < 2. Let x, y, a, b, and d be as in Case 1. Slightly modifying
the argument used in Case 1, we obtain the following analogues of estimates (2.4)
and (2.5):
Sα (x) − Sα (y) ≤ bα−1 x − y + (1 − bα−1 )x = bα−1 d + 1 − bα−1
CONFORMAL DIMENSION
5
and
Sα (x) − Sα (y), x − y− = x − y2 + (1 − bα−1 )
y, x − y− ≥ d 2 − bd + bα d.
Next, we must prove that
d − b + bα ≥ C(α)(1 − bα−1 + bα−1 d)
(2.8)
whenever 0 < b < 1 and |d − 1| ≤ b. When d = 1 + b, inequality (2.8) holds with
C(α) = 1. When d = 1 − b, (2.8) takes the form
1 − 2b + bα ≥ C(α)(1 − bα ).
(2.9)
By convexity of the function b → bα , we have bα ≥ 1 + α(b − 1); hence
1 − 2b + bα ≥ (2 − α)(1 − b)
and
1 − bα ≤ α(1 − b).
Therefore, inequalities (2.9) and (2.8) hold with C(α) = (2 − α)/α.
Case 3: α ≥ 2. Let V be R2 with the l ∞ -norm; that is, (x1 , x2 ) = max{|x1 |, |x2 |}.
Choose b ∈ (0, 1), let x = (1, 1 − 2b), and let y = (b, −b). Since x − y = 1 − b,
the functional ϕ(x1 , x2 ) = (1 − b)x2 belongs to F(x − y). Computations yield
Sα (x) − Sα (y) = (1 − bα , 1 − 2b + bα ) ≥ 1 − bα ≥ 1 − b
and
ϕ Sα (x) − Sα (y) = (1 − b)(1 − 2b + bα ) ≤ (1 − b)3 .
Since
lim
b↑1
inequality (2.3) fails.
ϕ(Sα (x) − Sα (y))
= 0,
Sα (x) − Sα (y)x − y
䊐
Remark 2.4
If V is finite-dimensional, then every quasi-symmetric mapping T : V → V is
surjective. This is no longer true in infinite-dimensional spaces; consider the unilateral
shift operator in the Hilbert space l 2 . It may be interesting to know whether every QSA
mapping T : V → V is surjective. Although various classes of accretive mappings
have been extensively studied (see [10], [26]), the author is unaware of any surjectivity
results for the classes defined by inequality (2.3) or its weaker version with ·, ·+ .
6
LEONID V. KOVALEV
Definition 2.5
A mapping T : V → V is called strongly accretive if there exists c > 0 such that
T (x) − T (y), x − y+ ≥ cx − y2 for all x, y ∈ V .
QSA mappings need not be strongly accretive. Indeed, consider Sα with α ∈ (0, 2)\{1}.
Since Sα (λx) − Sα (λy), x − y+ = λα+1 Sα (x) − Sα (y), x − y+ for all λ > 0, the
mapping Sα does not satisfy Definition 2.5.
2.6
The set of all η-QSA mappings from V to V is a convex cone. A pointwise limit of
η-QSA mappings is either QSA or constant. If T is QSA, then for any λ > 0 the
mapping T + λI is surjective.
LEMMA
Proof
The first two statements follow from the fact that the semi–inner product x, y− is
concave and continuous in x (see [10, Prop. 13.1]). The last statement follows from
the surjectivity of continuous strongly accretive mappings (see [10, Th. 13.1]).
䊐
3. Lebesgue-Bochner spaces
In what follows, I is the unit interval [0, 1] equipped with the Lebesgue measure.
Let Lp (I; V ), 1 ≤ p < ∞, be the Lebesgue-Bochner space of measurable functions
from I into V . Given f ∈ Lp (I; V ) and α ∈ (0, 1), define Tα,f : V → V by
Tα,f (x) = x +
1
Sα x − f (ζ ) dζ,
x ∈ V.
0
We write µf for the pushforward of the Lebesgue measure on I under f ∈ Lp (I; V ).
In this notation, Tα,f (x) = x + V Sα (x −z) dµf (z). By Lemma 2.6, Tα,f is a surjective
QSA mapping. The following lemma estimates the distortion of distances under Tα,f
in terms of µf .
3.1
Let α ∈ (0, 1), let p ∈ [1, ∞), and let f ∈ Lp (I; V ). Then for all x, y ∈ V ,
Tα,f (x) − Tα,f (y), x − y− ≥ r 2 + C(α)r α+1 µf B(x, r) , r = x − y, (3.1)
LEMMA
where C(α) > 0. As a consequence, Tα,f (x) − Tα,f (y) ≥ r + C(α)r α µf (B(x, r)).
CONFORMAL DIMENSION
7
Proof
For each z ∈ B(x, r), the mapping x → Sα (x − z) is η-QSA with η depending only
on α. By (2.2), we have
S (x − z) S (y − z) α
α
,
η(x − z/r) η(y − z/r)
x − zα
y − zα ,
.
= r max
η(x − z/r) η(y − z/r)
Sα (x − z) − Sα (y − z), x − y− ≥ r max
Since at least one of the distances x − z and y − z lies in the interval [r/2, 2r],
it follows that
Sα (x − z) − Sα (y − z), x − y− ≥
r α+1
.
2α η(2)
Integrating against µf over z ∈ B(x, r) and using the concavity of ·, x − y− , we
obtain
Sα (x − z) − Sα (y − z) dµf (z), x − y ≥ C(α)r α+1 µf B(x, r) .
−
B(x,r)
䊐
This implies (3.1).
Given f ∈ Lp (I; V ), let Dα f = Tα,f ◦ f . Then Dα f is a measurable function from
I into V . Lemma 3.2 shows that Dα maps Lp (I; V ) into itself. Although Dα is not
accretive in general, it does exhibit some expansive properties, such as (ii) and (iii) in
the following.
3.2
∈ (0, 1), let p ∈ [1, ∞), and let f ∈ Lp (I; V ). Then we have the following:
Dα : Lp (I; V ) → Lp (I; V ) is uniformly continuous;
Dα f (ξ1 ) − Dα f (ξ2 ) ≥ f (ξ1 ) − f (ξ2 ) for all ξ1 , ξ2 ∈ I.
Suppose, in addition, that f is a simple function of the form nk=1 xk χAk , where
Ak ⊂ I, 1 ≤ k ≤ n, are disjoint Borel sets of measure 1/n. Then for any
convex function : V → [0, ∞], we have
1
1
Dα f (ξ ) dξ ≥
f (ξ ) dξ.
(3.2)
LEMMA
Let α
(i)
(ii)
(iii)
0
0
Proof
(i)
It follows from the homogeneity of Sα that
Sα (x) − Sα (y) ≤ C(α)x − yα ,
x, y ∈ V .
(3.3)
8
LEONID V. KOVALEV
If f, g ∈ Lp (I; V ), then for every ξ ∈ I,
Dα f (ξ ) − Dα g(ξ )
≤ f (ξ ) − g(ξ ) +
1
Sα f (ξ ) − f (ζ ) − Sα g(ξ ) − g(ζ ) dζ
0
≤ f (ξ ) − g(ξ ) + C(α)f (ξ ) − g(ξ )α + C(α)
1
f (ζ ) − g(ζ )α dζ.
0
(ii)
(iii)
Choosing g = 0 yields Dα f ∈ Lp (I; V ). Using Minkowski’s and
Hölder’s inequalities, we conclude that Dα f − Dα gLp → 0 uniformly as
f − gLp → 0.
It follows from Lemma 3.1 that
Dα f (ξ1 ) − Dα f (ξ2 ) = Tα,f f (ξ1 ) − Tα,f f (ξ2 ) ≥ f (ξ1 ) − f (ξ2 ).
We assume that n > 1, for otherwise, Dα f = f . For 1 ≤ i, j ≤ n, let hij = xi −
xj α−1 (xi − xj ). For t ∈ R, let ft = f + tn−1 ni,j =1 hij χAi . Observe that
1
f0 = f , f1 = Dα f and that the function ψ(t) := 0 (ft (ξ )) dξ is convex.
Our goal is to prove that ψ(1) ≥ ψ(0), and for this it suffices to show that
ψ(t0 ) ≤ ψ(0) for some t0 < 0. Since is convex, it follows that
ψ(t) = n−1
n
n
n
xi + tn−1
hij ≤ n−2
(xi + thij )
j =1
i=1
= n−2
n
i=1
(xi ) + n−2
i,j =1
(xi + thij ) + (xj + thj i )
1≤i<j ≤n
with equality for t = 0. Note that hj i = −hij . Using the convexity of along
the line passing through xi and xj , we obtain
(xi + thij ) + (xj + thj i ) ≤ (xi ) + (xj ),
provided that t is negative and sufficiently small in absolute value. Summation
over i and j yields ψ(t0 ) ≤ ψ(0) for some t0 < 0, as desired.
䊐
The following is the main result of this section.
3.3
The mapping Dα : Lp (I; V ) → Lp (I; V ) is surjective for all α ∈ (0, 1) and all
p ∈ [1, ∞).
LEMMA
CONFORMAL DIMENSION
9
Proof
First, we prove that the range of Dα contains all simple functions in Lp (I; V ). To this
end, choose a finite-dimensional subspace H ⊂ V , and choose a finite partition of I
into Borel sets Ai , i = 1, . . . , n. Let S be the space of all functions from I to H which
are constant on each Ai . It is easy to see that S is an invariant subspace for Dα . Let
Sr = {f ∈ S : f Lp = r} be a sphere in S. Fix f ∈ Sr . Since the mapping Dα − I
is uniformly continuous and homogeneous of degree α < 1, there exists R > 2r such
that Dα g − gLp < R/2 for all g ∈ SR . It follows that tDα g + (1 − t)g = f for all
t ∈ [0, 1] and for all g ∈ SR . Since S is finite-dimensional, we can use the Brouwer
degree to conclude that f ∈ Dα (S).
Next, consider an arbitrary function f ∈ Lp (I; V ). There exists a sequence
fn → f , where each fn is a simple function as in Lemma 3.2(iii). Being convergent,
the sequence {fn } is relatively compact in Lp (I; V ). The converse of the AubinLions-Simon compactness theorem [19, Th. 1] yields the following:
(a)
{fn } is an equicontinuous family in Lp (I; V ); that is,
1−τ
lim sup
fn (ξ + τ ) − fn (ξ )p dξ = 0;
(3.4)
τ ↓0
n
0
{fn } is a bounded subset of Lp (I; W ) for some Banach space W that is
compactly embedded in V .
Let gn ∈ Lp (I; W ) be such that Dα gn = fn . Lemma 3.2(ii) and (3.4) imply that the
family {gn } is equicontinuous in Lp (I; V ). Furthermore, {gn } is bounded in Lp (I; W )
by Lemma 3.2(iii). By the Aubin-Lions-Simon compactness theorem (see [19],
[21]), the sequence {gn } is relatively compact in Lp (I; V ). Choosing a converging
subsequence and using the continuity of Dα , we obtain g ∈ Lp (I; V ) such that
Dα g = f .
䊐
(b)
Remark 3.4
When V is a Hilbert space and p = 2, the mapping Dα is strongly accretive, which
provides for a simpler proof of Lemma 3.3. This raises the following question: given a
separable Banach space V , is it possible to renorm L2 (I; V ) so that Dα : L2 (I; V ) →
L2 (I; V ) becomes strongly accretive?
4. Dimension-decreasing mappings
Dimension-increasing quasi-symmetric mappings appear in great abundance. Indeed,
let (X, dX ) be a metric space. For any α ∈ (0, 1), the snowflake Xα = (X, dXα ) is
quasi-symmetrically homeomorphic to X via the identity map, and H dim Xα =
α −1 H dim X (see [11]). Much deeper results concerning quasi-symmetric parametrizations of snowflakes can be found in [2] and [9].
10
LEONID V. KOVALEV
Dimension-decreasing quasi-symmetric mappings are harder to find. A naive
1/α
1/α
attempt to lower the dimension of X by replacing dX with dX fails since dX is not
necessarily a metric. Theorem 4.1 overcomes this obstacle at the cost of an additional
−1/α
factor µ B(x, dX (x, y))
, where µ is a finite measure supported on X or, more
generally, on an ambient Banach space.
THEOREM 4.1
Let µ be a Borel probability measure on V such that x dµ(x) < ∞. For any
α ∈ (0, 1), there exists a quasi-symmetric homeomorphism F : V → V such that for
all x ∈ V and for all r > 0,
−1/α F B(x, r) ⊂ B F (x), R , where R = min C(α)r 1/α µ B(x, r)
, r . (4.1)
Proof
Since every uncountable complete separable metric space is Borel isomorphic to I
(see [20, Ch. 15, Sec. 4]), one can find f ∈ L1 (I; V ) such that µ = µf . By Lemma 3.3,
−1
there exists g ∈ L1 (I; V ) such that Dα g = f . Let F = Tα,g
. By Lemma 3.2, the
mapping F is a quasi-symmetric homeomorphism of V onto itself. Given x ∈ V and
r > 0, let ρ > 0 be the smallest number such that F (B(x, r)) ⊂ B(F (x), ρ). For
any σ ∈ (0, 1), there exists y ∈ V such that x − y ≤ r and F (x) − F (y) ≥ σρ.
Applying Lemma 3.1 to F (x) and F (y), we obtain r ≥ σρ and
r ≥ C(α)(σρ)α µg B(F (x), σρ) → C(α)ρ α µg B(F (x), ρ) as σ → 1.
Since µg B(F (x), ρ) ≥ µ(B(x, r)), it follows that ρ ≤ C(α)r 1/α µ(B(x, r))−1/α , as
required.
䊐
4.2
Let E ⊂ V , and let s ∈ (0, 1). Suppose that there exists a collection of balls B(xij , rij ),
i, j ≥ 1, such that for every i ≥ 1,
COROLLARY
E⊂
∞
B(xij , rij )
and
j =1
∞
(1 + xij )rijs < 2−i .
(4.2)
j =1
Then for any ε > 0, there exists a quasi-symmetric homeomorphism F : V → V such
that
F (E) ⊂
∞
∞ B F (aij ), Rij ,
i=1 j =1
s/ε
where Rij = C(ε, s)rij .
(4.3)
CONFORMAL DIMENSION
11
Proof
Without loss of generality, we may assume that ε < (s −1 − 1)−1 . Let α = ε(s −1 − 1),
s
and let µ = c ∞
i,j =1 rij δ(xij ), where δ(x) is the unit point mass at x and c > 0 is such
that µ(V ) = 1. Apply Theorem 4.1.
䊐
Theorem 1.2 can now be proved by first restating the assumption dim E < s in terms
of the existence of certain coverings of E and then observing that the coverings of
F (E) provided by Corollary 4.3 yield dim F (E) ≤ ε.
Proof of Theorem 1.2
Let E ⊂ V be a set such that H dim E < s for some s < 1. Let Ek = E ∩ B(0, k),
k ≥ 1. For any integer i ≥ 1, the set Ek can be covered by a family of balls
s
−i−k
{Bij k : j ≥ 1} with centers in Ek and such that ∞
/(1+k). Rej =1 (diam Bij k ) < 2
placing two indices j, k in {Bij k : j, k ≥ 1} with a single index j , we obtain a collection
s
−i
{B(xij , rij ) : j ≥ 1} which covers E and verifies ∞
j =1 (1 + xij )rij < 2 . Therefore, (4.2) holds. Given ε > 0, we apply Corollary 4.2 to obtain a quasi-symmetric
∞ s
ε
mapping F and a covering (4.3) of F (E). Since ∞
j =1 Rij ≤ C(ε, s)
j =1 rij → 0 as
i → ∞, it follows that H dim F (E) ≤ ε.
Let us now indicate how this argument applies to other types of dimensions
mentioned in Theorem 1.2. If the lower Minkowski dimension of E is less than s < 1,
then E ⊂ B(0, R) for some R, and there exist positive numbers ρi → 0 and integers
Ni < 2−i ρi−s /(1 + R) such that E admits a covering (4.2) with rij = ρi , j ≤ Ni , and
rij = 0 for j > Ni . Applying Corollary 4.2, we conclude that the lower Minkowski
dimension of F (E) is at most ε.
If the upper Minkowski dimension of E is less than s, then we can set ρi = M −i
in the preceding paragraph, provided that M is sufficiently large. Using Corollary 4.2,
we find that the upper Minkowski dimension of F (E) is at most ε.
If the lower packing dimension of E is less than s, then E can be written as a
countable union ∞
k=1 Ek , where each set Ek is contained in B(0, k) and has lower
Minkowski dimension less than s. Therefore, for each k there exist positive numbers
−s
ρik → 0, i → ∞, and integers Nik < 2−i−k ρik
/(1 + k) such that Ek can be covered
by Nik balls of radius ρik . Taking the union of these coverings over k and using
Corollary 4.2, we conclude that the Minkowski dimension of F (Ek ) is at most ε for
each k ≥ 1. Therefore, the lower packing dimension of F (E) is at most ε.
Finally, if the upper packing dimension of E is less than s, then we can set
ρik = Mk−i in the preceding paragraph, provided that the numbers Mk are large
enough. It follows from Corollary 4.2 that the upper packing dimension of F (E) is at
most ε.
䊐
12
LEONID V. KOVALEV
Acknowledgments. I thank Albert Baernstein II, András Domokos, Juha Heinonen,
Ilya Krishtal, Jeremy Tyson, and Jang-Mei Wu for valuable discussions. I am grateful
to Jeremy Tyson and Jang-Mei Wu for arranging my visit to the University of Illinois
at Urbana-Champaign, where this research was started.
References
[1]
[2]
S. BANACH, Théorie des óperations linéaires, Chelsea, New York, 1955. MR 0071726
[3]
C. J. BISHOP and J. T. TYSON, Conformal dimension of the antenna set, Proc. Amer.
C. J. BISHOP, A quasi-symmetric surface with no rectifiable curves, Proc. Amer. Math.
Soc. 127 (1999), 2035 – 2040. MR 1610908
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
Math. Soc. 129 (2001), 3631 – 3636. MR 1860497
———, Locally minimal sets for conformal dimension, Ann. Acad. Sci. Fenn. Math.
26 (2001), 361 – 373. MR 1833245
M. BONK and B. KLEINER, Conformal dimension and Gromov hyperbolic groups with
2-sphere boundary, Geom. Topol. 9 (2005), 219 – 246. MR 2116315
M. BOURDON, Au bord de certains polyèdres hyperboliques, Ann. Inst. Fourier
(Grenoble) 45 (1995), 119 – 141. MR 1324127
———, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom.
Funct. Anal. 7 (1997), 245 – 268. MR 1445387
M. BOURDON and H. PAJOT, Cohomologie lp et espaces de Besov, J. Reine Angew.
Math. 558 (2003), 85 – 108. MR 1979183
G. DAVID and T. TORO, Reifenberg flat metric spaces, snowballs, and embeddings,
Math. Ann. 315 (1999), 641 – 710. MR 1731465
K. DEIMLING, Nonlinear Functional Analysis, Springer, Berlin, 1985. MR 0787404
J. HEINONEN, Lectures on Analysis on Metric Spaces, Universitext, Springer, New
York, 2001. MR 1800917
J. HEINONEN, P. KOSKELA, N. SHANMUGALINGAM, and J. T. TYSON, Sobolev classes
of Banach space-valued functions and quasiconformal mappings, J. Anal. Math.
85 (2001), 87 – 139. MR 1869604
W. HUREWICZ and H. WALLMAN, Dimension Theory, Princeton Math. Ser. 4,
Princeton Univ. Press, Princeton, 1941. MR 0006493
T. IWANIEC, private communication, October 2003.
S. KEITH and T. LAAKSO, Conformal Assouad dimension and modulus, Geom. Funct.
Anal. 14 (2004), 1278 – 1321. MR 2135168
L. V. KOVALEV and D. MALDONADO, Mappings with convex potentials and the
quasiconformal Jacobian problem, Illinois J. Math. 49 (2005), 1039 – 1060.
MR 2210351
P. MATTILA, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud.
Adv. Math. 44, Cambridge Univ. Press, Cambridge, 1995. MR 1333890
P. PANSU, Dimension conforme et sphère à l’infini des variétés à courbure négative,
Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177 – 212. MR 1024425
CONFORMAL DIMENSION
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
13
R. ROSSI and G. SAVARÉ, Tightness, integral equicontinuity and compactness for
evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2
(2003), 395 – 431. MR 2005609
H. L. ROYDEN, Real Analysis, 3rd ed., Macmillan, New York, 1988. MR 1013117
J. SIMON, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. (4) 146
(1987), 65 – 96. MR 0916688
P. TUKIA and J. VÄISÄLÄ, Quasisymmetric embeddings of metric spaces, Ann. Acad.
Sci. Fenn. Ser. A I Math. 5 (1980), 97 – 114. MR 0595180
J. T. TYSON, Sets of minimal Hausdorff dimension for quasiconformal maps, Proc.
Amer. Math. Soc. 128 (2000), 3361 – 3367. MR 1676353
———, Lowering the Assouad dimension by quasisymmetric mappings, Illinois J.
Math. 45 (2001), 641 – 656. MR 1878624
J. VÄISÄLÄ, Free quasiconformality in Banach spaces, I, Ann. Acad. Sci. Fenn. Ser. A
I Math. 15 (1990), 355 – 379; II, 16 (1991), 255 – 310; III, 17 (1992), 393 – 408.
MR 1087342; MR 1139798; MR 1190331
E. ZEIDLER, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear
Monotone Operators, Springer, New York, 1990. MR 1033498
Department of Mathematics, Washington University, St. Louis, Missouri 63130, USA; current:
Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, Texas
77843-3368, USA; lkovalev@math.tamu.edu