Mappings of BM O–bounded
distortion
Kari Astala
1
Tadeusz Iwaniec
Pekka Koskela
Gaven Martin
∗
Introduction
This paper can be viewed as a sequel to the work [9] where the theory of mappings of BM O–bounded distortion is developed, largely in even dimensions,
using singular integral operators and recent developments in the theory of
higher integrability of Jacobians in Hardy–Orlicz spaces. In this paper we
continue this theme refining and extending some of our earlier work as well
as obtaining results in new directions.
The planar case was studied earlier by G. David [4]. In particular he
obtained existence theorems, modulus of continuity estimates and bounds
on area distortion for mappings of BM O–distortion (in fact, in slightly more
generality). We obtain similar results in all even dimensions.
One of our main new results here is the extension of the classical theorem
of Painlevé concerning removable singularties for bounded analytic functions
to the class of mappings of BM O bounded distortion. The setting of the
plane is of particular interest and somewhat more can be said here because
of the existence theorem, or “the measurable Riemann mapping theorem”,
which is not available in higher dimensions. We give a construction to show
our results are qualitatively optimal. Another surprising fact is that there
are domains which support no bounded quasiregular mappings, but admit
∗
Research of all authors supported in part by grants from the N.Z. Marsden Fund. Also
the U.S. National Science Foundation (TI), DMS–9706611 and the Academy of Finland
(KA+PK), SA–34082.
1
bounded mappings of BM O–bounded distortion. Thus the space of quasiregular mappings is not uniformly dense.
Roughly a mapping is said to be of BM O bounded distortion if it has
a distortion function which is majorised by a function in BM O. We also
need some degree of integrability of the differential to begin with. For the
duration of this paper we use the notation
Φ(t) = tn log−1 (e + t),
0≤t<∞
(1)
for the Orlicz function defining the space Ln log−1 L, see Section 2.2. Here
are the precise definitions.
Let Ω be an open subset of Rn and let
f = (f 1 , f 2 , . . . , .f n ) : Ω → Rn
Definition 1.1 We say that f has finite distortion if, first of all
1,Φ
f ∈ Wloc
(Ω, Rn )
and secondly, there is a function K(x), 1 ≤ K(x) < ∞, defined a.e. in Ω
such that
|Df (x)|n = K(x)J(x, f ) a.e. Ω.
(2)
Above we denote by Df (x) : Rn → Rn the differential of f and by J(x, f ) its
Jacobian determinant, J(x, f ) = detDf (x) with |Df (x)| = sup|h|=1 |Df (x)h|.
The smallest K satisfying (2) is called the outer dilatation function of f . In
the plane it coincides with the usual linear dilatation function coming from
the Beltrami coefficient.
1,Φ
Definition 1.2 We say that f has BM O-bounded distortion if f ∈ Wloc
(Ω, Rn )
n
and there is a function M ∈ BM O(R ) such that
|Df (x)|n ≤ M (x)J(x, f )
a.e. Ω. .
(3)
We call M a BM O–distortion function for f . Let us emphasize that here we
only require the differential inequality (3) instead of an equation.
If K is bounded, then we may take M to be a constant (of BM O norm
equal to 0). We arrive at the usual class of quasiregular mappings if, in addi1,n
tion, f happens to be in the Sobolev space Wloc
(Ω, Rn ). There is a substantial
2
literature on this subject which is deep and has significant applications in
analysis and geometry. We refer the reader to the monograph [16] and the
reference therein as a starting point. It is important to notice that here, for
mappings of BM O–bounded distortion, we are relaxing both of these more
usual requirements. In particular we emphasise that having BM O–bounded
distortion is considerably weaker than assuming that the distortion function
is itself in BM O. The difference becomes important, especially in the complex plane, when studying extensions of conformal mappings. Note too that
for a BM O–function M , the product of M and the characteristic function of
a measurable set is clearly BM O–bounded, but it is not in general a BM O–
function. In this connection we should mention the following result [11]. A
necessary and sufficient condition that K = K(x) should be majorized in
Ω ⊂ Rn by a function M ∈ BM O(Rn ) is that
Z
Ω
eλK(x)
dx < ∞
1 + |x|n+1
(4)
for some positive number λ. As a matter of fact we have kM kBM O ≤ C(n)/λ
regardless of the value of the interal in (4)
We shall find interesting things happening if the BM O–norm of M is
small, or equivalently that λ at (4) is large. In other words, this occurs
when our mappings are in a sense close to quasiregular mappings. Similarly,
certain results fail if the BM O–norm of a distortion function is large. This
indicates that something quite unusual is going on.
It is also worthwhile noting that the class of mappings of BM O–bounded
distortion is invariant under a quasiconformal change of variables due to a
theorem of Reimann [15].
There is other related work on mappings of exponentially integrable dilatation in the plane. This starts with the work of G. David [4] and later P.
Tukia [21] who prove existence under various hypotheses on the dilatation.
Properties of solutions, such as the Stoilow factorisation theorem, were discussed by T. Iwaniec and V. Šverák [12] for mappings with locally integrable
dilatation function. More recent work concerning the regularity and other
properties of these mappings can be found in the work of Ryazanov, Srebro
and Yakubov [19] and Migliaccio and Moscariello [14]. Finally, continuity
properties of mappings with finite distortion (no additional conditions on
the dilatation) under minimal assumptions on the degree of integrability of
3
their derivatives have been recently established by T. Iwaniec, P. Koskela
and J. Onninen [10].
2
Notations and Definitions
Here, for completeness, we give the definition and basic properties of the
function spaces which will be used in the paper, as well as some terminology
which will prove useful.
2.1
Hardy Spaces and BMO
A measurable function a(x) supported in some ball B in Rn is called an
H1 –atom if it satisfies both of the conditions
1
|a(x)| ≤
a.e. x ∈ Rn
(5)
|B|
1 Z
aB =
a(x)dx = 0
|B| B
(6)
A function f ∈ L1 (Rn ) belongs to H1 (Rn ) if and only if it can be written as
P
a (possibly infinite) linear combination of H1 –atoms, f = ∞
k=1 λk ak , with
P∞
k=1 |λk | < ∞. The norm is then defined by
(
kf kH1 = inf
∞
X
|λk | : f =
k=1
∞
X
)
λ k ak
(7)
k=1
where the infimum is taken over all atomic decompositions of f . It is important to notice that such an f satisfies the moment condition
Z
Rn
f (x)dx = 0.
(8)
Next, for a measurable function g on Rn and a ball B ⊂ Rn we define the
average of g on B as
1 Z
g(x)dx
(9)
gB =
|B| B
If g ∈ L1loc (Rn ) and if the norm
kgkBM O
1 Z
= sup
|g(x) − gB |dx < ∞,
B |B| B
4
(10)
then we say g is of bounded mean oscillation, g ∈ BM O(Rn ).
There are two observations we wish to make. The first is that bounded
functions lie in BM O(Rn ), but they are not dense. Hence the C0∞ functions
on Rn also lie in BMO and are not dense. Thus standard approximation arguments deducing certain results from those for quasiregular mappings seem
doomed to failure. This is further emphasised since the regularity theory of
these mappings really does depend on the size of the BM O–norm of a distortion function [9]. There is a central fact to be noted here. It is the duality
theorem of Fefferman which states that BM O(Rn ) is the dual space of H1 (Rn )
[6]. The connection is that the differential inequality defining quasiconformal
or quasiregular mappings f is of the form |Df (x)|n ≤ M (x)J(x, f ), where
J(x, f ) is the Jacobian determinant. For “nice”mappings, the Jacobian is in
a Hardy space. Thus if M lies in BM O we can give meaning to the integrals
of |Df |n as a starting point for the theory, [9].
2.2
Orlicz and Zygmund Spaces
A continuous function P : [0, ∞] → [0, ∞] which is strictly increasing with
P (0) = 0 and P (∞) = ∞ is called an Orlicz function. It is customary to
call P a Young function if P is convex. Given a measurable set Ω ⊂ Rn , the
Orlicz space LP (Ω) consists of all measurable functions h on Ω such that
Z
Ω
P
|h(x)| λ
dx < ∞
(11)
for some λ = λ(h) > 0. In what follows we will usually assume that h takes
values in some finite dimensional normed space so |h(x)| stands for the norm
of h(x). The space LP (Ω) is a complete linear metric space [18]. In general
the Luxemburg functional
(
khkP = khkLP (Ω)
|h|
= inf λ > 0 : P ( ) ≤ P (1)
λ
Ω
Z
)
(12)
need not be a norm, but it is if P is a Young function. In this case LP (Ω) is
a Banach space.
Of particular importance in our applications are functions of the type
P (t) = tn logα (e + t).
5
(13)
For such functions as this, the spaces LP (Ω) are called Zygmund spaces.
The Orlicz–Sobolev space, denoted by W 1,P (Ω), consists of functions
1,1
whose distributional gradient in Wloc
(Ω) belongs to LP (Ω). We point out
that we do not assume that the function itself lies in LP (Ω), although this is
the case if Ω is a sufficiently regular bounded domain in Rn .
The Zygmund space LP (Ω) with P (t) = t log(e + t), also denoted L log L,
will play a special role in what follows when Ω is a bounded open region in
Rn . Here we can use the following integral expression for the norm
khkL log L
1 Z
|h|
|h| log e +
=
|Ω| Ω
|h|Ω
!
(14)
which is equivalent to the Luxemburg norm. More generally, for p ≥ 1 and
α ≥ 0 the nonlinear functional
"
[h]Lp logα L
1 Z
|h|
=
|h|p logα 3 +
|Ω| Ω
|h|p,Ω
!#1/p
1
1
p p
where |h|p,Ω = ( |Ω|
Ω |h| ) , is comparable with the Luxemburg norm via
constants depending only on p and α and not on Ω. As a matter of fact,
this is a norm if α = 1. We wish to record a duality result which will prove
useful. If ϕ is the Orlicz (Young) function
R
ϕ(t) = t logα (e + t)
(15)
defining the space L logα L, then the dual function space consists of those
functions g of 1/α–exponentially integrable type. More precisely,
g ∈ Expα
(16)
where Expα is defined by the Orlicz (Young) function t 7→ exp(t1/α ) − 1 with
the Luxemburg norm. We then have the following, see [18].
Lemma 2.1 If f ∈ L logα L and g ∈ Expα , then f g ∈ L1 and
kf gk1 ≤ C(n, α)kf kL logα L kgkExpα
6
(17)
3
Example
At this point it is worthwhile recalling the following example. We conjecture
that it provides optimal exponents for some of the results we will later prove.
Thus let f denote the radial stretching
f (x) =
x
1
log−γ ( )
|x|
|x|
(18)
defined in a neighbourhood of 0. We find that
J(x, f ) =
γ
|Df (x)|n =
γn+1
|x|n log |x|
γ
γn
|x|n log |x|
(19)
1
) is the dilatation function for f .
Thus the BM O–function Kγ (x) = γ1 log( |x|
Notice that kKγ kBM O → 0 as γ → ∞. There are several interesting facts we
wish to record concerning this function f .
1. (Higher Integrability): The differential of f has the following integrability properties.
|Df |n ∈ L logα L
for all 0 < α < α0 = nγ − 1 .
(20)
2. (Modulus of Continuity): f has the modulus of continuity estimate
|f (x) − f (y)| ≤ log−γ (
1
1
) = log−(α0 +1)/n (
)
|x − y|
|x − y|
(21)
3. (Volume Distortion): If E is a measurable set, then
|f E| ≤ C(n) log−nγ (
4
1
1
) = C(n) log−(1+α0 ) (
)
|E|
|E|
(22)
Modulus of continuity
In this section we provide an estimate on the modulus of continuity of a
monotone mapping in various Sobolev–Orlicz classes. 1 We begin with a
definition.
1
Added in September 1999: More general and thorough examination of the continuity
of monotone mappings is contained in [10].
7
Definition A mapping f : Ω → Rn is said to be monotone if for each
compact G ⊂ Ω and each pair x, y ∈ G we have
|f (x) − f (y)| ≤ max{|f (z) − f (w)| : z, w ∈ ∂G}
(23)
We note in particular that mappings of small BM O–distortion are open and
discrete and hence monotone [9].
In what follows P will denote an Orlicz function. There are two cases
which we care to differentiate. As we have mentioned the natural Sobolev set1,n
ting for the theory of mappings of finite distortion is in the space Wloc
(Ω, Rn ).
The cases we consider are those slightly above and slightly below this Sobolev
space.
4.1
Above Ln
For notational convenience we express the Orlicz function P = P (t) in the
form
P (t) = tn L(tn )
(24)
where L : [0, ∞] → [1, ∞] is nondecreasing. We further require that
∞
L(s)
ds = CL (n) < ∞
sn/n−1
1
Notice that the conditions above yield
Z
(25)
∞
L(s)
ds = ∞.
(26)
s
1
The canonical example to have in the back of your mind is the function
L(t) = logα (e + t), for α ≥ 0.
We set
Z τ 1−n
L(s)
`(τ ) =
ds, 0 < τ ≤ 1.
(27)
s
1
Note in particular that
lim `(τ ) = ∞.
(28)
Z
τ →0
Theorem 4.1 Let Ω = B(a, 3R) be a ball in Rn and let f ∈ W 1,P (Ω, Rn ) be
a monotone mapping. Then f has the modulus of continuity estimate
Z
C(n, L)
|Df |n
|f (x) − f (y)| ≤
|Df |n L
`(|x − y|/2R) Ω
|Df |nn,Ω
n
8
!
(29)
for all x, y ∈ B(a, R).
Proof. Because of homogeneity we may as well assume that y = 0 and
R = 12 . Then we must verify (29) when x lies in the unit ball B. We may
further assume that
1=
|Df |nn,Ω
1 Z
=
|Df |n .
|Ω| Ω
(30)
In particular this implies
1=
Z
1 Z
|Df |n ≤ 2 |Df |n L(|Df |n ).
|Ω| Ω
Ω
(31)
From the monotonicity of f we have
|f (x) − f (0)| ≤ max{|f (z) − f (w)| : |z| = |w| = r}
(32)
whenever |x| ≤ r ≤ 1. The Sobolev Embedding Theorem for spheres [13]
implies that for |z| = |w| = r we have
n
Z
|Df |n
(33)
Z
|f (x) − f (0)|n
≤ C(n)
|Df |n
r
S(r)
(34)
|f (z) − f (w)| ≤ C(n)r
S(r)
for almost all r and hence
again for almost every r. We now consider two sets
E = {y ∈ S(r) : |Df (y)|n < r1−n }
(35)
F = {y ∈ S(r) : |Df (y)|n ≥ r1−n }
(36)
and
and we split the integral over the sphere on the right hand side of (34)
accordingly. We thereby obtain
Z
|Df |n =
Z
S(r)
|Df |n +
E
≤ r
1−n
Z
|Df |n
F
Z
1
|E| +
|Df |n L(|Df |n )
1−n
L(r ) F
9
(37)
When we substitute this back into equation (34) we find
Z
L(r1−n )
1−n
|f (x) − f (0)|
≤ C(n)L(r ) +
P (|Df |)
r
S(r)
n
(38)
We next integrate this inequality with respect to r over the interval (|x|, 1)
to find
Z
|f (x) − f (0)|n
Z 1
Z
L(r1−n )
dr ≤ C(n)
L(r1−n )dr + P (|Df |)
r
|x|
B
1
|x|
(39)
and make the substitution s = r1−n to get
n
|f (x) − f (0)|
|x|1−n
Z
1
Z ∞
Z
L(s)
L(s)
ds ≤ C(n)
ds + P (|Df |)
s
sn/(n−1)
1
Ω
which, according to our notation, reads as
|f (x) − f (0)|n `(|x|) ≤ C(n)CL (n) +
Z
P (|Df |)
(40)
Ω
We then imply the simple consequence of our normalisation (31) to conclude
1 Z
P (|Df |)
|f (x) − f (0)| ≤ C(n, L)
`(|x|) Ω
n
(41)
which is precisely what we want. 2
If we return to consider the canonical example L(t) = logα (e + t) we have
`(τ ) =
Z
τ 1−n
1
≥
Z
1
τ 1−n
L(s)
ds
s
logα (e + s)
ds
e+s
1
1
log1+α (e + τ 1−n ) −
log1+α (e + 1)
1+α
1+α
1
1+α 1
log
≥
1+α
τ
=
and we obtain the following corollary
10
1,P
Corollary 4.1 For P (t) = tn logα (e + t), α ≥ 0, and f ∈ Wloc
(Ω, Rn ) a
monotone mapping, we have the modulus of continuity estimate
"Z
C(n, α)
|f (x) − f (y)| ≤
log(1+α)/n
2R
|x−y|
n
|Df | log
Ω
α
|Df |
e+
|Df |n,Ω
!#1/n
(42)
whenever x, y ∈ B(a, R) ⊂ B(a, 3R) = Ω.
Remarks. The estimates at (20) and (21) show that this bound is sharp.
That is, the exponent (1 + α)/n cannot be replaced by any larger constant.
Also note that when n = 2 G. David [4] has proved the following related
bounds for mappings of BM O–bounded distortion.
Theorem 4.2 Let f : C → C be a mapping of BM O–bounded distortion
with f (0) = 0, f (1) = 1. Then there are constants A and b > 0 such that for
x, y ∈ B(0, 2) we have the modulus of continuity estimate
−b/kM kBM O
|f (x) − f (y)| ≤ Alog |x − y|
4.2
Below Ln
Here we consider Orlicz functions of the type
P (t) =
tn
,
L(tp )
n−1<p<n
(43)
where L : [0, ∞] → [1, ∞] is a nondecreasing function with the further properties
1.
R∞
1
ds
sL(s)
=∞
2. The function t 7→ P (t1/p ) =
tn/p
L(t)
is convex.
Again, the canonical example is the function
L(t) = logα (e + t),
We next set
0 ≤ α ≤ 1.
τ −p
ds
sL(s)
1
Then we have the following modulus of continuity estimate
`(τ ) =
Z
(44)
11
(45)
Theorem 4.3 Let Ω = B(a, 3R) be a ball in Rn and let f ∈ W 1,P (Ω, Rn ) be
a monotone mapping. Then f has the modulus of continuity estimate
Z
C(n, p)
|Df |n
|f (x) − f (y)| ≤
`(|x − y|/2R) Ω L |Dfp|p
|Df |
n
(46)
p,Ω
for all x, y ∈ B(a, R).
Proof As before, we assume that y = 0 and R = 21 . Then we want to show
that
C(n, p) Z
|Df |n
n
(47)
|f (x) − f (0)| ≤
`(|x|) Ω L |Dfp|p
|Df |p,Ω
for all |x| ≤ 21 . Again the Sobolev Embedding Theorem for spheres implies
that for |z| = |w| = r
p
p p−n+1
|f (z) − f (w)| ≤ Cp (n) r
Z
|Df |p
S(r)
for almost all r with |x| ≤ r < 1, and hence
|f (x) − f (0)|p
Cp (n)p Z
≤
|Df |p
rp
ωn−1 rn−1 S(r)
(48)
by monotonicity of f . We note that
|f (x) − f (0)| ≤ Ap (n)|Df |p,Ω
(49)
which follows from (48) by integration by parts with respect to r ∈ (|x|, 1).
Now, because of homogeneity, we may assume that
|f (x) − f (0)| = max{Ap (n), Cp (n)}
(50)
and therefore that
Z
1
1
≤
|Df |p
(51)
rp
ωn−1 rn−1 S(r)
We may now apply Jensen’s inequality for the convex function defined at
Condition 2 to obtain
Z
1
1
P( ) ≤
P (|Df |)
r
ωn−1 rn−1 S(r)
12
(52)
which can be written as
Z
ωn−1
≤
P (|Df |)
rL(r−p )
S(r)
(53)
Next we integrate this inequality with respect to r over the interval (|x|, 1)
to find
Z 1
Z
Z
dr
ωn−1
≤
P
(|Df
|)
≤
P (|Df |)
(54)
|x| rL(r −p )
B(0,1)
Ω
and make the substitution s = r−p to get
ωn−1
Z
|x|−p
1
Z
ds
≤ p P (|Df |)
sL(s)
Ω
(55)
which, according to our definition of `, reads as
`(|x|) ≤
p
Z
ωn−1
Ω
|Df |n
L(|Df |p )
(56)
Then, in view of the normalisation we made at (50), we have |Df |p,Ω ≥ 1,
by (49), and hence
n
|f (x) − f (0)| `(|x|) ≤ C(n, p)
Ω
with C(n, p) =
2
p
ωn−1
|Df |n
Z
L
|Df |p
|Df |pp,Ω
(57)
max{Ap (n), Cp (n)}n , which is precisely what we want.
Returning to our canonical example L(t) = logα (e + t). We first compute
for 0 ≤ α < 1
τ −p
ds
(s + e) logα (s + e)
1
1
1
=
[log1−α ((e + p ) − log1−α ((e + 1)]
1−α
τ
1−α
≥ C log (1/τ )
`(τ ) ≥
Z
Then we have the following corollary
13
1,P
Corollary 4.2 For P (t) = tn log−α (e + t), 0 ≤ α < 1 and f ∈ Wloc
(Ω, Rn )
a monotone mapping, we have the modulus of continuity estimate
|f (x) − f (y)| ≤
"Z
C(n, p, α)
log(1−α)/n
2R
|x−y|
|Df |n log α
Ω
|Df |
e+
|Df |p,Ω
!#1/n
(58)
whenever x, y ∈ B(a, R) ⊂ B(a, 3R) = Ω.
In the case α = 1 the reader can easily verify the following corollary.
Corollary 4.3 For P (t) = tn log−1 (e + t) we have the modulus of continuity
estimate
n
|f (x) − f (y)| ≤
C(n, p)
log log 8 +
Z
2R
|x−y|
n
|Df | log
Ω
−1
|Df |
e+
|Df |p,Ω
!
(59)
whenever x, y ∈ B(a, R) ⊂ B(a, 3R) = Ω.
5
Distortion of Hausdorff Dimension
In this section we shall use the modulus of continuity estimates to measure
the distortion of Hausdorff dimension of sets in Rn under mappings of BM O–
bounded distortion and, more generally, under mappings in Sobolev–Orlicz
classes. These results appear to be new and of independent interest. It
is well known that quasiconformal mappings distort the classical Hausdorff
dimensions by bounded amounts, see [1] for the sharp result in the plane.
Here no such estimates are possible and we must look at finer measures of
dimension. Namely those obtained via logarithmic weight functions. These
estimates on the distortion of dimension, together with the existence theorem
in the plane, will provide us with our removablility theorem in the next
section.
We begin with a definitions and a lemma. A family of subsets Ej ⊂ Rn ,
j = 1, 2 . . . is said to have bounded overlap if there is a constant A ≥ 1 such
that for each i = 1, 2 . . . we have
#{j : Ei ∩ Ej 6= ∅} ≤ A
(60)
A typical situation where families of sets with bounded overlap occur is the
application of various covering theorems. We shall be using the Besicovitch
covering theorem [13].
14
Lemma 5.1 Let Ω ⊂ Rn have positive and finite measure and v be a function
for which
Z
|v|
|v| logα (e +
)<∞
(61)
|v|Ω
Ω
S
for some α ≥ 0. Suppose that E ⊂ Ω with E = ∞
j=1 Ej a union of measurable
sets with bounded overlap with |Ej | < ∞. Then
∞ Z
X
Z
|v|
|v|
) ≤ C(α, A) |v| logα (e +
)
|v| log (e +
|v|Ej
|v|E
Ej
E
j=1
Moreover,
R
E
α
|v| logα (e +
|v|
)
|v|E
(62)
→ 0 as |E| → 0.
Proof. We apply the elementary inequality logα (e + xy) ≤ 2α log(e + x) +
|v|E
|v|
(2α)α y to x = |v|
and y = |v|
and obtain
E
E
j
∞ Z
X
j=1
!
Z
∞ Z
∞
X
X
|v|
|v|
|v|E
α
α
α
|v| log (e +
) = 2
|v| log (e +
) + (2α)
|v|
|v|Ej
|v|E
|v|Ej
Ej
Ej
j=1 Ej
j=1
α
Z
|v|
α
≤ 2 A |v| log (e +
) + (2α) A |v|
|v|E
E
E
Z
|v|
)
≤ C(α, A) |v| logα (e +
|v|E
E
α
Z
α
establishing the first part of the lemma. The last claim follows easily. 2
The following lemma is a first effort to look at the distortion of Hausdorff
dimension under mappings of BM O–bounded distortion. We shall see later
that sets of Hausdorff dimension zero can be mapped to sets of large Hausdorff dimension. The more subtle measures of dimension we consider will be
Hausdorff measures with weights related to the improved integrability results
found in [9]. We recall the notation Hh (E) for the Hausdorff measure of a
set E with weight function h(t), see [7], [13]. The usual Hausdorff measures
in Rn are denoted Hβ for 0 ≤ β ≤ n.
Lemma 5.2 Let f : Ω → Rn be a mapping satisfying the modulus of continuity estimate
|f (x) − f (y)| ≤ C log
−s
2R
|x − y|
! "Z
3B
15
n
|Df | log
α
|Df |
e+
|Df |n,3B
!#1/n
(63)
whenever x, y ∈ B = B(a, R) ⊂ B(a, 3R) ⊂ Ω and for which
Z
n
|Df | log
|Df |
e+
|Df |n,Ω
α
Ω
!
<∞
(64)
Let h(t) = | log t|−sβ(n+β)/n , 0 ≤ t < 1, be a weight function. If E ⊂ Ω is
compact with Hh (E) = 0, then Hβ (f E) = 0.
Proof. For notational simplicity we set v = |Df |n . Let
F = {x ∈ E : lim sup | log r|sβ
Z
|v| logα (e +
B(x,r)
r→0
|v|
) > 1}
|v|B(x,r)
(65)
We first show that Hg (F ) = 0 for the weight function g(t) = | log t|−sβ ,
0 ≤ t < 1. Let 0 < < 1. For each x ∈ F find a radius 0 < rx < such that
Z
|v| logα (e +
B(x,rx )
|v|
|v|B(x,rx )
) ≥ | log rx |−sβ
(66)
We now use the Besicovitch covering theorem and select from the family
{B(x, rx )} a family of closed balls {Bj }∞
j=1 of bounded overlap satisfying the
S
inequality (66) above with F ⊂ Bj and
∞
X
χBj (x) ≤ C(n),
x ∈ Rn .
(67)
j=1
Then we have
∞
X
j=1
log
−sβ
1
rj
≤
∞ Z
X
|v| logα (e +
j=1 Bj
Z
≤ C S
|v|
)
|v|Bj
|v| logα (e +
Bj
S
|v|
)
|v|∪Bj
(68)
by Lemma 5.1 above. Since Bj lies within an neighbourhood of E and
S
as E is a compact set of measure zero we see that | Bj | → 0 as → 0. It
follows that Hg (F ) = 0. Let δ = dist(E, ∂Ω)/3. Since Hg (F ) = 0 we may
cover F by balls Uj , = 1, 2, . . . of radius tj < δ and such that for a given
> 0 we have
∞
X
δ
log−sβ < (69)
tj
j=1
16
Then from the modulus of continuity estimate we have
dia(f Uj ) ≤ C
≤ C
#1/n
"Z
|v|
|v| log (e +
)
|v|3Uj
3Uj
α
δ
log−s ( )
tj
#1/n
"Z
|v|
|v| log (e +
)
|v|Ω
Ω
log−s (
α
delta
)
tj
(70)
from which we conclude that Hβ (f F ) = 0 by raising both sides to the power
β, then summing and finally by letting → 0.
We now need to show that Hβ (f (E \ F )) = 0. To this end we cover E \ F
with balls Bj = B(xj , rj ), 0 < rj < 13 , j = 1, 2, . . . centered in E \ F and
such that
1/2
1. Uj = B(xj , 9rj ) ⊂ Ω
|v| log(e +
2.
R
3.
P∞
Uj
j=1
|v|
|v|Uj
) ≤ 2sβ log−sβ
1
3rj
log−sβ(n+β)/n ( r1j ) < .
Now we have from Condition 2. and the modulus of continuity
#1/n
"Z
1/2
3r
|v|
dia(f Bj ) ≤ C
|v| log(e +
)
log−s ( j )
|v|Uj
rj
Uj
1
1
≤ C log−sβ/n ( ) log−s ( 1/2 )
3rj
rj
≤ C log−(n+β)s/n (
1
)
rj
Now the result follows again by raising both side to the power β and then
summing over j. Lemma 5.2 is proved. 2.
When s = (1 + α)/n we combine this result with Corollary 4.1 to obtain
the following theorem for functions whose derivatives are in the Zygmund
class.
Theorem 5.1 Let P (t) = tn logα (e + t), α ≥ 0 and f ∈ W 1,P (Ω, Rn ) be a
monotone mapping. If E ⊂ Ω is compact with Hh (E) = 0 for the weight
function
2
h(t) = | log t|−(1+α)β(n+β)/n
then Hβ (f E) = 0.
17
We have shown in [9] that if n is even and f : Ω → Rn is a mapping
of finite distortion for which the distortion function has a BM O–bound of
sufficiently small norm, then f ∈ W 1,P (Ω, Rn ) for P (t) = tn log(e + t) and f
is open and discrete (and therefore monotone). We thus obtain the following
corollary.
Corollary 5.1 Let n = 2`. Then there is an 1 > 0 with the following
property.
1,Φ
Suppose that f ∈ Wloc
(Ω, Rn ), with Φ as in (1), has a distortion function
M for which kM kBM O < 1 . If E ⊂ Ω is compact with Hh (E) = 0 for the
weight function
2
h(t) = log−2β(n+β)/n (t)
then Hβ (f E) = 0.
When n = 2 and β = 1 this corollary reads as
Corollary 5.2 There is an 2 > 0 with the following property. Suppose that
1,Φ
f ∈ Wloc
(Ω, Rn ) is a mapping with a BMO–distortion function M for which
kM kBM O < 2 . If E ⊂ Ω is compact with Hh (E) = 0 for the weight function
h(t) = log−3/2 (t), then H1 (f E) = 0.
6
Distortion of Volume
It is clear that higher integrability properties of the Jacobian also imply
bounds on the distortion of volume under a Sobolev-Orlicz mapping. In
particular this is true for mappings of bounded BM O-distortion. For later
purposes we record these facts here.
Theorem 6.1 Let P (t) = tn logα (e + t), α ≥ 0, and f ∈ W 1,P (Ω, Rn ) be
a monotone mapping. Then the volume |E| of a compact subset E ⊂ Ω is
distorted at most by
|f (E)| ≤ C log−α (
18
1
).
|E|
Proof. Notice first that |f (E)| ≤ E |Jf (x)|dx for each measurable E ⊂
Ω. This follows easily using Corollary 4.1 and Lemma 5.1. Since in the
space Expα the Orlicz-norm of the characteristic function χE is equivalent
1
to logα ( |E|
), (17) gives
R
|f (E)| ≤
Z
E
|Jf (x)|dx ≤ C(n, α)kJf kL logα L kχE kExpα ≤ C log−α (
1
). 2
|E|
In even dimensions mappings of bounded BM O–distortion are contained
in the appropriate Sobolev-Orlicz spaces [9] and thus we have the following.
1,Φ
Corollary 6.1 Let n = 2` and suppose that the mapping f ∈ Wloc
(Ω, Rn )
has BM O–bounded distortion function M with norm kM kBM O < small
enough. If E ⊂ Ω is compact then
CΩ (n, M ) Z
J(x, f )dx
|f (E)| ≤
1
) Ω
log( |E|
The constant CΩ (n, M ) also depends on the distance of E to ∂Ω.
In the case of the complex plane and mappings of BM O–bounded distortion one can obtain asymptotically stronger bounds as shown by G. David
[4]. He proves that
|f (E)| ≤
CΩ (M ) Z
J(x, f )dx
1
) Ω
logb/ ( |E|
whenever f has a distortion function M with kM kBM O < ; b is a constant.
We expect that similar asymptotic bounds hold in all dimensions with b =
b(n) depending only on n.
7
Removable Singularities
In this section we shall prove a theorem analogous to the Painlevé theorem for analytic functions in the plane concerning removable singularities
for bounded analytic functions. The reader will no doubt be aware of the
significance of the vanishing linear measure of f E in Corollary 5.2. It allows
19
us to apply Painlevé’s theorem in a suitable setting (after using the factorisation theorem). Later, in the following section, we give examples to show
that our results are qualitatively optimal. In particular, we shall see that
some sets of Hausdorff dimension zero are not removable for bounded mappings of BM O–bounded distortion, although they are removable for every
bounded quasiregular mapping. This will in turn imply that mappings of
BM O–bounded distortion need not be uniform limits of quasiregular mappings.
Theorem 7.1 There is 3 > 0 with the following property. Let Ω ⊂ C be
a domain and E a compact subset with Hh (E) = 0 for the weight function
h(t) = log−3/2 (t). Suppose that f : Ω \ E → C is a bounded mapping with
a BM O–bounded distortion function M such that kM kBM O < 3 . Then f
extends to a mapping of BM O–bounded distortion on Ω.
Proof. In order to prove this result we will need to use a few notations and
results from [9]. The reader is referred to that paper for the details. First,
let µf be the Beltrami coefficient of f in Ω \ E. Since E has measure 0 we
can extend µf to µ defined on all of Ω by simply putting µ = 0 on E. Then
of course the dilatation function of f , K = (1 + |µ|)/(1 − |µ|), is bounded
above by a BM O–function of norm less than 3 . When 3 is sufficiently small
the Beltrami equation
∂f = µ∂f
(71)
has a homeomorphic solution [4], see also [9] [12] for further developments.
Thus there is f1 : Ω → C with Beltrami coefficient µ. Now µf = µf1 on
Ω \ E. From the Stoilow factorization theorem of [12] we find there is an
analytic function ϕ : f1 (Ω \ E) = f1 (Ω) \ f1 (E) → f (Ω \ E) such that
ϕ ◦ f1 = f : Ω \ E → C. Since ϕ is bounded and since f1 (E) has linear
measure zero from Corollary 5.2 we find that ϕ extends analytically over
f1 (E). This then defines f on Ω and the result is proven. 2
Corollary 7.1 If E ⊂ C is a compact subset of conformal capacity zero, then
E is removable for bounded mappings having a BM O–bounded distortion
function M for which kM kBM O < 3 .
The corollary is a simple consequence of the fact that if E has conformal
capacity zero, then Hh (E) = 0 for the weight function of Corollary 5.2, see
[13] and [7] for this and related material.
20
When the distortion function is bounded, this is a classical result in the
theory of quasiconformal mappings. However in the setting of quasiregular
mappings, much more is true. See [1, 2]
We now give a construction of a fairly general nature to show that our
results are qualitatively best possible. We give the construction in the plane,
but the careful reader will see where simple modifications will achieve the
corresponding result in all dimensions. For notational simplicity we denote
by
να (·) = Hhα (·), 0 < α < ∞
(72)
the Hausdorff measure with the weight function
hα (t) = | log t|−α ,
0≤t<1
(73)
We prove the following theorem.
Theorem 7.2 For each 0 < α < ∞ there is a set Eα ⊂ C and a homeomor1,Φ
phism fα : C → C in Wloc
(C, C) with a distortion function Mα ∈ BM O(C)
such that
1. limα→∞ kMα kBM O = 0,
2. να (Eα ) ≤ 1,
3. dim(fα Eα ) > 1.
Proof. The idea of proof goes back to [1]. However there are further innovations needed here.
First, we fix m disjoint disks Bj = B(zj , r), j = 1, . . . , m, all lying in
the unit disk D = {z : |z| < 1}. We choose m > 4 and r satisfying the
inequalities
π
< πmr2 < π
(74)
3
This is clearly possibe if m is sufficiently large. Inside each disk Bj we consider the smaller disks Bj0 = B(zj , σr), σ = √2m < 1, and choose similarities
ϕj such that
ϕj (D) = Bj0 .
(75)
21
Let us say in advance that the target set F = fα Eα will be the self-similar
Cantor set generated by the family of similarities ϕ1 , . . . , ϕm . Precisely we
define
F = ∩∞
(76)
l=1 ∪1≤j1 ,...,jl ≤m ϕj1 ◦ . . . ◦ ϕjl (D)
In particular, F does not depend on α. Since√the Lipschitz constant Lip(ϕj )
of each ϕj equals σr we have mLip(ϕj ) = 2 mr > √23 > 1. The Hausdorff
dimension of the set F is determined from the equation m(Lip ϕj )d = 1.
Hence
d = dim(F ) > 1.
(77)
Clearly, the size of the set Eα must depend on the weight function h =
| log t|−α . The construction of Eα and the mapping fα : C → C proceeds by an
induction. To this effect, we define a sequence {gl }l=1,... of Kl −quasiconformal
mappings gl : C → C, where
1 ≤ K1 < K 2 < . . .
(78)
For the sake of clarity the numbers Kl are revealed later. As a first step we
define


z
in C \ ∪m

j=1 Bj
 K1 −1
z−zj g1 (z) = r (79)
(z − zj ) + zj in Bj \ Bj0

1


a1 (z − zj ) + zj
in Bj0
where r1 = r and a1 = σ K1 −1 .
For the second step we introduce new disks Bj1 ,j2 = B(zj1 ,j2 , r2 ) by defining Bj1 ,j2 = g1 ◦ϕj1 (Bj2 ) and Bj0 1 ,j2 = g1 ◦ϕj1 (Bj0 2 ) for all pairs 1 ≤ j1 , j2 ≤ m.
All Bj1 ,j2 have the same radius r2 = r2 σ K1 . The mapping g2 : C → C is defined piecewise by


g (z)

 1
K2 −1
g2 (z) = a1 z−zrj1 ,j2 (z − zj1 ,j2 ) + zj1 ,j2

2


a2 (z − zj1 ,j2 ) + zj1 ,j2
in C \ ∪1≤j1 ,j2 ≤m Bj1 ,j2
in Bj1 ,j2 \ Bj0 1 ,j2
in Bj0 1 ,j2
(80)
where a2 = a1 σ K2 −1 .
For the l−th step, suppose that we are given the mapping gl−1 : C → C.
For each l−tuple 1 ≤ j1 , . . . , jl ≤ m we define the pairs of concentric disks
Bj1 ,...,jl = B(zj1 ,...,jl , rl ) ⊃ Bj0 1 ,...,jl by the rule
Bj1 ,...,jl = gl−1 ◦ ϕj1 ◦ . . . ◦ ϕjl−1 (Bjl )
22
Bj0 1 ,...,jl = gl−1 ◦ ϕj1 ◦ . . . ◦ ϕjl−1 (Bj0 l )
Thus the radii of Bj1 ,...,jl are the same and equal to rl = rσ Kl−1 rl−1 . The
mapping gl : C → C takes the form


g (z)

 l−1 Kl −1
gl (z) = al−1 z−zjr1 ...j2 (z − zj1 ...jl ) + zj1 ...jl

l


al (z − zj1 ...jl ) + zj1 ...jl
in C \ ∪1≤j1 ,...,jl ≤m Bj1 ,...,jl
in Bj1 ...jl \ Bj0 1 ...jl
in Bj0 1 ...jl
(81)
Kl −1
where al = al−1 σ
.
Having disposed of the disks Bj1 ...jl and the mappings gl : C → C we can
now define the set
Eα = ∩∞
(82)
l=1 ∪1≤j1 ,...,jl ≤m Bj1 ...jl
and the mapping fα : C → C
fα (z) = lim gl−1 (z)
l→∞
(83)
That the limit exists can be seen from the following observation. There is a
natural one-to-one relation between points of the set F , defined at (76), and
E. Indeed, to each z ∈ E there corresponds exactly one ∞−tuple (j1 , j2 , . . .)
such that
(84)
{z} = ∩∞
l=1 Bj1 ...jl
We then have
{f (z)} = ∩∞
l=1 ϕj1 ◦ . . . ◦ ϕjl (D) ∈ F
(85)
On the other hand, if z ∈ C \ E then for sufficiently large l0 , the sequence
{gl−1 (z)}l≥l0 is constant. Precisely, this happens when z ∈
/ ∪1≤j1 ,...,jl0 ≤m Bj1 ,...,jl0 .
It is also clear that the convergence at (83) is uniform and, therefore, f is a
homeomorphism.
However, to satisfy the assertions of the theorem we must carefully choose
the numbers Kl , l = 1, 2, . . ., depending on the parameter α > 0. The idea
is to make the sequence {Kl } grow exponentially with respect to l and at the
same time to ensure that
Kl
Pl
ν=1
Kν
→0
23
as α → ∞
(86)
Here are the explicit choices
1
l
(m α − 1)m α
Kl =
,
log(σ −1 )
l = 1, 2, . . .
(87)
which gives us the following formula
1
l
X
l
m α (m α − 1)
Kν =
log(σ −1 )
ν=1
(88)
We can now estimate the να −measure of the set Eα . It follows from the
recurrence relation rl = rσ Kl−1 rl−1 that
rl = r l σ
Pl−1
ν=1
Kν
,
l = 2, . . .
(89)
Moreover, for each l it holds that E ⊂ ∪1≤j1 ...jl ≤m Bj1 ...jl where all ml disks
have the same radius rl . Hence
να (Eα ) ≤ lim sup ml | log rl |−α
l→∞
=
l−1
−α
X
lim m l log r +
Kν log σ = 1
l
l→∞
ν=1
completing the proof of condition 2. in our theorem.
It remains to compute the distortion function K(z) for fα and show that
fα ∈ W 1,Φ (D), where we recall the Orlicz function Φ(t) = t2 log−1 (e + t). It
follows from the construction that
(
K(z) =
Kl for z ∈ El = ∪1≤j1 ...jl ≤m (Bj1 ...jl \ Bj0 1 ...jl ), l ∈ N
1 otherwise
(90)
In view of the condition (74), we also have the following estimate for the area
of the sets El :
|El | ≤ πml rl2 = πml r2l σ 2
≤ π exp(2
l−1
X
Pl−1
ν=1
Kν
Kν log σ)
ν=1
1
l
= π exp(2m α ) exp(−2m α ),
24
by formula (88). We want to show that
Z
eλK(x) dx < ∞
(91)
D
for all λ < λα :=
log(m/4)
1
m α −1
Z
El
. For this, we find that
1
l
eλK(x) dx ≤ πe2m α exp(λKl − 2m α )
Thus (91) holds if and only if the infinite series
l
verges.
1
But Kl =
1
m α (m α −1)
log(σ −1 )
√2
m
with σ =
P∞
l=1
(92)
l
exp(λKl − 2m α ) conl
and hence λKl − 2m α =
1
l
α
α − 1) < 2 log(
λ m α√−1
m − 2 m . The series converges if and only if λ(m
log
2
√
m
),
2
which is the same as λ < λα .
Recall the majorization result mentioned in the introduction. Accordingly, convergence of the integral (4) yields
K(z) ≤ Mα (z)
a.e.
(93)
for some function Mα ∈ BM O(C) such that
kMα kBM O ≤
C
.
λα
(94)
It is important to realize that the constant C here has nothing to do with
the value of the integral at (91). Consequently, C is independent of α. To
conclude with the assertion 1. of our theorem we simply observe that λα → ∞
as α increases to infinity.
Finally, the question arises whether fα belongs to the Orlicz-Sobolev
1,Φ
class Wloc
(C). We argue as follows. For each l = 1, 2, . . . we look at the
Kl −quasiconformal mapping fl = gl−1 : C → C. Fix an arbitrary bounded
domain Ω containing the unit disk D. Certainly, fl ∈ W 1,2 (Ω) ∩ W 1,Φ (Ω)
and we have the following uniform estimate for the integral of the Jacobian
Z
Ω
J(z, fl ) = |fl Ω| = |Ω|
On the other hand, the distortion inequality for fl implies:
|Dfl (z)|2 ≤ K(z)J(z, fl )
25
for all l = 1, 2, . . . This gives the Φ−norm of Dfl independent of l :
kDfl k2LΦ (Ω) ≤ Cλ (Ω)
Z
eλK(z)
Ω
Z
Ω
J(z, fl ) ≤ Cλ (Ω)|Ω|
Z
eλK(z) .
(95)
Ω
As fl → fα almost everywhere, we conclude that fα ∈ W 1,Φ (Ω) and that
kDfα k2LΦ (Ω)
≤ Cλ (Ω)|Ω|
Z
eλK(z) < ∞
(96)
Ω
for λ < λα . Theorem 7.2 is proven. 2
This result has a number of interesting consequences. We proved earlier
that sets of conformal capacity zero are removable for mappings of BM O–
distortion if the BM O norm of a distortion function was sufficiently small,
Theorem 6.1. Conversely, Theorem 6.2 shows this is not the case if the
BM O–norm of a distortion function is large (but finite). In fact, let ϕ
be a bounded nonconstant holomorphic function defined in C \ E0 ; such ϕ
exists since dim(E0 ) > 1. Choosing α sufficiently small gives a set Eα with
να (Eα ) = 0, and hence of capacity zero, which is not removable for the
mapping ϕ ◦ fα which is of bounded BM O–distortion.
Indeed this sort of construction, suitably modified, might be found in all
dimensions. In view of the construction above and the fact that the class of
BM O–bounded distortion mappings is preserved by postcomposition with
a quasiregular mapping, all we need is a bounded nonconstant quasiregular
mapping defined in the complement of a suitably regular Cantor set (say one
which is quasiconformally equivalent to E0 ) to take the place of ϕ. However
such mappings have been constructed by S. Rickman [17] in dimension 3 and
a close examination of Rickman’s construction shows that the Cantor set
in question is self similar and the quasiconformal image of E0 . Presumably
Rickman’s example can be constructed in all dimensions, cf [17]. However
this is not yet done.
One further consequence of our construction is the following rather surprising result (which, as noted above, seems valid in all dimensions).
Theorem 7.3 There are planar domains Ω ⊂ C, which support nonconstant
bounded mappings of g of arbitrarily small BM O distortion, with the further
property that any bounded quasiregular mapping defined in Ω is constant. In
particular g cannot be uniformly approximated by quasiregular mappings.
26
Proof. Given > 0, choose α so large that the mapping fα of Theorem
6.2 has BM O–distortion bounded by . Let ϕ be a bounded nonconstant
holomorphic mapping defined on C \ fα Eα = C \ E0 . Then set g = ϕ ◦
fα defined on Ω = C \ Eα . Since να (Eα ) < ∞, dim(Eα ) = 0 and Eα is
removable for all bounded quasiregular mappings. In particular any bounded
quasiregular mapping defined on Ω is constant. 2
In a very real sense this result shows that the quasiregular mappings play
a similar role within the space of mappings of BM O–bounded distortion as
that played by L∞ functions within the space of BM O–functions.
27
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[4] G. David, Solutions de l’equation de Beltrami avec kµk∞ = 1, Ann.
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(KA, PK)
(TI)
(GM)
University of Jyväskylä Syracuse University University of Auckland
Jyväskylä
Syracuse, NY
Auckland
Finland
U.S.A.
New Zealand
29