Beltrami operators in the plane
Kari Astala∗, Tadeusz Iwaniec†and Eero Saksman‡
Abstract
We determine optimal Lp -properties for the solutions of the general nonlinear elliptic system in the plane of the form
fz = H(z, fz ), h ∈ Lp (C),
where H is a measurable function satisfying |H(z, w1 ) − H(z, w2 )| ≤ k|w1 − w2 |
and k is a constant k < 1.
We will also establish the precise invertibility and spectral properties in
Lp (C) for the operators
I − T µ,
I − µT,
and T − µ,
where T is the Beurling transform. These operators are basic in the theory of
quasiconformal mappings and in linear and nonlinear elliptic partial differential
equations in two dimensions. In particular, we prove invertibility in Lp (C)
whenever 1 + kµk∞ < p < 1 + 1/kµk∞ .
We also prove related results with applications to the regularity of weakly
quasiconformal mappings.
1
Introduction
In this paper we study and make use of the strong interactions between the linear
and nonlinear elliptic systems in the plane, quasiconformal mappings and singular
= 12 (fx − ify ), fz = ∂f
= 12 (fx + ify ) and recall
integrals. To start with, set fz = ∂f
∂z
∂z
that general nonlinear systems Φ(z, fz (z), fz (z)) = 0 that are elliptic in the sense of
Lavrentiev [7] can be reduced to equations of the form
fz = H(z, fz ) + h(z),
∗
Supported by the Academy of Finland, projects 34082, 41933
Supported by NSF grant DMS-9706611
‡
Supported by the Academy of Finland, project 32837
†
1
(1)
where H : Ω × C → C satisfies the Lipschitz condition
|H(z, w1 ) − H(z, w2 )| ≤ k|w1 − w2 | with k < 1.
(2)
Here it is convenient to use the normalization
H(z, 0) = 0 a.e. z ∈ Ω.
Under these assumptions the solutions f : Ω → C to the homogeneous equation that
are in the Sobolev class W21 (Ω) are quasiregular mappings [25]. That is to say, they
satisfy the so-called Beltrami inequality
|fz | ≤ k|fz | a.e.
We see at once that the difference of any two solutions to the general Lavrentiev
system is a quasiregular mapping with the same constant k. It is in this context
that estimates for quasiregular mappings become central in establishing existence
and uniqueness results for basic boundary value problems for elliptic PDEs in the
complex plane.
In this work our purpose is to find the precise Lp -properties of the solutions to
the general nonhomogeneous equation (1). To illustrate this let us first consider the
linear case
Lµ1 ,µ2 f ≡ ∂z f − µ1 ∂z f − µ2 ∂z f = h, h ∈ Lp (C),
(3)
with smooth coefficients µ1 , µ2 ∈ C ∞ (C). The condition (2) of uniform ellipticity
takes now the form
sup {|µ1 (z)| + |µ2 (z)|} ≤ k < 1.
(4)
z∈C
In the plane any linear elliptic system, with two unknowns and two first order equations on the derivatives, reduces either to (3) or to Lµ1 ,µ2 f = h; the solutions to the
homogeneous equation are sense preserving in the former case while they are sense
reversing in the latter.
As is well known, for (smooth) elliptic differential operators all local smoothness
properties of the target h are inherited by the solution f and its gradient. However,
for the global Lp -problem this fails: For each p 6= 2 we present examples of uniformly
elliptic operators Lµ1 ,µ2 with C ∞ -coefficients such that for some h ∈ Lp (C) the system
(3) has no appropriate Lp -solutions, i.e. has no solutions with first derivatives in
Lp (C). See Section 6 and Example 17 below.
The relation between the ellipticity constant k and the degree of integrability p is
the central question here. The problem we are facing can be stated as follows: Fixing
the value of k < 1 find the exact range of those p’s such that (3) always admits
unique (up to a constant) solution f with partial derivatives in Lp (C). In terms of
the ellipticity bound k this is completely answered in our Theorem 1. The result
holds also for non-smooth bounded coefficients and, unexpectedly, the same precise
Lp -bounds extend to the general nonlinear elliptic system (1) as well.
2
In the study of these equations it is useful to consider fz as an unknown, and
hence we introduce the Beurling transform T via the relation
T(
∂f
∂f
) = ( ).
∂z
∂z
(5)
This identity holds for all f in the Dirichlet space (non-homogeneous Sobolev space)
E p (C), where
E p (C) = {f ∈ D0 (C) | Df ∈ Lp (C}.
(6)
Note that Wp1 (C) ⊂ E p (C). Via the Cauchy transform, solving the differential equation (3) becomes now equivalent to establishing the Lp -invertibility of the operator
I − µ1 T − µ2 T .
(7)
It is known that T corresponds to the Fourier multiplier ξ/ξ and, moreover, it has
the explicit representation as a Calderòn-Zygmund singular integral operator
1 Z f (w)dw ∧ dw
(T f )(z) = −
.
2πi C (z − w)2
These facts imply that T is bounded on every Lp (C) for 1 < p < ∞.
However, determining the precise value of the Lp –norm is a well known and longstanding open problem. If p0 = p/(p − 1) denotes the dual exponent, we have [24],
[13]
Conjecture : kT : Lp (C) → Lp (C)k = max{p − 1,
1
} = max(p, p0 ) − 1. (8)
p−1
Our work below will yield optimal spectral properties for the operators µT , µ ∈ L∞ ,
which parallel this conjecture.
Returning to equation (1) we consider the nonlinear differential operator
(Hf )(z) ≡ fz − H(z, fz ).
(9)
We are interested in finding the set of exponents p for which operator H : E p (C) →
Lp (C) has a continuous inverse. With the aid of the Beurling transform we turn H
into a nonlinear singular integral operator
(Bg)(z) ≡ g − H(z, T g).
(10)
and our question can be reformulated as determining for which p is the operator B
invertible in Lp (C).
3
Theorem 1 Assume that the measurable function H : C × C → C satisfies the
ellipticity condition
|H(z, w1 ) − H(z, w2 )| ≤ k|w1 − w2 | with k < 1.
(11)
together with the normalization H(z, 0) = 0 for all z ∈ C. Then the operator
B : Lp (C) → Lp (C) is invertible whenever
1
1+k <p<1+ .
k
Moreover, in this case B is a bi-Lipschitz homeomorphism of Lp (C). In particular,
for all g1 , g2 ∈ Lp (C)
0
Cp (k)kg1 − g2 kLp (C) ≤ kBg1 − Bg2 kLp (C) ≤ Cp (k)kg1 − g2 kLp (C) .
(12)
The novelty in Theorem 1 is the sharpness of the range of the exponent p; the result
fails for p ≤ 1 + k and for p ≥ 1 + 1/k, even in the simplest case when H(z, w) is
C ∞ -smooth and linear with respect to w. For instance, for each p ≥ 1 + 1/k there
are h ∈ Lp (C) ∩ C ∞ (C) and µ ∈ C ∞ (C) with kµk∞ = k which oscillate at ∞ so
that the non-homogeneous Beltrami equation ∂w −µ∂w = h admits no solutions with
∇w ∈ Lp (C), see Example 17.
As important special cases we obtain the optimal invertibility bounds for the linear
operators I − T µ, I − µT and T − µ. Recall further that quasiconformal mappings are
1,2
.
homeomorphic Wloc
(Ω)-solutions of the Beltrami equation ∂z f = µ ∂z f, kµk∞ ≤ K−1
K+1
−1
If µ is compactly supported and f normalized by f (z) = z + O(|z| ) near ∞, then
formula (5) gives
fz = 1 + (I − T µ)−1 (T µ).
An immediate consequence of Theorem 1 (or Theorem 3) is that every q.c. map
1
with maximal complex dilation kµk∞ = k belongs to Wp,loc
for each p < 1 + 1/k.
It is interesting to note that our proof works in precisely the converse direction: a
recent result on area distortion [3], c.f. Theorem 9 below, implies the correct Lp integrability for the partial derivatives of quasiconformal maps and we will use this
fact as our starting point in our proof in the next section.
Weakly K-quasiregular mappings in a domain Ω ⊂ C are, by definition, functions
1,q
f ∈ Wloc
(Ω) such that
maxα |∂α f (x)| ≤ Kminα |∂α f (x)|
a.e. x ∈ Ω.
1,q
Equivalently, they are solutions f ∈ Wloc
(Ω) of the Beltrami equation
∂z f (z) = µ(z) ∂z f (z) a.e. z ∈ Ω,
(13)
where kµk∞ ≤ k = K−1
and 1 ≤ q ≤ 2. In general, for q small enough a solution
K+1
of (13) need not be even continuous. However, if the solution f is contained in the
4
1,2
space Wloc
(Ω), then f is continuous, open, and discrete. In this case we simply call
f K-quasiregular.
The regularity theory of weakly quasiregular mappings asks for the range of q’s
1,q
such that every weakly K-quasiregular mapping in Wloc
(Ω) is in fact quasiregular.
We shall see, c.f. Theorem 20, that this happens if and only if the operators
I − µT
are injective on Lq (C) for all µ with kµk∞ ≤ K−1
. On the other hand, Theorem 6
K+1
below establishes the sharp bounds on the range of injectivity of I − µT on Lq (C).
One of the borderline cases here remains open; it turns out that this is closely related
to the properties of Muckenhoupt Ap -weights ω and to the following question:
Does the norm of the Beurling transform T on the weighted spaces Lpω ,
ω ∈ Ap , depend linearly on the Ap -norm kωkAp ?
For a detailed account see Section 7.
Similar questions concerning the invertibility of the operators (7) and (10) also
arise in connection with the study of distortion properties of quasiconformal mappings, removability properties of partial differential equations and so on [16], [18],
[20]. We call the operators (7) and their adjoints as well as the corresponding nonlinear operators (10) Beltrami operators. The above discussion makes it clear that
their invertibility properties and spectral behaviour are important in understanding
a number of different topics in analysis.
2
Spectral bounds and linear Beltrami operators
For the reader’s convenience we collect the statements of the spectral bounds and
the Banach space properties of the linear Beltrami operators here. The proofs and
related results are covered in later sections.
We begin by considering the spectral behaviour of the operator T itself, since
it will be instructive to compare this simple case with the more general Beltrami
operators. Let |λ| =
6 1. We claim that the operator T − λI is invertible on Lp (C) for
each p ∈ (1, ∞).
For this, let λ 6= 0 and denote by Sλ : Lp (C) 7→ Lp (C) the operator of change of
variables given by (Sλ h)(z) = h(z − λz). The claim follows by verifying the identity
(T − λ)−1 =
1
(λ + Sλ−1 T Sλ ).
2
1 − |λ|
Perhaps the easiest way to see this identity is by solving the equation
fz − λfz = h
5
for the function f . Set φ(z) = (Sλ f )(z) = f (z − λz). Chain rule yields φz (z) =
h(z − λz) and further (1 − |λ|2 )Sλ fz = (1 − |λ|2 )fz (z − λz) = φz (z) + λφz (z) =
T Sλ h + λSλ h. As (T − λ)−1 h = fz , this proves the identity.
Moreover, by the above argument the kernel of T + λ is nontrivial for all |λ| = 1.
We record these simple observations as
Proposition 2 Let p ∈ (1, ∞). Then the spectrum
σ(T : Lp (C) → Lp (C)) = {λ ∈ C : |λ| = 1}.
Thus the spectral behaviour of the Beurling transform resembles that of it’s onedimensional analogue, the Hilbert transform H. Namely, the relation H 2 = −I shows
that σ(H : Lp (R) → Lp (R)) = {i, −i} for all p ∈ (1, ∞). According to Proposition
2 the spectrum σ(T ) does not yield any direct information on the growth of the Lp norm kT kLp (C)→Lp (C) .
In contrast, we establish that the Lp -invertibility properties and the spectrum of
the operator µT , µ ∈ L∞ , indeed have a distinguished Lp behaviour, with bounds in
complete correspondence with (8).
Theorem 3 Let µ1 , µ2 ∈ L∞ (C) be such that k = k|µ1 | + |µ2 |k∞ < 1. Denote
p = 1 + 1/k and p0 = 1 + k. Then the (R-linear) operator
I − µ1 T − µ2 T
and its transpose I − T µ1 − T µ2 are invertible on Lq (C) for all q ∈ (p0 , p).
The above range for the exponents q is the largest possible, Theorem 3 fails whenever
q ≤ p0 or q ≥ p:
Theorem 4 For any p ∈ [2, ∞) there are µ ∈ L∞ with p = 1 + 1/kµk∞ such that
neither of the operators
I − T µ, I − µT
0
is invertible on Lp (C) or Lp (C). Here
1
p
+
1
p0
= 1.
It is clear that conversely a positive answer to conjecture (8) implies Theorem 3.
Theorem 3 is obtained as a direct corollary of Theorem 1. For this one notes
that the invertibility of the operators I − T µ1 − T µ2 follows by duality from the
corresponding results for their transposes I − µ1 T − µ2 T , which clearly satisfy the
conditions of Theorem 1. In turn, Theorem 4 will be deduced from Theorem 8 below.
In Theorem 4 the coefficient µ can be chosen to be a C ∞ -function, c.f. Section
6. Therefore it is only the behaviour of µ at ∞ which affects the invertibility of the
Beltrami operators. On the other hand, if the local regularity of the coefficient can
be controlled everywhere, including ∞, then it turns out that the Beltrami operators
I − µT are invertible in all Lp -spaces. An efficient and general way to describe this
phenomenon is to use the space V M O.
6
Theorem 5 Suppose that µ ∈ V M O(C) and that kµk∞ < 1. Then the operators
I − µT and I − T µ are invertible in Lp (C) for all p ∈ (1, ∞).
In [16] this result was shown for compactly supported µ ∈ V M O by applying arguments from the Fredholm-theory. For coefficients with noncompact support further
arguments are required. We shall reduce these to a general Liouville-type theorem,
which will be then proved in Section 5.
If instead of the invertibility one looks at the weaker properties such as the injectivity or surjectivity, then the behaviour of the operators differs essentially only at
the borderline cases.
Theorem 6 The Beltrami operator I − µ1 T − µ2 T is injective in Lp (C) whenever
1
< 1. For some µ1 , µ2 the injectivity fails when 1 > k|µ1 | +
k|µ1 | + |µ2 |k∞ ≤ p−1
1
|µ2 |k∞ > p−1 .
If k|µ1 | + |µ2 |k∞ < p − 1 < 1, then I − µ1 T − µ2 T is injective on Lp (C). For some
µ1 , µ2 the injectivity fails when 1 > k|µ1 | + |µ2 |k∞ > p − 1.
Hence only the limiting situation 1 + k = p < 2 remains open; consequently this
case remains open also in the regularity theory of weakly quasiregular mappings. We
expect that the injectivity does hold here, too, cf. Section 7.
One obtains the surjectivity properties of the Beltrami operators as a direct consequences of the above results and their proofs. For completeness we state these as a
separate theorem.
Theorem 7 Suppose µ1 , µ2 ∈ L∞ with k = k|µ1 | + |µ2 |k∞ < 1. Then the Beltrami
operator I − µ1 T − µ2 T is surjective on Lp (C) for all 1 + k < p < 1 + k1 .
For every p ≤ 1 + k and for every p ≥ 1 + k1 the surjectivity fails on Lp (C) for
some µ1 , µ2 with k = k|µ1 | + |µ2 |k∞ .
Finally, the results of this section can be interpreted as giving bounds for the
spectral radius of the Beltrami operators. In general no more can be said of the
spectrum itself.
Theorem 8 Let µ ∈ L∞ (C) be normalized so that kµk∞ = 1. Suppose 2 < q < ∞.
Then the spectrum
σ(µT : Lq (C) → Lq (C)) ⊂ {z ∈ C : |z| ≤ q − 1}.
(14)
Again this is optimal: there are compactly supported µ with kµk∞ = 1 such that the
equality holds in the above spectral inclusion.
Additional results on the Beurling transform and related topics are contained in
the papers [10], [29], [13], [14], [15], [3], [4]. We refer to [19] and the references there in
for results on higher-dimensional counterparts. In the extremal case of the space L2
the invertibility of the Beltrami operators and their counterparts leads to the study
of BMO-bounded coefficients µ, c.f. [20]
7
3
Jacobians of QC-maps as Ap-weights
The basic properties of planar quasiconformal mappings that are used in the sequel
can be found in [25]. In particular, recall that the mappings are a.e. differentiable
with positive Jacobian J(f ) ≡ |∂z f |2 − |∂z f |2 . We shall also use repeteadly the
consequence that pointwise a.e. all of the quantities |∂z f |2 , J(f ), and kDf k2 are
comparable, with bounds depending only on the dilatation K. Throughout this paper
the notation B = B(x, r) will stand for the open disk of center x and radius r while
|E| denotes the area of the measurable set E ⊂ C.
For the basic properties of Ap -weights we refer to [30]. Recall that the positive
weight w on C belongs to the class Ap = Ap (C) if
|w|Ap
1 Z
:= sup
w
|B| B
B
!
1 Z
w−1/(p−1)
|B| B
!p−1
< ∞.
(15)
Here 1 < p < ∞ and the supremum is taken over all disks B ⊂ C. The quantity |w|Ap
is referred to as the Ap -norm of the weight w.
According to a basic theorem of Coifman and Fefferman [9] regular CalderónZygmund operators, and the Beurling transform T in particular, are bounded on the
weighted spaces
Lpw (C)
= {f : kf k
Lpw
Z
=
p
|f | w dm
1
p
< ∞}
C
if the weight w belongs to Ap . We shall later apply this information in comparing the
derivatives uz and uz in Lpw .
Our aim in this section is to prove an auxiliary result (Theorem 12 below) which
expresses quasiconformal derivatives |∂z f | as Ap weights. The result, essential in the
proof of Theorem 3, is based on the following fundamental distortion estimate on
planar quasiconformal mappings, see [3].
Theorem 9 Assume that f is a K-quasiconformal map with f B(0, 1) = B(0, 1) and
f (0) = 0. Then there is a constant CK depending only on K such that
|f (E)| ≤ CK |E|1/K
holds for all measurable subsets E ⊂ B(0, 1).
For our present purposes it is useful to express Theorem 9 in an invariant form.
Corollary 10 There is a constant C(K) such that for any K-quasiconformal mapping of C, for any disk B ⊂ C and for any subset E ⊂ B we have
|f (E)| ≤ C(K) |f (B)|
8
|E| 1/K
|B|
.
Proof. First, by composing f with similarities we may assume that B = B(0, 1/2)
and f (0) = 0. Next, quasiconformal mappings of C satisfy [25] the quasisymmetry
condition
|f (y) − f (x)|
|y − x|
≤ γK (
) for distinct x, y, z ∈ C,
(16)
|f (z) − f (x)|
|z − x|
where the map γK is an increasing homeomorphism of [0, ∞) onto itself, depending
only on K. Using quasisymmetry we see that
(diam f B(0, 1) )2 ≤ C1 (K)|f B(0, 1)|.
In addition,
diam( f B(0, 1) ) ≤ C2 (K)dist( f B(0, 1/2), C \ f B(0, 1) ).
In other words, in the hyperbolic metric of the domain f (B(0, 1)), the hyperbolic
distance from 0 to any point in the set f B(0, 1/2) has a finite bound depending only
on K. If now Φ : f (B(0, 1)) → B(0, 1) is a conformal map with Φ(0) = 0, then by
the Koebe distortion theorem
1
|Φ0 (z)| ≥ C3 (K)
.
diam( f (B(0, 1)) )
for z ∈ f (B(0, 1/2)). Hence
|f (E)|
|f (E)|
≤ C1 (K)
≤ C4 (K)|Φ ◦ f (E)| ≤ C(K)|E|1/K
2
|f (B)|
(diam f B(0, 1) )
by Theorem 9.
The following immediate corollary will be needed in the proof of Theorem 3.
Corollary 11 There is a constant C(K) such that if B ⊂ C is a disk and f a
K-quasiconformal homeomorphism of C then
1 Z
p C(K) |f (B)| p2
(|fz | + |fz |)p ≤
.
|B| B
(1 + 1/k) − p |B|
(17)
for all p ∈ [1, 1 + 1/k), where k = K−1
. The estimate remains true also if B is
K+1
replaced by a K-quasidisk (i.e an image of a disk under a K-quasiconformal map of
the plane).
Proof. Consider first the case where B is a disk. Composing f with similarities,
we may now assume that |B| = |f (B)| = 1. Apply Corollary 10 to the set Et =
{|fz |2 + |fz |2 ≥ t} with t > 0. This gives
|Et | ≤ t−1
Z
Et
(|fz |2 + |fz |2 ) ≤ Kt−1
≤ t−1 C1 (K)|Et |1/K .
9
Z
Et
(|fz |2 − |fz |2 ) = Kt−1 |f (Et )|
Solving for |Et | from this estimate shows that |Et | ≤ C2 (K)tK/(1−K) . An easy computation yields then the claim. Finally, if B is a quasidisk the corollary is deduced
by observing that the quasisymmetry condition (16) gives a round disc B1 such that
B ⊂ B1 , |B1 | ≤ C2 (K)|B| and |f (B1 )| ≤ C3 (K)|f (B)|.
Reverse Hölder estimates are closely related to the Ap -weights. This can be expressed in various different ways; we shall make use of the following.
Theorem 12 Let f : C → C be a K-quasiconformal homeomorphism and let p ∈
[2, 1 + 1/k), where k = (K − 1)/(K + 1). Denote
ω = |Jf |1−p/2 .
Then ω ∈ A2 with |ω|A2 ≤
(18)
p C(K)
.
(1 + 1/k − p)
Proof. Let B be a ball. Denote by g = f −1 the inverse map. Then g is Kquasiconformal and we have ω = |Jg ◦ g −1 |p/2−1 . We apply Corollary 11 and a change
of variables in order to have the estimate
1 Z
1 Z
p C1 (K)
ω=
|Jg |p/2 ≤
|B| B
|B| f (B)
1 + 1/k − p
|f (B)|
|B|
!1−p/2
Next, a second application of Corollary 11 yields
1 Z −1
1 Z
p C1 (K)
ω =
|Jf |p/2−1 ≤
|B| B
|B| B
1 + k1 − (p − 2)
|f (B)|
|B|
!p/2−1
(if 0 ≤ p2 −1 ≤ 1 one simply applies Hölder’s inequality instead of Corollary 11) These
estimates and the definition of the A2 -class immediately give the theorem.
Remarks. The A2 -bounds of Theorem 12 are sufficient for our purposes in the sequel.
However, if one wants to determine the optimal Ar -class for ω when the dilatation
K and the exponent p are given it is enough to observe that the previous arguments
show for any disk B and for any K-quasiconformal map f of C that
|f (B)|
≤
c1 (t, K)
|B|
1 Z
|Jf |t
|B| B
!1/t
≤ c2 (t, K)
|f (B)|
|B|
whenever 12 (1 − k1 ) < t < 12 (1 + k1 ); as usually k = (K − 1)/(K + 1). This follows from
Corollary 11, with a change of variables when the exponent t is negative. Therefore,
reasoning as above one can show that ω = |Jf |1−p/2 is contained in the class Ar if
r>
1 + k(p − 1)
1
, when 2 ≤ p < 1 +
1+k
k
10
and
r>
1 − k(p − 1)
1
, when 1 − < p ≤ 2.
1−k
k
It is interesting to compare the A2 -results of Theorem 12 with the classical HelsonSzegö theorem, [30, p. 226]. This theorem asserts that a weight w on R belongs to
A2 if and only if one can write
w = eh1 +H(h2 ) ,
where h1 , h2 ∈ L∞ (R) with kh2 k∞ < π/2, and H denotes the ordinary Hilbert transform.
In our case it is not difficult to show that if we pass to the limits k → 0 and
p → ∞ in such a way that pk → 1 − ε, ε > 0, then the weight ω of Theorem 12
asymptotically approaches the expression ω = e(1−ε)Re(T µ) , kµk∞ = 1. Moreover the
corresponding A2 -norms stay bounded. Consequently, we obtain then a counterpart
of the Helson-Szegö theorem where H is replaced by the Beurling transform T .
We present here a more direct proof of this result based on the corresponding
asymptotic consequences of Theorem 9 that were given in [3]:
Theorem 13 Let µ, ν ∈ L∞ (C) be such that kµk∞ < 1. Then
eν+Re(T µ) ∈ A2 .
Proof. It is well known that a weight w ∈ A2 if and only if
1 Z ±(log w−(log w)B )
sup
e
< ∞,
B |B| B
where (log w)B denotes the mean value of log w over B. Thus in our case it is enough
to prove that
1 Z ReT µ−(Re(T µ))B
sup
e
<∞
(19)
B |B| B
if kµk∞ < 1. By the invariance properties of T we may assume that B = B(0, 1). It
is clear that (19) follows from the distribution inequality
|{z ∈ B(0, 1) : |Re(T µ(z)) − (Re(T µ))B(0,1) | > t}| < Ce−t ,
(20)
for normalized µ with kµk∞ = 1.
To prove (20) we write µ1 = µχB(0,2) and µ2 = µ − µ1 . Applying the simple
estimate
1
1 4
−
≤
for |z| ≤ 1 and |w| ≥ 2
2
2
(z − w)
(w) |w|3
11
we deduce that the oscillation of T µ2 on B(0, 1) is bounded by a universal constant.
Moreover, by [3, Corollary 5.1] we have that
|{z ∈ B(0, 2) : |Re(T µ1 (z))| > t}| < C 0 e−t .
Combining these facts yield (20) and proves the theorem.
Remark. The assumption kµk∞ < 1 above is necessary. For example, when µ =
(z/z)χB(0,1) , kµk∞ = 1 but since T µ = 1 + 2 log |z|χB(0,1) , exp(T µ) is not even
integrable.
Theorem 13 does not quite have a converse since the Fourier multiplier of T is even,
thus there are weights w ∈ A2 (C) which cannot be written in the form log w = ν +T µ
with µ, ν ∈ L∞ (C). This follows from Janson’s theorem [21].
4
Invertibility of the Beltrami operators
In this section we prove our main result, Theorem 1. In addition, we verify Theorem
8 and discuss in more detail the system of nonlinear first order partial differential
equations in the plane covered by Theorem 1. The proof of Theorem 1 is based on
a fundamental auxiliary result yielding a priori bounds for solutions of the linear
Beltrami operator I − µT , which we first establish and later show how the general
result can be reduced to this. Our strategy is to apply a quasiconformal change of
variables for solving the inhomogeneous equation
f − µT f = g.
This leads us in a natural way to consider the Beurling transform on weighted Lp spaces, with weights constructed from the Jacobians of quasiconformal mappings.
Lemma 14 Assume that µ ∈ L∞ (C) with kµk∞ ≤ k < 1. Suppose also that 1 + k <
p < 1 + 1/k. Then the operator I − µT is bounded below on Lp (C), i.e.
k(I − µT )gkp ≥ C0 kgkp , g ∈ Lp (C).
Proof.
(21)
Choose an arbitrary function g on C such that
g∈
C0∞ (C)
and
Z
g=0
(22)
C
and write
h = g − µT g.
(23)
Since such functions g are dense in Lp (C), it is enough toprove the claim under the
restriction (22). Furthermore, if w = iF −1 (ξ)−1 Fg(ξ) , i.e. if w is the Cauchy
12
transform of g, then w ∈ C ∞ (C) since (22) clearly implies that w is a rapidly decreasing Schwarz-function on C. Moreover, as ∂z w = g and ∂z w = T g we see that w
satisfies the non-homogeneous Beltrami equation
∂z w = µ∂z w + h.
(24)
We have thus reduced (21) to showing that k∂z wkp ≤ C0−1 khkp , i.e. to proving an
a’priori bound for the differential equation (24).
Set K = (1 + k)/(1 − k) so that 2K/(K + 1) < p < 2K/(K − 1). Choose a
K-quasiconformal homeomorphism f : C → C satisfying the Beltrami equation
fz = µfz .
(25)
The existence of f is exactly the assertion of the measurable Riemann mapping theorem [25, V.1]. Thus, setting
u = w ◦ f −1 ,
we calculate
wz = (uz ◦ f )fz + (uz ◦ f )fz
µwz + h = µ((uz ◦ f )fz + (uz ◦ f )fz ) + h.
(26)
An application of (25) enables us to eliminate the terms involving uz and leads to
(uz ◦ f )fz =
h
.
1 − |µ|2
(27)
Consequently,
Z
C
2 −p
p
|(uz ◦ f )fz | ≤ (1 − k )
Z
hp .
(28)
C
Next we use the Ap -properties of the derivatives |fz |.
Write ω =
2
2
2 −1
−1 p−2
|fz ◦ f |
and note that |fz | ≤ |fz | ≤ (1 − k ) Jf . A change of variables
gives
Z
Z
Z
|(uz ◦ f )fz |p ≤ (1 − k 2 )−1 |uz |p ω.
(29)
|(uz ◦ f )fz |p ≤
C
C
C
1−p/2
Here the weight ω is comparable to (Jf −1 )
and according to Theorem 12,
(Jf −1 )1−p/2 ∈ A2 ⊂ Ap for 2 ≤ p < 1 + 1/k. For 1 + k < p ≤ 2 observe that by
the definition (15) ω ∈ Ap if and only if ω −1/(p−1) ∈ Ap0 , where p0 = p/(p − 1) is the
0
conjugate exponent. As ω −1/(p−1) is comparable to (Jf −1 )p /2−1 ∈ A2 ⊂ Ap0 , p0 ≥ 2,
we see that ω ∈ Ap for all 1 + k < p < 1 + 1/k.
We are now able to use the theorem of Coifman and Fefferman [9] on the weighted
spaces Lp (ω). As the Beurling transform is a regular Calderón-Zygmund operator,
we may apply the result in the form
Z
C
p
|uz | ω ≤
λpω
13
Z
C
|uz |p ω,
(30)
where λω is the norm of T on the Banach space Lpω . Furthermore, λω has a bound
λω ≤ α(|ω|Ap ) < ∞ depending only on the Ap -norm of ω. The estimates give us
Z
C
|uz |p ω ≤ λpω
≤
Z
C
λpω
(1 −
|uz |p ω ≤ λpω
Z
k 2 )p
Z
C
|(uz ◦ f )fz |p
hp .
C
We now put these estimates into (26) to obtain
kgkp ≡ kwz kp ≤ k(uz ◦ f )fz kp + k(uz ◦ f )fz kp ≤ 2K 2 λω khkp .
(31)
This shows that the operator I − µT is bounded from below on Lp (C) for 1 + k <
p < 1 + k1 .
−1
Remark 15 The above calculation gives the bound C0 ≥ (2K 2 λω )
kT kLpω , for the constant C0 in (21).
, with λω =
We now apply the above to prove Theorem 1.
Proof of Theorem 1. Assume that the numbers k, p and the function H satisfy
the assumptions of the theorem. We first establish the bi-Lipschitz property (12)
for the map B. By hypothesis B maps the zero function to the zero function and,
moreover, for g1 , g2 ∈ Lp (C) there is the estimate
kBg1 − Bg2 kLp (C) ≤ kg1 − g2 kLp (C) + kkT g1 − T g2 kLp (C) ≤ (kCp + 1)kg1 − g2 kLp (C) ,
where Cp denotes the norm of T on Lp (C). Hence B is a well-defined Lipschitz map.
Assume then that g1 , g2 ∈ Lp (C) and set h = Bg1 − Bg2 . According to (11)
we may write H(z, T g1 ) − H(z, T g2 ) = µ(z)(T g1 − T g2 ), where the function µ =
µ(g1 , g2 ) : C → C satisfies kµk∞ ≤ k < 1. Hence the difference g = g1 − g2 solves the
nonhomogeneous Beltrami-equation
h = g − µT g.
We now invoke the apriori estimate (21), which shows that kgkLp (C) ≤ C(k, p)khkLp (C) .
When put together, we have proven the required bi-Lipschitz estimate (12).
In order to obtain the invertibility of the non-linear operator B, we observe that
since B is a bi-Lipschitz map it is enough to show that the image B(Lp (C)) is dense
e = h + H(g, T g) for
in Lp (C). To that end let h1 ∈ L2 (C) be arbitrary and denote Bg
1
2
2
g ∈ L (C). As T is an isometry on L (C) we obtain from condition (11) that Be is a
strict contraction on L2 (C). Hence the Banach fixed point theorem yields an element
e = g, that is, Bg = h . Now assume that h ∈ L2 (C) ∩ Lp (C). The
g ∈ L2 (C) with Bg
1
1
solution g ∈ L2 (C) for Bg = h1 also satisfies g ∈ Lp (C) according to the bi-Lipschitz
14
inequality (12). We thus infer that B(Lp (C)) contains the set L2 (C) ∩ Lp (C), which
is dense in Lp (C). This completes the proof of the theorem.
1
Next we turn to Theorem 8. Theorem 3 immediately shows that for kµk∞ ≤ p−1
the spectrum of µT on Lp (C) is contained in the closed unit disk. Thus one only
needs to construct a Beltrami coefficient µ for which the spectral inclusion (14) holds
as an equality.
Proof of Theorem 8 Let p > 2. As λ − µT = T −1 (λ − T µ)T , it suffices to
construct a compactly supported µ such that kµk∞ = k = 1/(p − 1) and
σ(T µ : Lp (C) → Lp (C)) = D
(32)
where D = {λ : |λ| < 1}. Let {λj }∞
j=1 be a dense subset of D. Choose disjoint balls
Bj = B(zj , rj ) such that rj > 0 and Bj ⊂ D for each j ≥ 1. We define µ by
∞
1 X
z − zj
χB (z).
µ(z) = −
λj
p − 1 j=1 z − zj j
In order to establish (32) it is enough to show that for each j ≥ 1 the operator λj −T µ
is not bounded from below on Lp (C).
Consider the auxiliary function
gα = (1 + α/2)|z|α χD ,
where the parameter α takes values in the open interval α ∈ (−2/p, 0). Since w =
z|z|α χD + (z)−1 (1 − χD ) has derivatives
1
(1 − χD )
z2
and w ∈ Wp1 (C) for α ∈ (− p2 , 0), we have that T −1 gα = wz . Let us define
wz = gα , wz = (α/2)z 2 |z|α−2 χD −
ν=−
Then
T −1 gα − νgα = −
1 z
χD .
p−1z
1
1 2+α
(1 − χD ) + (
+ α)z 2 |z|α−2 χD .
2
z
2 p−1
A direct computation gives
kT −1 gα − νgα kp
kgα kp
!p
=
2(pα + 2)
.
(1 + 2α)(p − 1)
Next, we note that the support of the function gα ((z − zj )/rj ) is contained in the
ball B j . Applying the affine invariance of T we may estimate
k(λj − T µ)gα ((z − zj )/rj )kp
kgα − T (νgα )kp
= λj
kgα ((z − zj )/rj )kp
kgα kp
−1
kT gα − νgα kp
Cp 2(pα + 2)
≤ λj
kT kLp (C)→Lp (C) ≤
→0
kgα kp
(1 + 2α)(p − 1)
15
as α → (−2/p)+ . This shows that λj − T µ is not bounded from below on Lp (C) and
therefore completes the proof of the Theorem.
Proof of Theorem 4. If p = 2 we may take µ ≡ 1, c.f. Proposition 1.
1
If p > 2, let µ be the dilatation function of Theorem 8; kµk∞ = p−1
. As shown
p
in the proof 1 ∈ σ(T µ) = σ(µT ) on L (C). Therefore neither I − µT nor I − T µ is
0
invertible on Lp (C). The same holds on Lp (C) for the transposes (I − µT )0 = I − T µ,
(I − T µ)0 = I − µT.
Remark. Example 17 below shows that one can even choose a smooth µ in the proof
of Theorem 4.
5
A Liouville Theorem for mappings with finite
distorsion
We shall next establish a result of Liouville type for mappings of finite distortion. We
will apply it in the later sections in proving the injectivity for the Beltrami operators in
various situations. However, since the result itself is of intrinsic interest we formulate
it in all dimensions n, n ≥ 2.
1,n
Definition A mapping f : Rn → Rn of Sobolev class Wloc
(Rn ) is said to be of finite
distortion if there is a measurable function K : Rn → [1, ∞) such that
|Df (x)|n ≤ K(x)J(x, f )
for a.e. x ∈ Rn .
(33)
Above |Df (x)| stands for the norm of the differential Df (x) : Rn → Rn and J =
J(x, f ) =det(Df (x)). The smallest such function K is called the outer dilation function of f.
For further information on this class of mappings see e.g. [20] [27] and their references.
In what follows we consider dilatations that satisfy the integral bound
Z
K n−1 (x)dx
B(0,R)
1/(n−1)
≤ K∞ ,
(34)
for all sufficiently large balls B(0, R) ⊂ Rn centered at the origin. The constant K∞
is, of course, independent of the ball B.
Combining the isoperimetric inequality with appropriate differential inequalities
leads to the following
Theorem 16 Let f : Rn → Rn be of finite distortion and assume that the outer
dilatation K of f satisfies the bound (34) for all R > R0 > 1. If moreover, J(x, f ) ∈
Lq (Rn ) with some 1 < q ≤ KK∞∞−1 , then f is constant.
16
Proof. Denote B = B(0, t) for t > 0 and abbreviate J = J(x, f ). By the isoperimetric inequality (see e.g. [27, Lemma 3.4]) and Hölder’s inequality we obtain for
almost all t ≥ 0 that
Z
Z
J ≤(
B
Z
n−1 n/(n−1)
|Df |
≤(
)
∂B
∂B
R
where we have denoted h(t) = (
∂B
|Df |n Z
)(
K
K
n−1 1/(n−1)
)
≤
Z
∂B
J h(t),
∂B
K n−1 )1/(n−1) . Consider the increasing function
φ(t) =
Z
J(x, f )dx.
B(0,t)
The previous estimate may be written in the form
t 0
φ (t)h(t) for a.e. t > 0.
n
φ(t) ≤
We first claim that
(35)
φ(t) = o(tn/K∞ ) as t → ∞.
(36)
To see this, fix an arbitrary bounded and measurable set E ⊂ Rn and consider balls
B = B(0, t) ⊃ E. Since J ∈ Lq (Rn ) by assumption, we have
φ(t) =
Z
B
J=
Z
J+
B\E
Z
1−1/q
J ≤ |B \ E|
Z
q 1/q
(
E
J )
1−1/q
+ |E|
B\E
Z
(
J q )1/q .
E
Multiplying by |B|−1+1/q and letting t grow to ∞ we conclude that
n(−1+1/q)
lim sup t
Z
φ(t) ≤ (
t→∞
Rn \E
J q )1/q ,
which clearly yields (36) since the set E was arbitrary and 1 − 1/q ≤ 1/K∞ .
In order to shorten further computations we pass to new variables: set
s = tn
and h(t) = K∞ H(tn ),
φ(t) = (ψ(tn ))1/K∞ .
Now (35) and (36) take the form
ψ(s) ≤ sψ 0 (s)H(s) for a.e. s > 0
(37)
ψ(s) = o(s) as s → ∞.
(38)
together with
In addition, our assumption (34) on the outer dilatation shows that
n−1
K∞
Z rn
n Zr
n−1 n−1
−n
n−1 n−1
ωn−1 t h (t)dt = r
K∞
H (s)ds.
≥ n
r ωn−1 0
0
17
An application of Hölder’s inequality yields the simple bound
1Z s
H(s)ds ≤ 1,
s 0
(39)
which is valid, say, for s ≥ s0 .
Let a ≥ s0 be arbitrary. It is enough to show that ψ(a) = 0. Since φ is absolutely
continuous we may integrate (37) to deduce
ψ(a) ≤ ψ(s) exp(−
Z
s
a
du
).
uH(u)
The desired conclusion follows from (38) as soon as we prove that
Z
s
a
du
≥ log(s) − C(a).
uH(u)
Rs
Set B(s) = a H(u)du. According to (39), 0 ≤ B(u) ≤ u and hence an application
of the elementary inequality H + 1/H ≥ 2 together with an integration by parts leads
to
Z
s
a
Z s
Z s
Z s
du
du Z s H(u)du
1
B(u)du
≥ 2
−
= 2 log(s/a) −
H(s) −
uH(u)
u
u2
a u
a
a s
a
Z s
du
≥ 2 log(u/a) − 1 −
= log(s) − (log(a) + 1).
a u
The proof is complete.
Remark. The above Liouville type theorem is sharp in a quite strong sense: For every
n ≥ 2 and K∞ > 1 there are K∞ -quasiconformal (hence non-constant) mappings
f : Rn → Rn , such that
J(x, f ) ∈ Lp (Rn ) for all p >
K∞
.
K∞ − 1
1
For example the mapping f (x) = xχB + x|x| K∞ −1 χB c satisfies all these requirements,
1
B = {x ∈ Rn : |x| < 1}. Note also that the K∞ -quasiconformal g(x) = x|x| K∞ −1 , x ∈
Rn , satisfies
K∞
J(x, g) ∈ weak − Lq for q =
.
K∞ − 1
6
Beltrami operators with coefficients in VMO
For elliptic differential operators the regularity of the coefficients always reflects the local smoothness and growth properties of the solutions of the corresponding equations.
For example, if the coefficient µ of the Beltrami equation fz = µfz is continuously
18
differentiable the same holds for f , and in particular the derivatives of f are locally
in Lp .
However, the oscillations of the coefficients, even at one point, can destroy the
p
L -regularity. The following example gives a µ ∈ C ∞ (C) such that a corresponding
nonhomogeneous Beltrami equation does not have appropriate Lp -solutions; this appears already at the borderline case kµk∞ = 1/(p − 1) of Theorem 3. Consequently,
the behaviour of the coefficients near infinity also play an important role in the global
Lp -estimates.
Example 17 Let p > 2 and define
µ(z) = z 2 (p + (p − 1)|z|2 )−1 .
Then µ ∈ C ∞ (C) and kµk∞ = 1/(p − 1). However, the operator I − µT : Lp (C) →
Lp (C) is not surjective.
In particular, for some φ ∈ Lp (C) the system wz − µwz = φ does not admit
solutions w ∈ E p (C).
Proof. Notice first that the quasiconformal diffeomorphism f : C → C, f (z) =
z(1 + |z|2 )−1/p satisfies the Beltrami equation fz = µfz and the derivatives fz and fz
are in weak − Lp (C), but not in Lp (C). Consider the function h,
h(z) = z(1 + |z|2 )−1/p log−1/p (1 + |z|2 ).
Define φ by the equation
hz − µhz = φ.
It is straightforward to see that φ ∈ Lp (C). Note, however, that hz , hz ∈
/ Lp (C).
We claim that the equation
Fz − µFz = φ
(40)
has no solution such that Fz , Fz ∈ Lp (C).
1,2
Assuming the contrary, we observe that F − h ∈ Wloc
(C) solves the homogeneous
Beltrami equation with complex dilatation µ. This implies (see e.g. [25, Theorem
VI.2.2]) that
F (z) = h(z) + Φ ◦ f (z),
where Φ is an entire analytic function. For large |z| our construction shows that
|h(z)| ≤ |z|1−2/p ≤ 2|f (z)|. On the other hand, from the Sobolev imbedding theorem
we deduce |F (z)| ≤ C|z|1−2/p . Combining these bounds proves that
|Φ(w)| ≤ (2C + 2)|w|
for large |w|.
19
According to the classical Liouville theorem Φ(w) = c1 w + c2 for some constants
c1 and c2 . Consequently, F = h + c1 f + c2 . However, calculating the derivatives we
obtain
−z 2
1 + log(1 + |z|2 ) Fz = hz + c1 fz =
.
1 c1 +
1
p(1 + |z|2 )1+ p
log1+ p (1 + |z|2 )
If here c1 6= 0, the constant term is dominant in the latter factor and then Fz ∈
/ Lp (C).
But if c1 = 0, then Fz = hz ∈
/ Lp (C). Therefore no choice of the constant c1 yields
the gradient of the solution in Lp (C).
On the other hand, a global control of the regularity of the coefficient µ, even in
the very weak sense of vanishing mean oscillations, that is, µ ∈ V M O(C), forces the
operators I − µT to become invertible on all Lp -spaces. The definition of the class
V M O seems to vary slightly in the literature; hence let us fix V M O(C) as the closure
of C0∞ (C) in BM O(C). It is easy to see that this is equivalent to taking the closure
of C(C) in BM O, where C is the Riemann sphere.
First of all the assumption µ ∈ V M O(C) with kµk∞ < 1 makes I −µT : Lp (C) →
p
L (C) a Fredholm operator with index zero, regardless of the value of p ∈ (1, ∞).
Since this will be needed below, for convenience of the reader we outline the proof,
c.f. for instance [16, p. 42–43].
Fix p ∈ (1, ∞). Notice first that Proposition 2 together with the spectral radius
formula shows that for every ε > 0, kT j kp→p ≤ (1 + ε)j as soon as j ≥ N = N (ε).
By choosing ε so that kµk∞ (1 + ε) < 1 we deduce the existence of an integer j0 such
that
kµj0 T j0 kp→p < 1.
This implies that the operator I − µj0 T j0 is invertible on Lp (C).
At this point one invokes the theorem of Uchiyama [31] stating that a commutator
of a classical Calderón-Zygmund operator in Lp (Rn ) and the multiplication by an
element in V M O is a compact operator on each Lp (Rn ), p ∈ (1, ∞) (observe that
our VMO agrees, modulo constants, with the space CMO employed by Uchiyama,
see [31, Lemma 3]). Thus µT − T µ is compact and by applying this repeatedly we
see that the difference K = µj0 T j0 − (µT )j0 is a compact operator on Lp (C). Writing
P = 1 + µT + (µT )2 + . . . (µT )j0 −1 it follows that
(I − µT )P = P (I − µT ) = I − (µT )j0 = (I − µj0 T j0 ) + K.
This shows that I − µT is a Fredholm operator. Finally, the continuous homotopy
t 7→ I − tµT , t ∈ [0, 1], shows that its index is zero.
In [16] it was already observed that for compactly supported µ the kernel of the
operator I − µT is trivial and since the index is zero the operator becomes invertible.
We now extend this result by removing the restrictive assumption on the support of
µ.
20
Proposition 18 Let 2 ≤ p < ∞ and suppose that µ ∈ V M O(C) satisfies kµk∞ < 1.
Then the operator I − µT is injective on Lp (C) .
Proof. It involves no loss of generality in assuming that p > 2. Showing that the
kernel of the operator I − µT : Lp (C) → Lp (C) is empty amounts to proving that
every solution of the Beltrami equation
fz = µfz
(41)
with fz , fz ∈ Lp is constant.
Assume therefore that f has partial derivatives in Lp and solves (41), 1 < p < ∞.
From the definition of VMO we get a constant µ0 , |µ0 | < 1, such that
lim
Z
R→∞
B(0,R)
|µ(z) − µ0 |dm(z) = 0.
(42)
Next, consider the linear map φ(z) = z − µ0 z and write g = f ◦ φ so that g ∈ E p (C)
e z , where
satisfies the Beltrami equation gz = µg
µe =
µ ◦ φ − µ0
.
1 − µ0 µ ◦ φ
e ∞ < 1 and µ
e satisfies the relation analogous to (42), but
Clearly µe ∈ V M O with kµk
with µ0 = 0 in this case.
The outer dilation function K, defined by |Dg(z)|2 ≤ K(z)Jg (z), satisfies the
2
pointwise estimates 1 ≤ K(z) = 1 + 2|µ(z)|(1 − |µ(z)|)−1 ≤ 1 + 1−k
|µ(z)|. Therefore
lim
Z
R→∞
K(z)dm(z) = 1
B(0,R)
and since Jg ∈ Lq (C), 1 < q = p2 < ∞, by assumption Theorem 16 implies that g is
constant. Therefore the kernel of I − µT must be trivial.
Proof of Theorem 5. The above argument shows that for all p ∈ (1, ∞) the
operator I − µT is Fredholm on Lp with index zero and by Proposition 18 it is also
injective when p ≥ 2. This proves that I − µT is invertible for 2 ≤ p < ∞.
For the case 1 < p ≤ 2 we note as in the proof of Theorem 3 that I − T µ = T (I −
0
µT )T −1 is invertible on Lp (C), 2 ≤ p0 < ∞, and so is its transpose (I −T µ)0 = I −µT
on Lp (C).
7
Injectivity in the borderline cases and weakly
quasiregular maps
One more aspect of the Beltrami operators we wish to consider is their injectivity.
We begin by studying their relations to the regularity theory of weakly quasiregular
mappings.
21
Lemma 19 Suppose that the Beltrami operator I − µT : Lq (C) → Lq (C) is injective
for some 1 < q ≤ 2, where µ has compact support with kµk∞ < 1. Then each weakly
1,q
quasiregular mapping f ∈ Wloc
(Ω) in a domain Ω ⊂ C with complex dilatation µ|Ω is
quasiregular.
Proof. It is enough to prove that for each φ ∈ C0∞ (Ω) the function g = φf has
partial derivatives in L2 (C). According to our assumptions gz , gz ∈ Lq (C) and we
have the nonhomogeneous Beltrami equation
gz − µgz = f (φz − µφz ) ∈ L2 (C).
(43)
On the other hand, since I − µT is invertible in L2 (C) we can solve this equation also
for a function ge whose derivatives are in L2 (C), that is to say
gez − µgez = f (φz − µφz ), gez , gez ∈ L2 (C),
Furthermore, since φ and µ have compact support it follows that gez ∈ Lq (C) and
gez = T gez ∈ Lq (C) . Observe that (g − ge)z − µ(g − ge)z = 0 in Lq (C).
The assumed uniqueness or the injectivity of the Beltrami operator I − µT now
implies that ge = g modulo a constant, establishing the Lemma.
On the general level of all weakly K-quasiregular mappings we obtain the following.
Theorem 20 Let k = K−1
and suppose 1 < q ≤ 2. Then the following are equivalent
K+1
for any subdomain Ω ⊂ C.
a) The operators I − µT are injective on Lq (C) whenever kµk∞ ≤ k.
1,q
b) Every weakly K-quasiregular map f ∈ Wloc
(Ω) is quasiregular
Proof. That a) implies b) follows from Lemma 19, since quasiregularity is a local
property.
K−1
Conversely, if I − µT is not injective and kµk∞ ≤ k = K+1
, then w − µT (w) = 0
1,q
q
for some nonzero w ∈ L (C). The Cauchy transform gives us a function f ∈ Wloc
(C)
satisfying the homogeneous Beltrami equation fz − µfz = 0, with w = fz .
The function f is now a weakly K-quasiregular mapping of C. If f were strongly
1,2
quasiregular, i.e. if f ∈ Wloc
(C), then the basic methods of elliptic differential
2q
equations are applicable: Letting q∗ denote the Sobolev conjugate q∗ = 2−q
of q and
using the Harnack inequality for the components of the function f − f (0) (see for
instance [11, 6.2 and Theorem 14.39]) it follows that for |z| = r and for R > 2r
|f (z) − f (0)|q∗ ≤
Z
C
|f (w) − f (0)|q∗ dm(w)
|B(0, R)| B(0,R)
22
while the Sobolev imbedding theorem gives
Z
B(0,R)
q∗
|f (w) − f (0)| dm(w) ≤ Cq
Z
B(0,R)
!q∗/q
q
(|fz | + |fz |) dm(w)
.
Since the derivatives of f are globally in Lq (C) and f is non-constant, letting R → ∞
1,2
yields a contradiction. Thus f ∈
/ Wloc
(C).
This example can also be used to construct a weakly K−quasiregular mapping
1,q
in Wloc
(Ω) which is not quasiregular. To this effect we cover the entire plane by the
domains Ωα,β = {αz + β : z ∈ Ω}, where α, β ∈ C. We see that at least one of the
restrictions f|Ωα,β is not quasiregular. A change of variables finally gives a mapping
for which b) fails.
Examples of weakly quasiregular mappings show that they can be discontinuous
with bad singularities [18]: Let D = {z : |z| < 1} and choose a countable number of
S
disjoint disks B(xi , ri ) ⊂ D such that the measure |D \ i B(xi , ri )| = 0. Define
1+1
K
f (z) = ri
1
|z − xi |1− K
+ xi ,
z − xi
z ∈ B(xi , ri ),
(44)
on D \ i B(xi , ri ) set f (z) = z and for |z| ≥ 1 let f (z) = 1/z. Even if f has a ’pole’
at each xi , a calculation shows that f is weakly K− quasiregular and contained in
1,q
2K
, see [18].
all Sobolev classes Wloc
(C) with q < K+1
On the other hand, Theorem 3 implies that the Beltrami operator I − µT is
injective on Lq (C) if 1 + kµk∞ < q < 1 + 1/kµk∞ . Combining these facts with
Theorem 20 leads to
S
Theorem 21 Let 1 < K < ∞. Then every weakly K-quasiregular mapping, con1,q
2K
< q ≤ 2, is quasiregular on Ω.
tained in a Sobolev space Wloc
(Ω) with K+1
1,q
2K
For each q < K+1 there are weakly K-quasiregular mappings f ∈ Wloc
(C) which
are not quasiregular.
The regularity part of Theorem 21 as above was shown in [3] as a consequence
of the area distortion estimates; here we emphasize the conceptual connection to the
injectivity properties of the corresponding Beltrami operators I − µT . This approach
2K
becomes especially interesting at the borderline case q = K+1
.
1,q
2K
We conjecture that all weakly K-quasiregular mappings f ∈ Wloc
with q = K+1
,
are in fact quasiregular. Equivalently, we expect that the Beltrami operators I −
µT are injective on Lq (C) at the critical exponent q = 1 + kµk∞ , q < 2. This
open problem has interesting connections and reductions which we now discuss before
continuing with further injectivity results.
In particular, we are led to consider how the norm kT kLpω of the Beurling transform
on the weighted spaces Lpω depends on the Ap -norm of the weight ω. Examples suggest
that kT kLpω depends linearly on kωkAp . But this is not yet known in general. It turns
2K
out that this would settle the borderline cases q = K+1
in Theorem 21.
23
Proposition 22 Let 1 < K < ∞ and assume that the Beurling transform satisfies
the bounds
kT φkLpω ≤ C(p)kωkAp kφkLpω
(45)
for all ω ∈ Ap and φ ∈ Lpω , with a constant C(p) depending only on p, 1 < p < ∞.
1,q
2K
Then every weakly K−quasiregular mapping f ∈ Wloc
with q = K+1
is quasiregular.
Proof. Let q < 2. We begin with a duality argument. One has to show that if (45)
is true then I − µT or I − T µ = T (I − µT )T −1 is injective on Lq (C) for all µ with
q = 1 + kµk∞ . On the other hand this is equivalent to showing that the transpose
operator (I − T µ)0 = I − µT : Lp (C) → Lp (C) has dense range in the weak topology
of Lp , 1q + p1 = 1.
Therefore fix φ ∈ C0∞ (C). According to Theorem 3, φε = (I − (1 − ε)µT )−1 φ ∈
Lp for each 0 < ε < 1. Furthermore,
φε − µT φε = φ − εµφε .
By the density of C0∞ in Lp it is enough to show that our assumptions imply εφε
tends weakly to 0 as ε → 0.
For this we claim that if one has the linear dependence (45) on the Ap -norm then
one obtains
k(I − (1 − ε)T µ)−1 kLp (C)→Lp (C) ≤
C(q)
ε
for all small ε > 0.
(46)
If the estimate (46) could be proven then it would give the uniform bound
kεφε kp ≤ C(p)kφkp .
On the other hand, since k(1 − ε)T µkL2 ≤ (1 − ε)kµk∞ =
kφε k2 ≤
(47)
1−ε
,
p−1
p−1
kφk2 .
p−2
This shows that εφε → 0 in L2 and combined with the uniform bound (47) it also
implies that εφε → 0 weakly in Lp (C).
Thus to conclude the proof we have to show that the assumption (45) implies (46).
This takes us back to our proof of Theorem 3; we must check how the quasiconformal
estimates there reflect on the norm estimates of the Beltrami operators. Indeed,
let f be the quasiconformal homeomorphism with dilatation (1 − ε)µ and let ωε be
the weight ωε = |fz ◦ f −1 |p−2 . According to Remark 15, the norm of the operator
(I − (1 − ε)µT )−1 : Lp (C) → Lp (C) is bounded by 2K 2 λωε , where λωε = kT kLpωε is
24
the norm of the Beurling transform on Lpωε . But as kµk∞ = 1/(p − 1), Theorem 12
shows that
p C(K)
1 + kµk∞
kωε kAp ≤ kωε kA2 ≤
, K=
.
ε
1 − kµk∞
Thus (46) is indeed a consequence of the linear dependence of kT kLpωε on kωε kA2 .
Remarks.
a) The best general result known in the direction of (45) is due to Buckley [8] and
yields for kT kLpω bounds of power type with exponent strictly bigger than 1. He also
gave examples with a linear dependence on kωkAp .
b) The missing borderline case in the regularity of weakly quasiregular mappings
would also follow from the conjecture at (8). In fact, one only has to note that the
injectivity of I −µT on Lq (C), q = 1+kµk∞ , is a consequence of (46) and that clearly
the conjecture at (8) implies (46).
As a last related aspect on weak quasiregularity we see that in the VMO-setting
one again obtains strong regularity consequences; Theorem 5 and Lemma 19 have the
following immediate corollary.
1,q
Corollary 23 Assume that q > 1 and f ∈ Wloc
(Ω) has dilatation µ and that f is
weakly quasiregular. If µ is in V M O(Ω), then f is quasiregular.
Let us then turn to the other aspects of injectivity of Beltrami operators. The
case of the higher borderline exponent p = 1 + 1/kµk∞ reduces to Theorem 16.
Proposition 24 Suppose that p > 2 and f : C → C solves the equation
fz − µ1 fz − µ2 fz = 0, where k|µ1 | + |µ2 |k∞ ≤ 1/(p − 1). If both fz , fz are in Lp (C),
then f is a constant.
Proof. Since f satisfies the Beltrami equation fz −µfz = 0 with µ = µ1 +(fz /fz )µ2 ,
p
1
kµk∞ ≤ p−1
, the outer dilatation function of f is bounded by K = p−2
. As then
K
p = K−1 , Theorem 16 implies the claim.
Next for non-injectivity, the functions f in (44) established this for p < 1 + kµk∞
and I − µT . Similarly, if we let
−2k
g(z) = zχD + z|z| k+1 (1 − χD ),
(48)
D = {z : |z| < 1}, then gz , gz ∈ Lp (C) for p > 1 + k1 and (I − µ1 T )gz = 0 for
µ1 = −k zz χDc .
Combining these facts we conclude with the
Proof of Theorem 6. Theorem 16 and Proposition 24 show that I − µ1 T − µ2 T
is injective on Lp (C) if 1 + k < p ≤ 1 + 1/k, k = k|µ1 | + |µ2 |k∞ . The non-injectivity
part was established in (44) and (48).
Acknowledgements. The authors wish to thank Gaven Martin and Pekka Koskela for
helpful discussions on the topics of this paper.
25
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AFFILIATIONS:
K. ASTALA: Department of Mathematics, University of Jyvaskylä, FIN-00014 Jyvaskylä, Finland. E-mail : [email protected]
T. IWANIEC: Department of Mathematics, Syracuse University, Syracuse NY13244,
USA. E-mail : [email protected]
E. SAKSMAN: Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), FIN00014 University of Helsinki, Finland. E-mail : [email protected]
28