Planar Mappings of Finite Distortion

Computational Methods and Function TheoryVolume 10 (2010), No. 2, 663–678

Planar Mappings of Finite Distortion

Pekka Koskela

(Communicated by Peter Duren)

Abstract. We review recent results on planar mappings of finite distortion.This class of mappings contains all analytic functions and quasiconformal map-pings.

Keywords. Sobolev mapping, quasiconformal mapping, Jacobian.

2000 MSC. 26B10, 30C65, 28A5, 46E35.

1. Introduction

A mapping f : Ω → C, where Ω ⊂ C is a domain, is said to be a mappingof finite distortion if f belongs to the Sobolev space W 1,1

loc (Ω; C), the Jacobiandeterminant Jf = J(·, f) = det(Df) of f (Df being the differential matrix of f)is locally integrable in Ω, and there is a measurable function K(z) ≥ 1, finitealmost everywhere, such that f satisfies the distortion inequality

(1.1) |Df(z)|2 ≤ K(z)Jf (z), for almost every z in Ω.

The smallest K(z) ≥ 1 that satisfies the distortion inequality we denote by Kf ;that is,

(1.2) Kf (z) =

⎧⎨⎩|Df(z)|2J(z, f)

, if J(z, f) > 0

1, if J(z, f) = 0.

Above f ∈ W 1,1loc (Ω; C) simply means that both the real and imaginary parts

of f belong to the usual W 1,1loc (Ω): they and their first order distributional partial

derivatives are locally integrable. Switching to a suitable representative, one mayassume that f is absolutely continuous on almost all lines in Ω and thus one canas well consider the classical partial derivatives that exist almost everywhere.Also, we used the notation |A| to refer to the operator norm of a matrix A.

Received April 1, 2010, in revised form November 1, 2010.Published online November 17, 2010.Supported partially by the Academy of Finland grants no. 120927, 131477.

ISSN 1617-9447/$ 2.50 c© 2010 Heldermann Verlag

664 P. Koskela CMFT

When Kf ∈ L∞(Ω), we recover the class of quasiregular mappings, also calledmappings of bounded distortion. Recall that a quasiregular mapping always has acontinuous representative and that a continuous, injective quasiregular mappingis called quasiconformal. The simple example

f(z) =z

|z|(1 + |z|)shows that this need not be the case for a general mapping of finite distortion,even when Kf ∈ L1

loc(Ω). It is not hard to modify this example so as to checkthat the existence of a continuous representative may fail also when Kf ∈ Lp

loc(Ω)for all p < ∞.

With some more effort, one may reach exp(Kf log−2(e + Kf )) ∈ Lploc(Ω) for all

p < ∞ but not exp(Kf ) ∈ Lploc(Ω) for any p > 0, [38], [45]. In what follows, we

say that f has locally exponentially integrable distortion if f has finite distortionand exp(λKf ) is locally integrable for some λ > 0. The critical integrabilitycondition for the existence of a continuous representative can be identified asexp(ψ(Kf )) ∈ L1

loc(Ω) for some sufficiently regular increasing ψ with∫ ∞

1

t−1ψ′(t) dt = ∞.

Thus, one essentially needs the local exponential integrability of the distortionfunction in order to guarantee that a mapping of finite distortion has a continuousrepresentative. This degree of integrability turns out to be critical for many otherdesired properties as well.

This review paper is organized as follows. Section 2 deals with existence anduniqueness for the associated Beltrami equation. In Section 3 we deal with topo-logical properties, in Section 4 with the optimal regularity, and in Section 5 withthe optimal modulus of continuity. Section 6 deals with homeomorphic map-pings of finite distortion and Section 7 with dimension distortion. In Section 8,we discuss geometric properties, and Section 9 gives other possible definitionsfor mappings of finite distortion. Section 10 deals with boundary values. In thefinal section, Section 11, we point out some applications of the results and of themethods developed to study these mappings.

2. The Beltrami equation

The Beltrami equation∂f(z) = μ(z)∂f(z)

is of central importance in the study of planar quasiconformal mappings.

Let us recall the notation used above. We write z = x + iy, where x, y are realand f(z) = u(z) + iv(z) with u, v real-valued, and further

∂xf(z) = ux(z) + ivx(z),

∂yf(z) = uy(z) + ivy(z).

10 (2010), No. 2 Planar Mappings of Finite Distortion 665

Then

∂f(z) =1

2(∂xf(z) − i∂yf(z)),

∂f(z) =1

2(∂xf(z) + i∂yf(z)).

We conclude that the derivatives ∂f(z), ∂f(z) are defined if the partial derivativesat z exist, both for the real and imaginary parts of f . To be a solution for theequation for a given μ simply means that the pointwise equation should holdalmost everywhere.

If f is (real) differentiable at a point z, then the maximal directional derivativeof f at z corresponds to the argument

1

2

(arg ∂f(z) − arg ∂f(z)

)and has the size

|∂f(z)| + |∂f(z)|.Similarly, the minimal directional derivative of f at z corresponds to the argu-ment

π

2+

1

2(arg ∂f(z) − arg ∂f(z))

and has the size ∣∣|∂f(z)| − |∂f(z)|∣∣ .

Thus|Jf (z)| =

∣∣|∂f(z)|2 − |∂f(z)|2∣∣ .

If f furthermore satisfies the above Beltrami equation, we conclude that

|Df(z)|2 ≤ K(z)|Jf (z)|,where

K(z) =1 + |μ(z)||1 − |μ(z)|| .

Moreover, Jf (z) > 0 if |μ(z)| < 1.

The following existence theorem for the Beltrami equation is very powerful. Itfollows by combining results in [3, pp. 557, 565, 576]. Also see [39]. For pre-decessors of this result we refer to the historical discussion in [3]. Especially wewish to highlight the paper [11].

Theorem 2.1. Suppose that μ : C → B(0, 1) is such that exp(λK(z)) ∈ L1loc(C)

for some λ > 0. Then the Beltrami equation

∂f(z) = μ(z)∂f(z)

has a continuous, injective solution f : C → C of finite distortion. Moreover,any other solution g of finite distortion is of the form g(z) = h ◦ f(z) (almosteverywhere), where h : f(C) → C is holomorphic.

666 P. Koskela CMFT

Here K(z) is defined by the formula above; by the results of Menchoff [59](also see [16]), f is almost everywhere (real) differentiable. Notice that f is notclaimed to be onto. This holds if μ has compact support and, more generally, ifexp(λK(z))(1 + |z|)−4 is integrable.

The proof for the existence essentially consists of truncating μ to bound itsnorm away from one, solving the truncated equation that has a quasiconformalsolution, normalizing the solution suitably, and passing to the limit. The crucialpoint is to obtain an equicontinuous family, and thus one expects that existenceholds under somewhat weaker assumptions. This is indeed the case, see forexample [56, 7, 39, 21, 9]. For the factorization result, the size condition cannotbe essentially relaxed and the integrability condition is indeed essentially sharpif only the size of K(z) is taken into account; recall the existence of “cavitating”maps from the introduction.

Finally, let us point out that each mapping f of finite distortion is a solutionto a Beltrami equation obtained from the ∂ and ∂ derivatives of f . Recallingthat the necessary partial derivatives exist almost everywhere, one simply definesμ(z) = 0 if ∂f(z) = 0 and μ(z) = ∂f(z)/∂f(z) otherwise. The associated μ(z)satisfies

|μ(z)| ≤ Kf (z) − 1

Kf (z) + 1

(at the points of differentiability).

3. Topological properties

It immediately follows from the uniqueness result from the previous section thatmappings of locally exponentially integrable distortion are continuous, and ifnon-constant, both open and discrete. Here continuity naturally refers to the ex-istence of a continuous representative, meaning that one can modify our mappingin a set of measure zero so as to make it continuous. All this may fail if we relaxthe exponential integrability [45]. However, if one assumes a priori regularityfor the mapping in question, then one may substantially relax the integrabilitycondition.

Theorem 3.1. Suppose that f : Ω → C has finite distortion and |Df |2 is locallyintegrable. Then f is continuous. If Kf is locally integrable, and f is non-constant, then it is both discrete and open.

For this result we refer the reader to [40]. In fact, it was proven in [40] that(locally) f = h ◦ g, where g is a homeomorphism and h is holomorphic.

It is believed that the local integrability of Kf in Theorem 3.1 (while keeping theregularity assumption or even under local integrability of |Df |2 log−1(e + |Df |))can be relaxed, say, to the local integrability of Kf log−1(e + Kf ). For a preciseconjecture, see [3].


What then about limits of sequences of mappings of finite distortion? It shouldcome as no big surprise that suitably normalized families form compact families.For this see [37], [38].

4. Optimal regularity

Let f have locally exponentially integrable distortion in Ω. Then both exp(λKf )and Jf are locally integrable, for some λ > 0. Our a priori regularity assumption

on f is that f ∈ W 1,1loc (Ω; C), but one immediately concludes that f ∈ W 1,p

loc (Ω; R2)for all p < 2 using the distortion inequality

|Df(z)|2 ≤ Kf (z)Jf (z),

Holder inequality and the integrability properties of Kf and Jf . Using the ele-mentary inequalities

ab ≤ exp(a) + b log(e + b)

for a, b > 0 and |Jf (z)| ≤ |Df(z)|2, one easily improves this to

|Df |2 log−1(e + |Df |) ∈ L1loc(Ω).

On the other hand, the simple examples

f(z) =z

|z| log−s

(e +

1

|z|)

,

s > 0, for which exp(λKf ) ∈ L1loc(B(0, 1)) if and only if λ < 2s and

|Df |2 logt(e + |Df |) ∈ L1loc(B(0, 1))

if and only if t < 2s − 1, show that L2-derivatives cannot be hoped for.

The following theorem from [1] shows that the above simple examples are essen-tially critical for the regularity of mappings of locally exponentially integrabledistortion. For predecessors of this result see [11, 35, 36, 13].

Theorem 4.1. Suppose the function f : Ω → C has finite distortion and thatexp(λKf (z)) ∈ L1

loc(Ω) for some λ > 0. Then, for every β < λ,

|Df |2 logβ−1(e + |Df |) ∈ L1loc(Ω),

Jf logβ(e + Jf ) ∈ L1loc(Ω).

Theorem 4.1 gives us the optimal degree of integrability for Jf for mappings oflocally exponentially integrable distortion and one does not expect for any im-provement on the integrability of the Jacobian in terms of the distortion whenthe exponential integrability is substantially relaxed. See however [10, 19]. Onthe other hand, if f is a continuous, non-constant mapping of finite distortion,then already local integrability of Kf guarantees that Jf (z) > 0 almost every-where [46]. Thus one expects for some kind of an integrability result on J−1

f as

668 P. Koskela CMFT

soon as f is sufficiently regular and Kf is at least locally integrable. The follow-ing result that combines estimates from [11, 48] gives a rather optimal conclusion.Also see [31].

Theorem 4.2. Suppose that f : Ω → C is non-constant and has finite distortion.If f is homeomorphic and exp(λK) ∈ L1

loc(Ω), then

exp

(C

(λ log

(e +

1

Jf

))1/2)

∈ L1loc(Ω).

If |Df |2 ∈ L1loc(Ω) and Kf ∈ Lp

loc(Ω) for some 1 ≤ p < ∞, then

logp

(e +

1

Jf

)∈ L1

loc(Ω).

Regarding Theorem 4.1, one can also relax the a priori assumption Jf ∈ L1loc(Ω),

see [13]. Indeed, there is a constant C so that every mapping f ∈ W 1,1loc (Ω, C)

with |Df |2 log−Cλ−1(e+|Df |) for which exp(λKf ) ∈ L1loc(Ω) satisfies Jf ∈ L1

loc(Ω)and is hence a mapping of finite distortion. Here one assumes that Df(z) = 0almost everywhere in the set {z ∈ Ω: Jf (z) = 0} and Kf is then defined as inthe introduction. This conclusion gives information on removable singularities.It is an interesting open problem to determine the optimal value for C. This alsoapplies to Theorem 4.2.

5. Modulus of continuity

As pointed out earlier, mappings with locally exponentially integrable distortionare continuous. The optimal modulus of continuity turns out to be logarithmic,as one expects from the examples in Section 4. These examples also indicate theoptimal modulus of continuity for the inverse in the homeomorphic setting. Thefollowing result combines estimates from [33, 60].

Theorem 5.1. Suppose that the function f : Ω → C has finite distortion withexp(λKf ) ∈ L1

loc(Ω). Then f is locally uniformly continuous with a modulus of

continuity of the form log−λ/2(C/t). If f is injective, fix z0 ∈ Ω and some r with0 < r < d(z0, ∂Ω). Then for all |z − z0| ≤ r/6

|f(z) − f(z0)| ≥ d exp

(−C

λlog

(Ir

|z − z0|))

,

where C is an absolute constant,

d =1

2d

(f(z0), ∂f

(B

(z0,

r

3

))), and I2 =

1

πr2

∫B(z0,r)

exp(λKf ).


Above, the lower bound in the injective case holds for a fixed C for all λ andthere is another constant C ′ for which the estimate fails for all λ. The value ofthe optimal constant C is not known.

If f is additionally quasiconformal in a disk, then the modulus of continuity inthe first part of Theorem 5.1 can be improved on, see [69].

Theorem 5.2. Suppose that the function f : B(0, 2) → C has finite distortionwith exp(λKf ) ∈ L1(B(0, 2)) and that f is quasiconformal on B(0, 1). Then fis uniformly continuous in B(0, 1) with a modulus of continuity of the formlog−λ(C/t).

In Theorem 5.1 and in Theorem 5.2 we assume an (exponential) integrabilitycondition on the distortion of our mapping. Since each (exponentially) integrablefunction is always slightly better, depending on the function itself, integrablethan a priori assumed, it would seem plausible that, even though the powers ofthe logarithms in Theorem 5.1 and in Theorem 5.2 for f are sharp, one couldslightly improve the indicated moduli of continuity. Surprisingly, this is not thecase in the setting of Theorem 5.1 [8]. We would like to know if this extends to thesetting of Theorem 5.2 or if one could improve the given modulus of continuity.

Notice that the indicated modulus of continuity in Theorem 5.1 for the inverse ofa homeomorphism of exponentially integrable distortion is substantially betterthan that for the mapping itself. This indicates that one might have a logarithmicmodulus of continuity for the inverse mapping under a less restrictive integrabilitycondition on Kf . This turns out to be true [51].

Theorem 5.3. Suppose that an injective f : Ω → C has finite distortion withKf ∈ Lp

loc(Ω) for some 1 ≤ p < ∞. Then f−1 is locally uniformly continuous in

f(Ω) with a modulus of continuity of the form log−p/2(C/t).

6. Homeomorphisms of finite distortion

Recall from Section 5 that already local integrability of Kf for a homeomorphicmapping of finite distortion yields a modulus of continuity for the inverse map-ping. Thus it appears that the inverse mapping could be more regular than whatone first expects. The following result from [28] shows that this is indeed thecase.

Theorem 6.1. Let f : Ω → C be a continuous, injective mapping of finite dis-tortion. Then f−1 : f(Ω) → Ω is a mapping of finite distortion. Moreover, if Kf

is locally integrable, then f−1 ∈ W 1,2loc (f(Ω); C).

There is even more symmetry than indicated in the first part of Theorem 6.1.Indeed, let f ∈ W 1,1

loc (Ω; C) be continuous and injective. Then, by [59, 16], f isalmost everywhere (real) differentiable, and it follows that either Jf ≥ 0 almost

670 P. Koskela CMFT

everywhere or that Jf ≤ 0 almost everywhere. Assume that the former holds.

Then f−1 ∈ W 1,1loc (f(Ω); C) if and only if f has finite distortion. For this see

[28, 32]. This conclusion shows the naturality of the finite distortion condition.Somewhat surprisingly, the inverse of any homeomorphism of the class W 1,1

loc isof locally bounded variation, but one needs the finite distortion condition toguarantee that the distributional derivatives are locally integrable functions [30].

The statements of Theorem 6.1 indicate that integrability of Kpf for some p with

0 < p < 1 should give better than L1- but weaker than L2−derivatives for f−1.This hope turns out to be futile [28], but the following estimate from [28] givesthe optimal such result under a stronger a priori regularity assumption on f .

Theorem 6.2. Let 1 < r ≤ ∞, 1 < q < 2 and set

a =

⎧⎨⎩

(q − 1)r

r + q − 2, if 1 < r < ∞,

(q − 1), if r = ∞.

Suppose that f : Ω → C is a continuous, injective mapping of finite distortionsuch that f ∈ W 1,r

loc (Ω, C) and Ka ∈ L1loc(Ω). Then f−1 ∈ W 1,q

loc (f(Ω), C).

Notice that we have not given any estimate on the integrability degree of thedistortion of f−1 above. The optimal result under exponential integrability ofKf can be found in [18]. For further estimates see [17].

Theorem 6.3. Let f : Ω → C be a continuous, injective mapping of finite distor-tion. If exp(λKf ) is locally integrable, then Kf−1 ∈ Lp

loc(f(Ω)) for all 0 < p < λ.

Recall that local integrability of the distortion function for an injective mappingof finite distortion implies that the inverse has L2-derivatives. In fact, this esti-mate comes as an integral identity [29]. This means that minimization of Kf inL1 for injective mappings amounts to minimizing the Dirichlet energy for the in-verse mappings. In order to have uniqueness for minimizers with given boundaryvalues, it is convenient to use the mean Hilbert-Schmidt norm:

‖A‖ =1

2tr(AtA).

Suppose that f has finite distortion. Analogously to the definition of Kf wedefine Kf simply by replacing the operator norm |Df(z)| by the mean Hilbert-Schmidt norm ‖Df(z)‖. The following result is from [4].

Theorem 6.4. Let Ω be a bounded convex domain and let f : Ω → C be acontinuous, injective mapping of finite distortion with Kf ∈ L1(Ω). Assumethat f extends to a homeomorphism between the closures of Ω, f(Ω). Then theminimization problem ∫

Ω

Kg, g = f on ∂Ω


over all homeomorphic mappings g : Ω → f(Ω) of finite distortion has a uniquesolution g. This extremal mapping is a smooth diffeomorphism whose inverse isharmonic in f(Ω).

7. Dimension distortion

Quasiconformal mappings preserve the class of sets of area zero. This neednot be the case for general injective mappings of finite distortion. However,as mentioned in Section 4, already local integrability of the distortion functionguarantees the, almost everywhere, strict positivity of the Jacobian. With somework, one can check that this implies that sets of positive area get mapped tosets of positive area. In the other direction, local exponential integrability ofthe distortion function is again crucial: under this assumption, sets of area zeroget mapped to sets of area zero [43], and exponential integrability cannot besubstantially relaxed for this conclusion. Regarding quasiconformal mappings,actually even Hausdorff dimension distortion can be controlled: if f : C → C isK-quasiconformal and E has Hausdorff dimension 0 < s < ∞, then the Hausdorffdimension of f(E) is bounded away both from zero and two, in terms of s and K.The sharp estimates are obtained from the optimal degree of regularity of a K-quasiconformal mapping. The situation of mappings of finite distortion is harder,but one indeed obtains an optimal generalized upper bound using Theorem 4.1[53].

Theorem 7.1. Let f : Ω → C be a continuous mapping of finite distortion withexp(λKf ) locally integrable for some λ > 0. Set hs(t) = t2 logs(1/t) for s ∈ R.Suppose that E ⊂ C satisfies dimH(E) < 2. Then Hhs(f(E)) = 0 for all s < λ,where Hhs is the generalized Hausdorff measure associated to hs.

As pointed out in [53], the exponent s in Theorem 7.1 is optimal. Regarding lowerbounds, one could ask for an optimal gauge function h so that Hh(f(E)) > 0whenever the Hausdorff dimension of E is strictly positive and exp(λKf ) is locallyintegrable. See [70] for a partial result. There are many other natural questionsone could ask. In addition to Theorem 7.1 and the above result, the only furtherresults we know of can be found in [2, 63].

8. Geometric properties

One calls a Jordan domain Ω ⊂ C a quasidisk if it is the image of the unit diskunder a quasiconformal mapping f : C → C. If f is K-quasiconformal, we saythat Ω is a K-quasidisk. Another possibility is to require that f is additionallyconformal in the unit disk B(0, 1). It is essentially due to Kuhnau [55] that f isa K-quasidisk if and only if Ω is the image of B(0, 1) under a K2-quasiconformalmapping f of the entire plane that is conformal in B(0, 1), see [15].

672 P. Koskela CMFT

The situation is different for mappings of finite distortion. Let us consider themodel domain

(8.1) Ωs = {x + iy : 0 < x < 1, |y| < x1+s} ∪ B(xs, rs),

where xs = s + 2, rs =√

(s + 1)2 + 1, and s > 0. From [50, 52, 67] we have thefollowing results.

Theorem 8.1. Let f : C → C be a homeomorphism of finite distortion for whichf(B) = Ωs for some fixed s > 0. Assume that exp(λKf ) ∈ L1

loc(C) for someλ > 0. Then necessarily λ ≤ 2/s. Conversely, for every s > 0 there existsa homeomorphism f : C → C of finite distortion such that f(B) = Ωs andexp(λKf ) ∈ L1

loc(C) for all λ < 2/s. If f is additionally required to be qua-siconformal in B(0, 1), then the critical bound for λ is 1/s.

Notice the difference to the setting of quasiconformal mappings above: insteadof the switch from K to K2 under the additional conformality condition, oneessentially switches from λ to λ/2. One might expect this to be the case ingeneral, but this turns out not to hold [20]. For this, let us introduce a secondclass of model domains:

(8.2) Δs = B(x′s, r

′s) \ {x + iy : x > 0, |y| < x1+s},

where x′s = −s and r′s =

√(s + 1)2 + 1.

Theorem 8.2. Let s > 0. Given λ < 2/s, there is a homeomorphism f : C → C

of finite distortion with f(B(0, 1)) = Δs and exp(λKf ) ∈ L1loc(C). If f : C → C

is a homeomorphism of finite distortion so that f(B(0, 1)) = Δs and f is K-quasiconformal in B(0, 1), then Kf /∈ Lp

loc(C) for p > K/s.

Based on Theorem 8.2, one concludes that it is not possible to determine if aJordan domain is the image of the unit disk under a homeomorphism of theplane that has locally exponentially integrable distortion, simply studying theRiemann mapping function.

The above two theorems should be viewed as experiments towards trying tounderstand the geometry of the image of a disk under entire homeomorphisms oflocally exponentially integrable distortion. Even in these model cases, it would beworthwhile to examine the situation under other assumptions on the distortion,see [51] for results under assumptions on Kf−1 . Furthermore, our model domainsare very special, and one should study more examples.

9. Boundary values

The classical Caratheodory theorem for conformal mappings states that eachconformal mapping of the disk onto a Jordan domain extends to a homeomor-phism between the closures of the domains. This also holds under exponentialintegrability of the distortion function [54].


Theorem 9.1. Let f : B(0, 1) → Ω ⊂ C be a homeomorphism and such thatexp(λKf ) ∈ L1(B(0, 1)) for some λ > 0. Then Ω is a Jordan domain if and onlyif f extends to a homeomorphism between B(0, 1) and Ω.

Recalling the discussion from the introduction on continuous representatives, itshould come as no surprise that exponential integrability can only be slightlyrelaxed in Theorem 9.1. For further results on boundary extensions see [54, 58].

Given a (sense-preserving) homeomorphism f : S(0, 1) → S(0, 1) and some t with0 < t < π/2, set

(9.1) δf (θ, t) = max

{ |f(ei(θ+t)) − f(eiθ)||f(eiθ) − f(ei(θ−t))| ,

|f(eiθ) − f(ei(θ−t))||f(ei(θ+t)) − f(eiθ)|

}.

Then f has a quasiconformal extension to the entire plane if and only if δf (θ, t)is uniformly bounded both in θ and in t. The following result from [68], also see[66], gives a sufficient condition for a locally exponentially integrable distortionextension.

Theorem 9.2. Suppose that exp(β sup0<t<π/2 δf (θ, t)) ∈ L1loc(S(0, 1)) for some

β > 0 for a homeomorphism f : S(0, 1) → S(0, 1). Then f extends to a homeo-morphism g : C → C of locally exponentially integrable distortion.

The size condition on δf above can be replaced by existence of a constant C sothat δf (θ, t) ≤ C log(1/t) for all 0 ≤ θ < 2π and all 0 < t < π/2, see [68]. Neitherof these two size conditions are necessary, not even close. In fact, necessarily onlyδf (θ, t) ≤ exp(C log(1/t)).

Regarding self maps of the disk we have the following result from [4].

Theorem 9.3. Let h : S(0, 1) → S(0, 1) be a homeomorphism. Then f extends toa homeomorphism f : B(0, 1) → B(0, 1) of finite distortion with Kf ∈ L1(B(0, 1))if and only if ∫ 2π

0

∫ 2π

0

∣∣log |h(eiθ) − h(eit)|∣∣ dθ dt < ∞.

One further knows that the unique L1-minimizer f for Kf in Theorem 9.3 isquasiconformal if and only if h is bi-Lipschitz [4]. For more on the minimizationproblem see [57]. It is not known if Theorem 9.3 has analogues under otherintegrability conditions on Kf .

10. Alternate definitions

Quasiconformality also allows for a metric characterization. Given a homeomor-phism f : Ω → Ω′ and 0 < r < d(z, ∂Ω), set

Hf (z, r) =Lf (z, r)

lf (z, r),

674 P. Koskela CMFT

where

Lf (z, r) := sup{|f(z) − f(w)| : |z − w| ≤ r},lf (z, r) := inf{|f(z) − f(w)| : |z − w| ≥ r}.

Let

hf (z) = lim infr→0

Hf (z, r),

Hf (z) = lim supr→0

Hf (z, r).

Then f is K-quasiconformal if and only if hf (z) < ∞ outside a set of σ-finitelength and hf (z) ≤ K almost everywhere [14, 27, 41, 42]. Regarding mappingsof finite distortion, we have the following conclusion from [49].

Theorem 10.1. Let f : Ω → Ω′ be a homeomorphism. Suppose that hf (z) < ∞outside a set of σ-finite length and that hf ∈ L2

loc(Ω). Then f has finite distortionand Kf ∈ L2

loc(Ω). Moreover, if we replace hf with Hf , then L2loc(Ω) gets replaced

by L1loc(Ω).

In fact, one has Kf (z) ≤ hf (z) almost everywhere and thus Theorem 10.1 givesa sufficient metric condition also for f to have locally exponentially integrabledistortion (replace hf ∈ L2

loc(Ω) with exp(λhf ) ∈ L1loc(Ω)). It is not known if

one can relax L2loc(Ω) to L1

loc(Ω) in Theorem 10.1. This would follow if one couldprove almost everywhere (real) differentiability even in this setting. The abovedoes not lead to a metric characterization: σ-finiteness is necessary for the claimbut this may fail for a mapping of (locally exponentially) integrable distortion.

Besides of the analytic and metric definitions of quasiconformality, one also hasthe geometric definition. Towards this end, let us recall the definition of themodulus of a curve family.

A Borel function ρ : C → [0,∞] is said to be admissible for a path family Γ if∫γρ ds ≥ 1 for each γ ∈ Γ. We define the modulus mod(Γ) of Γ as

inf

{∫Ω

ρ2(z) dA :ρ : C → [0,∞] is an admissible Borel function for Γ

}.

By modKf (z)(Γ) we mean the Kf (z)-weighted modulus, where instead of the

integral∫

ρ2(z) dA we take the infimum over∫

ρ2(z)Kf (z) dA. Recall from The-orem 6.1 that the inverse of a homeomorphism of locally integrable distortionhas locally L2-integrable distributional derivatives. This together with standardarguments (see e.g. [47]) and results from [47] give the following invariance prop-erty.

Theorem 10.2. Let f : Ω → Ω′ be a homeomorphism of locally integrable distor-tion. Then mod(f(Γ)) ≤ modKf (z)(Γ) for each path family in Ω. Alternatively,this inequality holds if f is a continuous, non-constant mapping of locally expo-nentially integrable distortion.


In the non-injective case, one actually has better inequalities that take the mul-tiplicity of f into account, see [47].

The homeomorphic case in Theorem 10.2 allows for a converse: by [12, 65] everyhomeomorphism that satisfies the asserted modulus inequality for some locallyintegrable weight function K belongs to the Sobolev class W 1,1

loc (Ω, C). Conse-quently, from the almost everywhere (real) differentiability of Sobolev homeo-morphisms [59] (also see [16]) and usual modulus estimates one easily concludesthat f must be of finite distortion with Kf (z) ≤ K(z) almost everywhere. Thereis also an inequality in the opposite direction for homeomorphisms of locallyintegrable distortion. Indeed, one has the inequality

mod1/Kf (z)(Γ) ≤ mod(f(Γ)),

which easily follows from the definition. Here mod1/Kf (z)(Γ) is defined by replac-

ing∫

ρ2(z) dA with∫

ρ2(z)K−1f (z) dA.

Let us finally comment on so-called mappings of BMO-bounded distortion.These are mappings of finite distortion that satisfy Kf (z) ≤ M(z) for somefunction M of bounded mean oscillation. This is the same class as mappingswith locally exponentially integral distortion, see [3, pp. 545–546].

11. Applications

Instead of describing in detail the applications of mappings of finite distortionand the techniques developed for them to other fields, we simply list the areasand papers that we are aware of.

First of all, mappings of locally exponentially integrable distortion have beenapplied to complex dynamics, especially in connection with conjugation problems[22, 23, 24, 25, 62]. For applications to partial differential equations and non-Newtonian fluids, see [61, 3, 6]. The most recent applications are related toThurston earthquake maps [34] and to inverse problems [5].

References

1. K. Astala, J. Gill T., S. Rohde, E. Saksman, Optimal regularity for planar mappings offinite distortion, Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010), 1–19.

2. K. Astala, T. Iwaniec, P. Koskela, G. Martin, Mappings of BMO-bounded distortion, Math.Ann. 57 no.4 (2000), 703–726.

3. K. Astala, T. Iwaniec, G. Martin, Elliptic Partial Differential Equations and Quasiconfor-mal Mappings in the Plane, Princeton University Press, Princeton and Oxford, 2009.

4. K. Astala, T. Iwaniec, G. Martin, J. Onninen, Extremal mappings of finite distortion,Proc. London Math. Soc. (3) 91 no.3 (2005), 655–702.

5. K. Astala, M. Lassas, L. Paivarinta, The borderline between invisibility and visibility forCalderon’s inverse problem, in preparation.

6. M. Bildhauer, M. Fuchs, X. Zhong, A lemma on the higher integrability of functionswith applications to the regularity theory of two-dimensional generalized Newtonian fluids,Manuscripta Math. 116 no.2 (2005), 135–156.

676 P. Koskela CMFT

7. M. A. Brakalova, J. A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math. 76(1998), 67–92.

8. D. Campbell, S. Hencl, A note on mappings of finite distortion: Examples for the sharpmodulus of continuity, preprint.

9. Z. Chen, μ(z)-homeomorphisms in the plane, Michigan Math. J. 51 no.3 (2003), 547–556.10. A. Clop, P. Koskela, Orlicz Sobolev regularity of mappings with subexponentially inte-

grable distortion, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat.Appl. 20 no.4 (2009), 301–326.

11. G. David, Solutions de l’equation de Beltrami avec ‖μ‖∞ = 1, Ann. Acad. Sci. Fenn. Ser.A I, Math. 13 no.1 (1988), 25–70.

12. Fang Ainong, The ACL property of homeomorphisms under weak conditions, Acta Math.Sinica (N.S.) 14 no.4 (1998), 473–480.

13. D. Faraco, P. Koskela, X. Zhong, Mappings of finite distortion: The degree of regularity,Adv. Math. 190 no.2 (2005), 300–318.

14. F. W. Gehring, The definitions and exceptional sets for quasiconformal mappings, Ann.Acad. Sci. Fenn. Ser A I Math. 281 (1960), 1–28.

15. F. W. Gehring, K. Hag, Reflections on reflections in quasidisks. Papers on analysis, Rep.Univ. Jyvaskyla Dep. Math. Stat. 83 (2001), 81–90.

16. F. W. Gehring, O. Lehto, On the total differentiability of functions of a complex variable,Ann. Acad. Sci. Fenn. Ser. A I 272 (1959), 1–9.

17. F. Giannetti, A. Passarelli di Napoli, Bi-Sobolev mappings with differential matrices inOrlicz Zygmund classes, J. Math. Anal. Appl. 369 no.1 (2010), 346?-356.

18. J. Gill, Integrability of derivatives of inverses of maps of exponentially integrable distortionin the plane, J. Math. Anal. Appl. 352 (2009), 762–766.

19. , Planar maps of subexponential distortion, Ann. Acad. Sci. Fenn. Math. 35 (2010),1–2.

20. C. G. Guo, P. Koskela, J. Takkinen, Generalized quasidisks and conformality, preprint.21. V. Gutlyanskii, O. Martio, T. Sugawa, M. Vuorinen, On the Degenerate Beltrami Equation,

Trans. Amer. Math. Soc. 357 no.3 (2005), 7875–900.22. P. Haissinsky, Chirurgie parabolique, C. R. Acad. Sci. Paris Ser. I Math. 327 no.2 (1998),

195–198.23. , Chirurgie croisee, Bull. Soc. Math. France 128 no.4 (2000), 599–654.24. , Deformation J-equivalente de polynomes geometriquement finis, Fund. Math. 163

no.2 (2000), 131–141.25. P. Haissinsky, L. Tan, Convergence of pinching deformations and matings of geometrically

finite polynomials, Fund. Math. 181 no.2 (2004), 143–188.26. J. Heinonen, P. Koskela, Sobolev mappings with integrable dilatations, Arch. Rational

Mech. Anal. 125 no.1 (1993), 81–97.27. , Definitions of quasiconformality, Invent. Math. 120 no.1 (1995), 61–79.28. S. Hencl, P. Koskela, Regularity of the inverse of a planar Sobolev homeomorphism, Arch.

Ration. Mech. Anal. 180 no.1 (2006), 75–95.29. S. Hencl, P. Koskela, J. Onninen, A note on extremal mappings of finite distortion, Math.

Res. Lett. 12 no.2–3 (2005), 231–237.30. , Homeomorphisms of bounded variation, Arch. Ration. Mech. Anal. 186 no.3

(2007), 351–360.31. S. Hencl, P. Koskela, X. Zhong, Mappings of finite distortion: reverse inequalities for the

Jacobian, J. Geom. Anal. 17 no.2 (2007), 253–273.32. S. Hencl, G. Moscariello, A. Passarelli di Napoli, C. Sbordone, Bi-Sobolev mappings and

elliptic equations in the plane, J. Math. Anal. Appl. 355 no.1 (2009), 22–32.


33. D. A. Herron, P. Koskela, Mappings of finite distortion: gauge dimension of generalizedquasicircles, Illinois J. Math. 47 no.4 (2003), 1243–1259.

34. J. Hu, M. Su, Thurston unbounded earthquake maps, Ann. Acad. Sci. Fenn. Math. 32no.1 (2007), 125–139.

35. T. Iwaniec, P. Koskela, G. Martin, Mappings of BMO-distortion and Beltrami-type oper-ators. Dedicated to the memory of Tom Wolff, J. Anal. Math. 88 (2002), 337–381.

36. T. Iwaniec, P. Koskela, G. Martin, C. Sbordone, Mappings of finite distortion: LnlogχL-integrability, J. London Math. Soc. (2) 67 no.1 (2003), 123–136.

37. T. Iwaniec, P. Koskela, J. Onninen, Mappings of finite distortion: compactness, Ann. Acad.Sci. Fenn. Math. 27 no.2 (2002), 391?-417.

38. T. Iwaniec, G. Martin, Geometric function theory and nonlinear analysis, Oxford Mathe-matical Monographs, Clarendon Press, Oxford 2001.

39. , Beltrami equation, Mem. Amer. Math. Soc. 191 (2008), x+92 pp.40. T. Iwaniec,V. Sverak, On mappings with integrable dilatation, Proc. Amer. Math. Soc.

118 (1993), 181–188.41. S. Kallunki, P. Koskela, Exceptional sets for the definition of quasiconformality, Amer. J.

Math. 122 (2000), 735–743.42. , Metric definition of μ-homeomorphisms, Michigan Math. J. 51 (2003), 141–152.43. J. Kauhanen, P. Koskela, J. Maly, Mappings of finite distortion: Condition N, Michigan

Math. J. 49 (2001), 169–181.44. , Mappings of finite distortion: Discreteness and openness, Arch. Ration. Mech.

Anal. 160 (2001), 135–151.45. J. Kauhanen, P. Koskela, J. Maly, J. Onninen, X. Zhong, Mappings of finite distortion:

Sharp Orlicz-conditions, Rev. Mat. Iberoamericana 19 (2003), 857–872.46. P. Koskela, J. Maly, Mappings of finite distortion: the zero set of the Jacobian, J. Eur.

Math. Soc. 5 no.2 (2003), 95–105.47. P. Koskela, J. Onninen, Mappings of finite distortion: Capacity and modulus inequalities,

J. Reine Angew. Math. 599 (2006), 1–26.48. , Mappings of finite distortion: decay of the Jacobian in the plane, Adv. Calc. Var.

1 no.3 (2008), 309–321.49. P. Koskela, S. Rogovin, Metric definition of μ-homeomorphisms, Ann. Acad. Sci. Fenn.

Math. 30 (2005), 385–392.50. P. Koskela, J. Takkinen, Mappings of finite distortion: formation of cusps, Publ. Mat. 51

no.1 (2007), 223–242.51. , Mappings of finite distortion: formation of cusps III, Acta Math. Sin. 26 (2010),

223–242.52. , A note to “Mappings of finite distortion: formation of cusps II”, Conform. Geom.

Dyn. 14 (2010), 184–189.53. P. Koskela, A. Zapadinskaya, T. Zurcher, Mappings of finite distortion: generalized dimen-

sion distortion, J. Geom. Anal. 20 no.3 (2010), 690?-704.54. L. V. Kovalev, J. Onninen, Boundary values of mappings of finite distortion, Future trends

in geometric function theory, Rep. Univ. Jyvaskyla Dep. Math. Stat., Univ. Jyvaskyla,Jyvaskyla 92 (2003), 175–182.

55. R. Kuhnau, Moglichst konforme Spiegelung an einer Jordankurve, Jber. Deut. Math.-Verein 90 (1988), 90–109.

56. O. Lehto, Homeomorphisms with a given dilatation, Lecture Notes in Math. 118 (1970),58–73.

57. G. J. Martin, The Teichmuller problem from mean distortion, Ann. Acad. Sci. Fenn. Math.34 (2009), 233–247.

678 P. Koskela CMFT

58. O. Martio, V. Ryazanov, U. Srebro, E. Yabukov, On Q-homeomorphisms, Ann. Acad. Sci.Fenn. Math. 30 (2005), 49–69.

59. D. Menchoff, Sur les differentielles totales des fonctions univalents, Math. Ann. 105 (1931),78–85.

60. J. Onninen, X. Zhong, A note on mappings of finite distortion: the sharp modulus ofcontinuity, Michigan Math. J. 53 no.2 (2005), 329–335.

61. , Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm.Super. Pisa Cl. Sci. (5) 6 no.1 (2007), 103–116.

62. C. L. Petersen, S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegeldisk, Ann. of Math. (2) 159 no.1 (2004), 1–52.

63. T. Rajala, A. Zapadinskaya, T. Zurcher, Generalized dimension distortion under mappingsof sub-exponentially integrable distortion, preprint.

64. V. Ryazanov, U. Srebro, E. Yakubov, On ring solutions of Beltrami equations, J. Anal.Math. 96 (2005), 117–150.

65. R. Salimov, ACL and differentiability of Q-homeomorphisms, Ann. Acad. Sci. Fenn. Math.33 (2008), 195–301.

66. S. Sastry, Boundary behaviour of BMO-qc automorphisms, Israel J. Math. 129 (2002),373–380.

67. J. Takkinen, Mappings of finite distortion: formation of cusps II, Conform. Geom. Dyn.11 (2007), 207–218.

68. S. Zakeri, On boundary homeomorphisms of trans-quasiconformal maps of the disk, Ann.Acad. Sci. Fenn. Math. 33 no.1 (2008), 241–260.

69. A. Zapadinskaya, Modulus of continuity for quasiregular mappings with finite distortionextension, Ann. Acad. Sci. Fenn. Math. 33 no.2 (2008), 373–385.

70. , Generalized dimension compression under mappings of exponentially integrabledistortion, preprint.

Pekka Koskela E-mail: [email protected]: University of Jyvaskyla, Department of Mathematics and Statistics, P.O. Box 35(MaD), FIN-40014, Jyvaskyla, Finland.

Documents

Planar Mappings of Finite Distortion