Quasiconformal mappings in the plane

21

80

II. Distortion Theorems for Quasiconformal Mappings

conditions on cp such that there exists a quasiconformal mapping w : G ~ G' with the boundary values w(z) = cp(z).

6.2. A general boundary condition. In order to find a necessary condition, we suppose that the boundary value problem has a K-quasiconformal solution. If Q = G(Zl' Z2' za, Z4) denotes a quadrilateral which consists of G and the vertices Zi and Q' = G'(Wl> w2' w a, w4) is the image of Q under w, then by Lemma 1.5.1 we have

M(Q)jK < M(Q') < K M(Q) .

(6.1)

For every choice of the vertices Zi E C the image points Wi = cp(zJ must therefore lie on C' so that the double inequality (6.1) holds. The condition (6.1) assumes a weaker form, better suited to our applications, if we fix the vertex Z4 and choose the other three vertices such that Q is conformally equivalent to a square. Then M(Q) = 1, and so Q' must satisfy the condition

1jK < M(Q')

~

K .

(6.2)

On the other hand, we shall show in 6.5 that this necessary condition is also sufficient in the following sense: it guarantees the existence of a quasiconformal, though not necessarily K-quasiconformal solution. Thus the following basic result is true: Theorem 6.1. Let cp be a homeomorphism between the boundaries of the ] ordan domains G and G' which preserves the positive orientation, and Q = G(zv Z2' za, Z4) a quadrilateral. If there exists a K-quasiconformal mapping w : G ~ G' with the boundary values cp, then (6.1) holds for the quadrilateral Q' = G'(CP(Zl)' CP(Z2)' cp(za), CP(Z4))'

Conversely, if (6.2) holds for the family of quadrilaterals Q with unit module and fixed Z4' then there exists a quasiconformal mapping w : G ~ G' with the boundary values cp and with a maximal dilatation bounded above by a number depending only on K. In the proof of this theorem, which will be completed in 6.5, there is no loss of generality in supposing that G and G' are half-planes; this follows from the conformal invariance of the module and the boundary correspondence theorem quoted in 1.2.2. In this special case condition (6.2) can be given in an explicit form as we shall now show.

6.3. Necessary condition for the half-plane. Let G and G' be upper half-planes. To transform condition (6.2) to a simple form we set Z4 = 00 and normalize all our mappings so that they preserve the point at infinity.

81

§ 6. Boundary Values of a Quasiconformal Mapping

The quadrilateral Q = G(zl> Z2' Z3' (0) is conformally equivalent to a square if Z3 - Z2 = Z2 - Z1' We then write Z2 = X, Z1 = X - t, Z3 = X t, where t 0 since the orientation Zl' Z2' Z3 is positive with respect to G. For the module of the quadrilateral Q' = G'(wl> w2' w3 ' (0) we have by

>

+

1.2,

If we set this value into (6.2) we obtain for the boundary values of a K-quasiconformal mapping w: G --->- G', w( (0) = 00, the double

inequality 1/.-1

r

(nK) <= 1/V 2

+

w(x t) - w(x) w(x t) - w(x - t)

+

< 1/.-1 =r

(!!-) 2K

(6.3) .

To reduce this condition to a symmetric form we write 1

A(K)

=

(p,-l(n K/2))2 -

(6·11

1.

Then f-l (1/V1 +A(K))=:nKI2, and so f-l (1/V1

+ A(K)) f-l (1/V-1+-A-:--:-(1-IK---:C)) = :n /4. 2

By (2.7) this equation remains true if A(1IK) is replaced by 1j).(K). Thus (6.5) A(1IK) = 1/A(K) . Expressing f-l-l(:n K12) in (6.3) in terms of A(K) and applying (6.5) we obtain the following form of the first part of Theorem 6.1, in the special case of half-planes: Theorem 6.2. The boundary values of a K-quasiconformal self-mapping w of the upper half-plane preserving the point 00 satisfy the double inequality _1_

<

A(K) =

+

w(x t) - w(x) < w(x) - w(x _ t) =

A(K)

(6.6)

for all real x and t, t =1= O. The inequality (6.6) is sharp. This follows immediately from the fact that a square can be mapped K-quasiconformally onto any quadrilateral whose module is either K or 11K (d. the remark at the end of 1.3·4).

6.4. The distortion function A. Before dealing with the converse of Theorem 6.2 we give some estimations concerning the distortion function A, which will also occur in another connection in § 9.

82

II. Distortion Theorems for Quasiconformal Mappings

From (2.4) we obtain for the inverse of fl the functional equation fl

Because 1/V1

-1

(1 - V1-

(,u-l(X))2 ,u I(X)

_

(2 x) -

)2

.

+ A(K) = fl-1(n KI2), A satisfies the equation 1 + A(2 K) = (V1 + A(K) + VA(K)r

(6.7)

for every K. Since A(1) = 1 we obtain from (6.7)

A(2) = 16

vz.

+ 12

Repeated application of (6.7) yields an expression for A(2 n ) for every positive integer n. Using the estimations of fl(r) set out in 2.4 we can derive an approximation formula for the asymptotic behaviour of A(K) as K --+ 00. To this end insert r = 1/V1 A(K) in (2.13). Then fl(r) = Jt Kj2 and so

+ 2 (V1 + A(K) + VA(K)) where 0 < ~ < (1 + A(K))-3/ 2

= enK/2

+~ ,

(6.8)

•

We write 1 V,1 + A(K) = -4"e

nK 2 /

where t is to be estimated for K 2 (V1

+ e-

~

nK 2 /

+ A(K) + VA(K)) > e

3nK 2 / ,

(6.9)

1. Were t ~ 0 we could infer that

+ A(K) + VA(K))

This contradicts (6.8), and hence t

+ t e-

> o.

:::;; enK/2

.

It follows that

+ 4 t e~ < 64 eA comparison with (6.8) shows that t < 16. Since 0 < t < 16 we obtain from (6.9) the asymptotic expansion 2 (V1

nK 2 /

3nK 2 / ,

3nK 2 / .

(6.10) where the remainder is positive.

6.5. Solution of the boundary value problem. Following Beurling and Ahlfors [1J we now construct a quasiconformal self-mapping of the upper half-plane with given boundary values. By Theorem 6.2 the construction is: only possible if the boundary values satisfy a condition of form (6.6). The following theorem shows that this condition is also

83

§ 6. Boundary Values of a Quasiconformal Mapping

sufficient, although the maximal dilatation of the constructed mapping turns out to be larger than the value arising from (6.6). The result can be immediately generalized to the case dealt with in Theorem 6.1. We thus obtain the required proof for the sufficiency of condition (6.2). Theorem 6.3. Let cp be a function which is continuous and strictly increasing on the real axis and which satisfies the condition 1

-< e=

+

cp(x t) - cp(x) cp(x) - cp(x - t)


(6.11)

>

for every real x and t, t O. Then there exists a quasiconformal mapping w of the upper half-plane onto itself, which has the boundary values w(x) = cp(x) and a maximal dilatation less than a bound depending on (!. Proof: The function w= u

+iv =

~ (iX + (3) + ~i (iX - (3) = ~ (1 2 2 2

+ i) (iX -

i PR),

where t

t

iX(X, y) =

J cp(x + t y) dt ,

(3(x, y) =

o

J cp(x

- t y) dt,

(6.12)

o

is defined and continuous in the whole finite Z = x +i y-plane. Since cp is strictly increasing, w carries every point of the upper half-plane y 0 into the upper half-plane v O. On the real axis, w(x) = cp(x), and at conjugate points Z = x i y, Z = x - i Y we have w(z) = w(z).

>

>

+

We have to prove that w is a quasiconformal mapping of the finite plane. Then the image domain will also consist of the whole finite plane (d. the remark at the end of 1.8.1), and it follows from the above that w maps the upper half-plane onto itself and has the right boundary values cp(x). We show first that 1£i is a homeomorphism of the finite plane. It is enough to prove that w assumes every value at no more than one point. By Lemma 1.1.1 the continuity of the inverse w- l of w then follows from the one-to-one character and continuity of w. Let Zl = Xl + i Yl> Z2 = x 2 + i Y2 be two points such that W(Zl) = W(Z2)' It follows from the above that Zl and z2lie in the same half-plane bounded by y = O. Since w(z) = w(z) we may suppose that Yl and Y2 are positive.

84

II. Distortion Theorems for Quasiconformal Mappings

We write (6.12) in the form

<X(X, y) =

x

x+y ,.

1

J q;(~) d~ ,

y

Jq;(~) d~

f3(X, y) = :

(6.13)

.

x-y

x

+

We see that the mean values of q; are equal on the intervals (Xl' Xl YI) and (x2, x2 + Y2) as well as on (Xl - Yl> Xl) and (x 2 - Y2' x2). Since q; is strictly increasing, one of the first mentioned intervals must be contained in the other, and the same holds for the last two. If for example x 2 ~ Xl it follows that X2 Y2 ~ Xl YI and x 2 - Y2 < Xl - YI' This implies that Zl = Z2' and so the mapping W is a homeomorphism.

+

+

Finally we show that w is quasiconformal in the upper half-plane. From (6.13) it follows that iX and f3 are continuously differentiable and have the partial derivatives

iXx(X, y) = iXy(X, y)

1

Y (q;(x + y) 1

= -

y

(q;(x

+ y) -

1

f3Ax, y) =

y

iX(X, y)) , f3y(X, y) =

y

(q;(x - y) - f3(x, y)).

f3y

<0 .

- q;(x)) ,

1

(q;(x) - q;(x - y)) ,

Since q; is strictly increasing we have

<Xx> 0,

f3x

> 0,

> 0,

iX y

(6.14)

The Jacobian J = (iX y f3x - iXx f3 y)j2 of w is therefore positive. Hence all points of the upper half-plane are regular for w. To obtain an upper bound for the dilatation quotient D Iw,l)j(lwzl -lw,1) of w (d. 1.9.4) we write

+

Iw 12 + Iw-1z 2 D<2' = IWzl2 - IWzl2 tX:t

= -

{3x

tXx /{3:t

tX y {3y •

=

tX

2

x

it follows that

(Iw z\

+ tXy2 + {32x + {32y tX y

{3:t - tXx {3y

+ tX~/(tXx (3x) + {3x/tXx + {3~/({)I.x (3x) - (3x/{3y + tX,,/tXy

To estimate the partial derivatives of assumption (6.11). From

q;(X+: y)-q;(x)

=

iX

(6.15)

and f3 we make use of the

~e (q;(x+y)

-q;(x+:

Y))

85

§ 6. Boundary Values of a Quasiconformal Mapping

Between iXx and iX y we have therefore the double inequality 1

iXx(X, y)

> iXy(X, y) =

;

f

(rp(x

+ y)

- rp(x

+ t y)) dt

o

1( + y) -rp (1 )) <Xx(x, y) x+"2 Y ~ 2(1 +e)

~ 2y rp(x

.

(6.16)

In the same way we obtain

1((

1)

(

f3x(x,Y»-f3y(x'Y)~2Y rp x-"2 Y -rp x-y)

)2: 2(Jx(x, y) (1+e)'

(6.17

)

Finally it follows from (6.11) that

(6.18) Considering the estimations (6.16-18) we deduce from {6.15) that

D(z) < 8 I] (1

+ 1])2.

(6.19)

By Theorem 1.3.1, w is quasiconformal in the upper half-plane. To complete the proof, we conclude from symmetry property w(z) = w(z) that (6.19) holds in the lower half-plane also. By Theorem 1.8.3 w is therefore a quasiconformal mapping of the whole plane, with a maximal dilatation not exceeding 8 I] (1 + 1])2. Remark. Beuding and Ahlfors [1J showed that under condition (6.11) the boundary value problem has a solution with maximal dilatation :::;; e2 • Reed [1J has proved that the above mapping w is 8 e-quasiconformal.

6.6. Boundary value problem for a boundary component of a multiply connected domain. In the case of multiply connected domains we restrict ourselves to the problem of finding a mapping whose boundary values are given on a single boundary component. Let D be a domain having the Jordan curve C as a free boundary curve; C is then a boundary component of D (d. 1.1.9). Let G' be a Jordan domain and f a homeomorphism of C onto the boundary curve C' of G'. We shall investigate the conditions under which there exists a quasiconformal mapping w of D into G' (i.e. onto a subdomain of G') which has the boundary values w = f on C. The problem goes back to the above boundary value problem for Jordan domains. In one direction this is clear: If G denotes the domain bounded by C and containing D, and if w is a quasiconformal mapping of G onto G' with the boundary values w = f on C, then the restriction

86

II. Distortion Theorems for Quasiconformal Mappings

of w to D is the required solution. What is interesting is that the following converse is also true (see ViiisiiHi [2J): Theorem 6.4. Let D and D' be domains with the free boundary curves C and C', respectively, and w : D ~ D' a K-quasiconformal mapping which can be extended to a homeomorphism of Due onto D' U C'. Let G and G' be the components of the complement of C and C' which contain D and D', respectively. Then there exists a quasiconformal mapping of G onto G' which has the same boundary values as w on C and whose maximal dilatation is less than a bound depending only on K and the domain D. We remark that this theorem is closely connected with the problem of quasiconformal continuation to be dealt with in § 8. It states that the boundary problem is solvable for the whole domain G if there exists a solution in a neighbourhood of the boundary of G.

Proof of Theorem 6.4: By means of a conformal mapping the theorem can be reduced to the case when G is the unit disc and C the unit circle. Since C is a free boundary curve of D there exists a number r < 1 such that the set {zlr < Izi 1} is contained in D. Let ZI' Z2' Za and Z4 be four points on C which lie in such a manner that the module of the quadrilateral Ql = G(z}> Z2' Za, Z4) is equal to one. By Theorem 6.1 it remains to find an upper bound, depending only on rand K, for the module of the quadrilateral Q; = G'(w}> w2 , wa' w4), where Wi denotes the boundary value of w at Zi'

<

To do this we join the endpoints of the longer a-side of QI' say ZI and Z2' to the circle Izi = r by means of radial segments. They divide the annulus r Izl < 1 into two Jordan domains, of which one has Za and Z4 as boundary points. Denote by Q2 the quadrilateral consisting of this domain and the vertices ZI' Z2' Za, Z4 (Fig. 9).

<

Zz

Fig. 9

The image W(Q2) of Q2 has the same b-sides as Q;, and by the monotonicity theorem in 1.4.6 its module is greater than M(Q;). Since w i:3

87

§ 6. Boundary Values of a Quasiconformal Mapping

K-quasiconformal it follows that

M(Q;)

< K M(Q2)

.

(6.20)

To find an upper bound for M(Q2) we make use of Lemma 1.4.1. We denote by sa(Qi) and Sb(Qi) the distances ~tween the a- and b-sides of Qi both measured in Qi' i = 1, 2. Since Zl Z2 is the longer a-side of Q1' it follows that Sb(Q1) = IZ3 - z41 and

Hence, we obtain (6.21) Since the majorant (4.7) for M(Q) introduced in Lemma 1.4.1 is a decreasing function of salsb' it follows from (6.21) that

(6.22)

On the other hand, interchanging the roles of the a- and b-sides of Q1 we deduce from Lemma 1.4.1 that

or (6.23 ) Setting this estimate 22 in (6.22) we obtain an upper bound for M(Q2) which depends only on r. By (6.20) it follows that M(Q;) is bounded above by a number depending only on rand K. From Theorem 6.1 we infer the existence of the desired quasiconformal mapping of G having the same boundary values as w on C. 22 In view of later applications we remark that (6.23) hOlds for every quadrilateral with module one.

88

II. Distortion Theorems for Quasiconformal Mappings

§ 7. Quasisymmetric Functions 7.1. Definition of a quasisymmetric function. Let qJ be a continuous, strictly increasing function defined on the bounded or unbounded interval I of the (finite) x-axis. We call qJ k-quasisymmetric on I if there exists a positive constant k such that 1

-< k =

qJ(x

+ t)

- qJ(x)


=

qJ(x) - qJ(x -t)

(7.1)

>

+

tEl and t 0 . A function qJ is called quasisymmetric for x, x - t, x on I if qJ is k-quasisymmetric on I for some k. 23

By Theorems 6.2 and 6-3 a finite real function qJ is quasisymmetric on the real axis if and only if there exists a quasiconformal mapping of the upper half-plane onto itself with the boundary values qJ. It follows immediately that the inverses and compositions of functions quasisymmetric on the real axis are also quasisymmetric.

>

A linear polynomial qJ(x) = a x + b, a 0, is 1-quasisymmetric on every interval. Conversely, it is easy to see that every function which is 1-quasisymmetric on an interval is linear. The k-quasisymmetry of a function is obviously invariant with respect to 1-quasisymmetric mappings. In other words, if qJ is k-quasisymmetric on I and qJI' qJ2 are 1-quasisymmetric (thus linear) then the composition qJ2 qJ qJI is k-quasisymmetric on qJ;l(I). 0

0

The quasisymmetry of a function qJ on the whole x-axis cannot be characterized by the local properties of qJ, i.e. it is not enough to assume (7.1) only for small values of t. An example is provided by the exponential function, which is e-quasisymmetric on every segment of length 2 but not quasisymmetric on the whole x-axis. On the other hand, on bounded intervals the quasisymmetry of a function can be deduced from its local properties. In fact if (7.1) holds t ~ b, then we have for 0 t :s;:: 2 b, for 0

<

qJ(x + t) - qJ (x qJ (x

+ ~) -

<

+ ~) :s;:: k (qJ (x + :) qJ(x)

:s;:: k 2(qJ(X) - qJ (x -

qJ(x) )

~ k (qJ(X) -qJ (x - ~)):s;:: k2(qJ(X-~) -qJ(x- t)).

By addition we obtain

+

qJ(X t) - qJ(x) < qJ(x) - qJ(x - t) = 23

+)),

k2

The term quasisymmetric is due to Kelingos [1].

,

89

§ 7. Quasisymmetric Functions

<,l.lld a similar reasoning yields the lower bound 1/k 2• By repeating this process n times we arrive at the following result: If cp is continuous and strictly increasing on the x-axis and if (7.1) holds for 0 t < b, then cp is k2 "-quasisymmetric on every interval of length 2,,+1 b.

<

Remark. The results can also be expressed in the following form: If cpo is continuous and strictly increasing on the x-axis and (7.1) holds for 0 t < band t 2"b, where n is a positive integer, then cp is k 2 "_ quasisymmetric on the whole x-axis.

>

<

7.2. Extension of a quasisymmetric function. Apart from the boundary problem solved in 6.5, we are interested in the case when the boundary values are given only on a segment I of the real axis. By Theorem 6.3 a boundary problem of this type is solvable if the given boundary values can be extended to a function which is quasisymmetric on the whole x-axis. The following lemma states that this is always so if the boundary function is quasisymmetric on I.

Lemma 7.1. Every function cp which is k-quasisymmetric on an interval

I = {xla ::s x < b} can be extended to a K-quasisymmetric function on the whole x-axis. The constant K is less than a number depending only on k. Proof: Since k-quasisymmetry is invariant with respect to linear (polynomial) transformations there is no loss of generality in assuming that a = rp(a) = 0 and b = rp(b) = 1. We can then extend rp by means of the formulae rp(- x) = - rp(x) , (7.2) rp (x + 2) = rp(x) + 2 to a continuous strictly increasing function on the whole x-axis. We prove that this extension satisfies an inequality of form (7.1) for all x and t, t> 0, with a constant K in place of k. First, let t < 1/2. In view of (7.2) we may suppose that x - t < x t. Then x t ~ max (x, t - x) ~ (x t)/3 and

+ + 2 rp(x + t) ~ rp(x) - rp(x -

+

t) = rp(x)

+ rp(t -

x)

< 0 ::s x

~ rp (+ (x + t)). (7.3)

By hypothesis rp is k-quasisymmetric on the segment {xIO::;:: x Since rp(O) = 0 we therefore have

k~ rp (

~

++

+~ (rp (

::;:: k (rp

(~

(x

(x (x

< 1}.

t) )

+ t)) -

rp

+ t)) -

rp

(+ (+

(x

+ t))) ~ rp(x + t)

(x

+ t)))::;:: k

2

rp

- rp

(+ (x + t))

(+ (x + t)).

(7.4)

90

II. Distortion Theorems for Quasiconformal Mappings

It follows that fj?(x

+ t) ~ (1 + k + k

2

)

fj?

(+ (x + t)) .

(7.5)

Since

+ t) ~ fj?(x + t)

fj?(x

- q;(x)

> fj?(x + t) -

fj?

(~

(x

+ t))

we deduce from (7.3-5) that 1

2 k 2 (1

+ k + k2) ~

+

qJ(x t) - qJ(X) qJ(x) _ qJ(x _ t)

< 1

+ k + k2 .

This double inequality was proved under the hypothesis that t

(7.6)

< 1/2.

In the case t ::::::: 2 it follows from (7.2) that both fj?(x + t) - fj?(x) and fj?(x) - fj?(x - t) lie between t/3 and 3 t. Then we have 1

"9 <

+

qJ(x t) - qJ(x) qJ(x) - qJ(X - t)

< 9.

(7.7)

From the inequalities (7.6) and (7.7) and from the remark at the end of 7.1 we conclude that fj? is K-quasisymmetric on the x-axis with K - max (9 4 , 2 4 (k 2 k 3 k 4)4).

+ +

7.3. HOlder continuity of a quasisymmetric function. Let fj? be a k-quasisymmetric function on the x-axis. By Theorem 6.3 there exists a quasiconformal mapping w of the upper half-plane onto itself with the boundary values fj?(x). The mapping can be extended to the whole plane by reflection, and it follows from Theorem 4.3 (d. the remark at the end of 4.2) that w, and consequently fj?, are uniformly Holder continuous with respect to the spherical metric. In every compact subset of the finite plane the ratio of the euclidean and spherical distances is bounded. Since fj?( (0) = 00, fj? therefore satisfies a condition (7.8)

<

in every bounded interval I, where ql> 0 ql ~ 1, depends only on k, while the finite constant C1 also depends on I and the function fj? To estimate Ifj?(x) I for large values of Ixl we note that the inverse of fj? is also quasisymmetric and therefore Holder continuous with respect to the spherical metric. Since the spherical distance between x and 00 is of the same order as 1/[xl for large lxi, there corresponds to every r 0 a finite C2 such that

>

1

j;/ holds for Ixl only on k.

> r, where 0 < q2 <

C2

< IqJ(x)q'l

(7.9)

1. It is easy to see that q2 too depends

91

§ 7. Quasisymmetric Functions

If finally we take two real numbers x, y such that y belongs to a bounded interval I and x is arbitrary, then it follows from (7.8) and (7.9) that

Itp(x) - tp(y) I < C (Ix - y[q, where C depends on I and

+ Ix -

ylllq,) ,

(7.10)

tp.24

7.4. Approximation of a quasisymmetric function. As above let tp be a k-quasisymmetric function on the x-axis. For every positive integer n we introduce the integral 00

J cn tp(r) e- n(r-z)' dr ,

tpn(z) =

-00

where Cn

l

nr

= CIe- , drt =

V: .

It follows from inequality (7.10) that this integral converges uniformly in every compact set of the finite plane and defines an analytic function tp.. of the complex variable z = x i y.

+

The functions tp.. have the following properties: 1 0. The restriction of tp.. to the x-axis is k-quasisymmetric. For we have 00

+ t)

- tpn(x) = cn f (tp(x -00 for all real x and t. tpn(x

+ t + r)

- tp(x

+ r)) e- nr' dr

2°. The derivative of tp.. is positive at every point of the x-axis. Indeed for real x we have 00

tp:(x)

=

2n Cn f tp(r) (r - x) e- n(r-x)' dr -00 00

= 2 n cn J (tp(r) - tp(x)) (r - x) e-n(r-x)' dr. -00

The last expression here is positive since tp(x) is increasing.

30. As n ~ 00, tp..(x) converges uniformly to tp(x) on every bounded segment of the real axis. This assertion can be proved as follows: If x lies in a bounded interval I, then by (7.10)

Itp..(x) - tp(x) I = cn '11- 1 / 3

:::;; c..

J

_'11- 1/3

Itp(x

+ r) -

1_[ (tp(x + r) -

tp(x)) e- nr' drj 00

tp(x) I e-nr'dr

+ 2 cn C f

(r q, + r llq ,) e- nr' dr.

n- 1/3

24 Despite the Holder continuity a quasisymmetric function need not be absolutely continuous (d. the remark in IV.1.4).

92

II. Distortion Theorems for Quasiconformal Mappings

Since cp is uniformly continuous on every bounded interval the first integral on the right tends uniformly to zero as n -7 00. This is also true for the second integral' as we see from the following inequality, valid for q 0 and n (q(2)3,

>

>

00

00

J rq e- nr' dr = J e-nr'+qlogr dr n- 1D

n-1D

< JOO =

2 nr 2 n 2/ 3 -

~

q n 1/ 3

e- nr' + qlogr dr = n- q/3 . (2 n2/3 _ 2 n1/3)-1 e-n1/3 •

n- 1/3

Thus the functions CPn have all the properties 1 °- 30. This approximation to a quasisymmetric function by a sequence of analytic quasisymmetric functions will be applied in order to solve a problem on conformal mappings.

7.5. A sewing theorem for conformal mappings. We now look at a purely function-theoretical problem. It concerns mapping two Jordan domains with a common boundary curve conformally onto the upper and lower half-planes in such a way that a given relation holds between the boundary values of the mappings. The following theorem states that quasisymmetry of the relation is q sufficient condition; for a partial converse see 8.6. Sewing theorem. 25 Let cp be a quasisymmetric function on the real axis. Then the upper and lower half-planes can be mapped conformaUy onto disjoint Jordan domains such that the boundary values fl and f2 satisfy the relation fl(X) = Mcp(x)). Proof: By applying the approximation process of 7.4 we can carry out the proof in two steps. We first suppose that cp can be extended to an analytic function in the finite plane which has the properties 1° and 2° of 7.4. We then construct two conformal mappings fl and f2 of the upper and lower half-planes, respectively, which satisfy the condition Mx) = f2(cp(X)) on an arbitrarily given interval f. In the second step an arbitrary quasisymmetric function is approximated by analytic functions of the above type and the interval f tends to the whole x-axis. Thus let cp be an analytic function in the whole plane, having the properties 1° and 2°. There is no loss of generality in assuming that cp(f) = f. In fact, if we can find the required mapping in the case cp(I) =1= I, we 26 Lehto-Virtanen [1J, Pfluger [3J. The sewing theorem will be used in the proof of the so-called existence theorem in V.1, whereas Pfluger's proof depends on an application of the existence theorem.

93

§ 7. Quasisymmetric Functions

°

need only write ij;(z) = a ljJ(z) + b with a> and b chosen so that if(I) = I. If 11 and 12 satisfy the relation Mx) = Mif(x)) then flex) = h(ljJ(xl) where 72(Z) = Ma z + b). For every v = 0,1, 2, ... , let 51,. be the subdomain of the upper halfplane whose boundary consists of I and a circular arc which meets I at its end-points in angles (2/3 n. Let 52,. denote the reflection of 51,. in I. From some v = n onwards the domains 51,. will be mapped by IjJ conformally on a Jordan domain lying in the upper half-plane. If this were not so, there would exist in every 51, • either two points Zl' Z2 such that IjJ(Zl) = IjJ(Z2) or a point Z where the imaginary part of IjJ is nonpositive. These points would have limit points on I. This leads to

r

>°

a contradiction since IjJ maps I topologically and since 1jJ' (x) implies that every point x E I has a disc neighbourhood whose upper half IjJ maps conformally into the upper half-plane.

For v = n we now define two conformal mappings II,n, 12,n of 5 1,n and 5 2 ,n, respectively, by setting ,

h n(z) =

ljJ(z) ,

Then II,n maps 5 1,n onto a domain lying in the upper half-plane while 12,n maps the sub-domain 5 2,n of the lower half-plane onto itself. Thus the image domains are non-intersecting Jordan domains. It follows from the definition of II,n and 12,n that the boundary condition fl,n(x) = 12,n(ljJ(x)) is satisfied on i. If the above holds for n = 0, then 5 1,n coincides with the upper halfplane. In this special case the pair of functions 11,n> 12,n is thus a solution of our problem. The general case can be reduced to this in the following way:

We start from some value of v, v ~ 1, for which there exist two conformal mappings II,.: 51,. -+ 5;,.,/2,.: 5 2,.-+ 5;,. satisfying the above conditions, i.e. 5;,. and5;,. are disjoint Jordan domains and the boundary values of the mappings satisfy h.(x) = 12,.(IjJ(x)) on 1. If we can construct corresponding mappings of 51, .-1 and 5 2,'-1' repeating the process v times leads to the case where the domains considered are the upper and lower half-planes, respectively. Let T I ,. be the sub-domain of 51,. whose boundary consists of I and the circular arc CI ,. which meets I at its endpoints in an angle (3/4) (2/3)" n. Denote the reflections of T I ,. and CI ,. in I by T 2 ,. and C2 , .. respectively.

II. Distortion Theorems for Quasiconformal Mappings

94

We consider the sets D v = T l ,v U I U T 2,v and D~ = T;,v U I' U T~,v, where T;,v = li,v(Ti ) and I' denotes the Jordan arc 11,v(1) = 12,v(I). (Fig. 10). Then D; is a simply connected domain whose boundary consists of the Jordan arcs 11,v(Cl ,v) = C;,V, 12,v(C2,v) = C~,v and their common endpoints. The Jordan arc I' joins these endpoints. 26 ,//

-----

................

"-

/

;/ /

,),,11-'

'\ \

/

\

{

I

\

S1,1I

I

\

J

\

\

/

/

\

/ (

/

I

I \ \

,)2,11

I /

\

SZ,1I-1

\,

"-,

""'------

/

•

\ \

/

........ /

Fig. 10

Let I be a conformal mapping of D~ onto the unit disc. The composite functions hV-l = 1 0 11,v and 12,v-l = 1 0 12,v then map T l ,v and T 2,v conformally onto disjoint Jordan domains and satisfy the boundary condition 11, v-I (x) = 12, v-I ((cp(x)) on 1. The circular arcs Cl,v and C2,v are mapped onto the arcs I(C;,v) and I(C~,v) respectively, lying on the circumference of the unit disc. Hence 11, v-I and 12, v-I can be continued by reflection to conformal mappings of 51, v-I and 52, v-I' The image domains 11, v-I (51, V-I) and 12, v-I (52, V-I) are disjoint and their common boundary consists of I(I') and its reflection in the unit circle. The mappings 11, v-I and 12, v-I thus possess the required properties, and we have proved the theorem in the case when cp is an analytic function with the properties 1° and 2° listed in 7.4. Let us now consider the general case when cp is an arbitrary k-quasisymmetric function on the real axis. By 7.4 we can approximate cp by a sequence of analytic functions CPn' n = 1,2, ... , having the properties 1°_3°. By the first part of the proof, there exist for each n = 1, 2, . .. two conformal mappings In,1 and In,2 which map the upper half-plane HI 26

The change from the domains

S;,v U r U S~,v

Si,v

to the

need not be a Jordan domain.

Ti,v

is necessary, since the set

§ 7. Quasisymmetric Functions

95

and the lower half-plane H 2 onto disjoint Jordan domains Gn,1 and Gn,2, respectively, and satisfy In,I(X) = In,2(rp(X)) on the interval In = {xl- n S; X s; n}. Using a linear mapping we can ensure that In,I(O) = 0, In,I(1) = 1 and In,I(- 1) = - 1. We show that In,l and In,2 then converge to conformal mappings as n ---* (x). Since all functions rp.. are k-quasisymmetric on the real axis there exist by Theorem 6.3 K-quasiconformal self-mappings wn of HI with the boundary values rp.. (x), where K is a number depending on k. By reflection each w.. can be extended to a K-quasiconformal mapping of the plane. This will still be denoted by w n . The mapping W n' n = 1, 2, .... defined hy the formula

W .. (z)

lnl(Z) for zEHlUI.. , { In,2(W.. (Z)) for . Z E H 2 U In'

='

(7.11)

is a homeomorphism of the slit-domain H n = HI U In U H 2. By Theorem IX3 on the removability of analytic arcs it follows that W .. is K-quasiconformal. We now make use of the convergence theorems proved in § 5. The rnappings W.. are defined in every bounded domain G from some n = no onwards and satisfy the normalization conditions Wn(O) = 0, W n(1) = 1 and W n(- 1) = - 1. It follows from Theorem 5.1 that {Wnln 2 no} is a normal family in G. Hence, it contains a subsequence which converges uniformly to a limit function W in every bounded domain. By Theorem 5-3, W is K-quasiconformal. Since the point at infinity is removable (Theorem 1.8.1), W can be extended to a K-quasiconformal mapping of the whole plane. As regards the mappings wn and their inverses w;l, these converge on the real axis to rp and rp-\ respectively. From Theorem 5.1 we deduce therefore that the families {w n } and {W;I} are normal. By Theorem 5.3 we can choose a subsequence such that as n ---* (X) all three sequences W n , w n and W;I tend to K-quasiconformal mappings W, wand w- l of the plane, the first converging uniformly in every bounded domain and the last two uniformly in the whole plane. If we write W = 11> W 0 w- l = 12' then it follows from (7.11) that In,i, i = 1, 2 converge uniformly to Ii as n ---* (X) in every bounded subset of Hi' Thus the restictions of the functions 11 and 12 to HI and H 2 are conformal mappings which map HI and H 2 onto the disjoint Jordan domains W(Hl ) and W(H2). From lim rpn(x) = rp(x) it follows that

Il(X)

=

lim In,I(X) = lim In,2(rpn(X)) = 12(rp(X))

for every real x, and the sewing theorem is proved.

96

II. Distortion Theorems for Quasiconformal Mappings

§ S. Quasiconformal Continuation 8.1. Continuation to the exterior of a compact set. In this section we are concerned with extending a quasiconformal mapping w : G ~ G' to a quasiconformal mapping of a domain G1 containing G as a proper subset. Such an extension is called a quasiconformal continuation. The reflection principle discussed in 1.8.4 gives us an example of quasiconformal continuation. This method has the advantage that the maximal dilatation remains unchanged. Here we shall prove the existence of a quasiconformal continuation under much more general hypotheses but cannot determine the best possible bound for the maximal dilatation of the extended mapping. We begin with the following extension theorem which, apart from its intrinsic interest, is important in later applications. Theorem 8.1. Let Wo : G ~ G' be a K-quasiconformal mapping and F a compact subset of the domain G. Then there exists a quasiconformal mapping of the whole plane which coincides with Wo in F and whose maximal dilatation is bounded by a number depending only on K, G andF. Proof. We cover the given set F by finitely many closed discs lying in G. Let these discs be joined in G by Jordan arcs such that a connected compact set F 1 ( G is formed. The components of the complement of F 1 are simply connected domains (d. 1.1.9). We choose a finite number Ulf U 2 , ••. , Un of these domains such that their union covers the compact set - G. Finally, by Lemma 1.1.4, we can separate the closed set - G n U i from - U i by a closed polygon Ci for each i. Denote by D i the Jordan domain lying in U i and bounded by C i . The complement of U D i is a compact set F lying in G and containing F. It the theorem is proved for F instead of F, then it is automatically true for F, since all ways of choosing F depend on F and G only. Hence, there is no loss in generality in assuming from now on that the complement of F consists of n disjoint Jordan domains bounded by closed polygons. The required mapping is then obtained by applying Theorem 6.4 n times in the following way.

Since the boundary curves Ci of D i lie in G, each of the sets G U D i is a domain. As a first step in the proof we show that there exists a quasiconformal mapping of G U D 1 which coincides with Wo in F. To this end we note that the sets G n D 1 and G n (- D 1) are domains. This can be inferred from Lemma 1.1.3 in the same way as in 1.1.8.

97

§ 8. Quasiconformal Continuation

The polygon Cv which is a free boundary curve of G n Dv is mapped by W o onto a Jordan curve C;. Let denote the domain bounded by C; which contains the set W o (G D I). By Theorem 6.4 there exists a quasiconformal mapping WI : D I --->having the boundary values WI = W o on CI and a maximal dilatation K I which is less than a bound depending only on K and G D I.

n

D;

D;

n

The mapping

WI(Z) w(z) = { wo(z)

for for

Z E

D1

Z E (-

D I)

nG

is a homeomorphism of the domain G U D I and has a maximal dilatation at most max (K, K I ) in G U D I - CI . From Theorem 1.8.3 on the removability of analytic arcs it now follows that the maximal dilatation of w in G U D I is also less than the same bound. Since w coincides with W o in F, the first part of the proof is complete. Since the boundary of DIlies in F, F I = F U D I = F U D I is a compact subset of the domain G U D I and its complement consists of the domains D 2 , • • . ,Dn . In the same way as above we can construct a quasiconformal mapping of G U D I U D 2 which coincides with w in F I and therefore with W o in F. By repeating this method n times we obtain the required mapping of the whole plane.

8.2. Quasiconformal arcs and curves. By analogy with the concept of analytic arc we can define quasiconformal arcs and curves: A Jordan arc or a Jordan curve C is called quasiconformal if there exists a quasiconformal mapping of a domain G ) C which carries C into a line segment or a circle. The definition can be stated in a simpler form if C is compact. In fact, by Theorem 8.1 we have:

Every closed quasiconformal arc and every quasiconformal curve is the image of a line segment or a circle under a quasiconformal mapping of the whole plane. A subarc of a quasiconformal curve is obviously quasiconformal. Conversely let C be a closed quasiconformal arc and w a quasiconformal mapping of the plane which carries a line segment I onto C. If I lies on the line L, then C lies on the quasiconformal curve w(L), and we conclude: A closed Jordan arc is quasiconformal if and only if it is a subarc of a quasiconformal curve. All analytic arcs and curves are quasiconformal by definition. We give some further examples in 8.10.

98

II. Distortion Theorems for Quasiconformal Mappings

8.3. Continuation over a boundary arc. Quasiconformal arcs and curves play an important role in the study of the conditions under which a quasiconformal mapping can be continued across the boundary of a given domain. Theorem 8.2. Let G and G' be two domains having C and C' as either Iree boundary arcs or Iree boundary curves. Further let w : G ->- G' be a quasiconlormal mapping under which C and C' correspond. II C and C' are quasiconlormal, then w can be continued to a quasiconlormal mapping 01 a domain Gl containing G U c.

Prool: Let I and g be quasiconformal mappings of certain neighbourhoods U and U' of C and C', respectively, such that C and C' are carried onto a line segment or a circle. It follows from the discussion on free boundary arcs in 1.1.9 that there exists in I(U) a domain D which is symmetric with respect to I(C), contains I(C) and is divided by I(C) into two parts D l and D 2 such that l-l(D]) lies in G and l-l(D 2 ) in - G. Let D', D; and D; be the corresponding sets for the mapping g. We can choose D so small that w(t-llDl )) is contained in g-l(D;). The reflection principle can then be applied to the mapping g w 01- 1 in D. It follows that w can be extended to a quasiconformal mapping of the domain Gl = G U l-l(D) and the theorem is proved. 0

Let us apply Theorem 8.2 to the special case when G and G' are n-tuply connected domains with quasiconformal boundary curves. By 1.1.8 the boundary curves correspond pairwise under the mapping w. According to Theorem 8.2 w can therefore be extended to a quasiconformal mapping of a domain Gl which contains the closure of G. If we combine this result with Theorem 8.1 we deduce (Springer [lJ): Theorem 8.3. Let G and G' be two n-tuply connected domains whose boundary curves are quasiconlormal. Then every quasiconlormal mapping w : G ->- G' can be extended to a quasiconlormal mapping 01 the whole plane.

8.4. Quasiconformal reflection. The property of a Jordan curve of being quasiconformal can also be characterized in a simple way with the help of the anti-quasiconformal mappings mentioned in 1.3.2, which are composed of a quasiconformal mapping and a reflection. Let C be a Jordan curve bounding the domains Gl and G2 • An anti-quasiconformal mapping of Gl onto G2 which leaves the points of C invariant is called a quasiconlormal rellection in C (Ahlfors [3]). If C is quasiconformal there exists a quasiconformal mapping w of the plane which carries Gl onto the upper half-plane HI' Then the mapping C/J :Gl ->-G 2 , C/J(z) = w-l(w(z)f, is a quasiconformal reflection in C.

99

§ 8. Quasiconformal Continuation

If, conversely, C permits a quasiconformal reflection (/J, then, starting with a conformal mapping f : HI ---+ G1 , we define a mapping w by the

formulae w(z) = f(z) for z E HI' w(z) = (/J(t(z)) for Z (f HI' Then w is a quasiconformal mapping of the plane which maps the real axis onto C, and C is therefore quasiconformal. Thus we have the following result: A Jordan curve admits a quasiconformal reflection if and only if it is quasiconformal.

8.5. Characterization of quasiconformal curves by module conditions. Let C be a Jordan curve bounding the domains G1 and G2 • We choose four arbitrary points Zi on C such that the sequence Zv Z2' zs, Z4 defines the positive orientation with respect to G1 . The quadrilaterals QI = G1(Zl' Z2' zs, Z4) and Q2 = G2(Z4' zs, Z2' Zl) are called coniugate with respect to C. If C is quasiconformal there exists a quasiconformal mapping w of the whole plane which carries C into the real axis. The images W(QI) and W(Q2) of the above defined quadrilaterals are placed symmetrically with respect to the real axis and their canonical mappings are reflections of one another. Hence M(W(Ql)) = M(W(Q2))' Since w is quasiconformal it follows that the quotient M(Q2)/M(Ql) has a fixed upper bound.

This necessary condition is also sufficient: Theorem 8.4,27 A Jordan curve Cis quasiconformal if and only if for all coniugate quadrilaterals Qv Q2 with M(QI) = 1, the modttle of Q2 has a fixed upper bound.

Proof: The necessity follows from the above argument. To prove the sufficiency we denote by G~ the reflection of G2 in the real axis. It follows from the hypothesis that the correspondence Z ---+ z between the boundaries of G1 and G~ satisfies a condition of type (6.2). Hence there exists a quasiconformal mapping w: G1 ---+ Gf with boundary values w(z) = Z. Then the mapping is a quasiconformal reflection in C and the assertion follows from the result in 8.4.

w

8.6. Characterization of a quasiconformal curve by means of conformal mappin~s. Here we remind the reader of the function-theoretical sewing theorem proved in 7.5. It follows from the next theorem (Tienari [1]) that the sufficient condition given in the sewing theorem is also necessary under the additional assumption that the image of the real axis is a quasiconformal curve. As in 7.5 let fl : HI ---+ GI and f2: H 2 ---+ G2 be conformal mappings with the same value at infinity. 27

cf. Pfluger

[3J.

100

II. Distortion Theorems for Quasiconformal Mappings

Theorem' 8.5. The Jordan curve C is quasiconlormal il and only il 011 is quasisymmetric on the real axis.

1";1

Prool: If C is quasiconformal then by Theorem 8.}, 12 can be quasiconformally continued throughout the plane. Thus there exists a quasiconformal mapping WI: HI ~ G1 with the boundary values 12' The composite function W;-I 011 is then a quasiconformal mapping of the upper half-plane onto itself which preserves the point at infinity and has the boundary values 1;1 011' Hence 1;1 011 is quasisymmetric on the real axis. Conversely, if 1;1 011 is quasisymmetric on the real axis, there exists a quasiconformal mapping WI: HI ~ HI with the boundary values 1;1 011' Hence 11 W;-I has the same boundary values on the real axis as 12' and these functions define a homeomorphism of the plane. Since an analytic curve is removable, this mapping is quasiconformal. It maps the real axis onto C, which is therefore quasiconformal. 0

8.7. Metric characterization of quasiconformalcurves. Let us consider a Jordan curve C and two finite points z1' Z2 on it. They divide C into two arcs, and we consider the one with the smaller euclidean diameter. If the quotient of this diameter and the distance IZ1 - z21 is bounded for all finite points Zl' Z2 on C, we say that C is of bounded turning. This geometrically intuitive property turns out to be equivalent to the quasiconformality of C (d. Ahlfors 0]): Theorem 8.6. A Jordan curve isquasiconlormal il and only il it is 01 bounded turning. Prool: We suppose first that C is of bounded turning. The proof that it is then quasiconformal depends on Theorem 8.4. Let G1 and G2 be the domains bounded by C and Q1 = G1(z1' Z2' Z3' Z4) a quadrilateral of module one. We shall derive an upper bound for the module of the conjugate quadrilateral Q2 = G2(Z4' Z3' Z2' Zl)' We denote by sa and Sb the distances between the a- and b-sides of Q1 measured in Q1 (d. 1.4.3), and by da and db the corresponding distances in the plane. Then clearly Sb ~ db' Since M(Q1) = 1 the inequality (6.23) of 6.6 holds so that salsb 10- 3 • Hence

>

sa> 10- 3 db'

(8.1)

Since C is of bounded turning there exists a number h with the following property: If C is divided into two arcs by any two points z1' Z2 then at least one of these arcs lies inside a circle of diameter :::;: h IZ1 - z21. In particular, a b-side of Q1 (and of Q2) lies inside a circle of diameter h da .

101

§ 8. Quasiconformal Continuation

This fact, together with (8.1), yields (8.2) For if not, one b-side lies inside a circle of diameter ~ 10- 3 db/'ll:. The other b-side, which is at a distance db from this one, must lie outside the circle. It follows that a-sides can be joined in QI by a circular arc of length ~ 10- 3 db' This contradicts (8.1), and (8.2) is true. The module of Q2 can now be estimated by means of Theorem 1.4.1First we obtain as above a disc Iz - zol h db /2 which contains an a-side of Q2' We define

<

for

1

Iz-zol<-h 2

d. 10 -3 db b+ -h' n

elsewhere. It follows from (8.2) that the e-length of an arc joining the a-sides of Q2 is at least 10- 3 db/(n h). Hence by Theorem 1.4.1 we have

M(Q) < 2

=

n (h db/2

+

10-6

10-

3

2 db/(n h))2 _ 6 (n h - 10 n 2

d~/(n h)2

+ 10

_3)2

The quasiconformality of C follows from Theorem 8.4. Now let us suppose that Cis quasiconformal. Then for some K, there is a K-quasiconformal mapping w of the plane which maps C onto the real axis. To prove that C is of bounded turning we choose two finite points which divide C into the subarcs CI , C2 . We show the existence of a number h, independent of Zl and Z2' such that one of the arcs Ci lies inside a circle of diameter h IZI - z21. Zl' Z2

Let Pi be the point of Ci which is at the maximum distance from Zl' There is no loss of generality in assuming that the distances IPI - zll and IP2 - zll are greater than IZ2 - zll, since otherwise the assertion holds trivially for every h 2. If d = min (IPI - zll, IP2 - Zll) we can then construct the annulus B = {z I d> Iz - zll IZ2 - zll}; this has the module

>

M(B)

>

=

log IZ2

d -

Zl

(8·3)

I

z;

The image domain B' = weB) separates the points W(ZI) , = W(Z2) from the points P; = W(PI)' P; = W(P2)' All these points lie on the real axis and we can normalize w so that = 0 P; and P; =""00.

z;

z;

< < z;

102

II. Distortion Theorems for Quasiconformal Mappings

Thus B' separates 0, z~ from p;, 00. Its module can therefore be estimated by Teichmiiller's module theorem (d. 1-3). This gives z~

Since ft is decreasing M(B')

z~ ) + p~

.

< 2 ft (1 tV2) and it follows from (2.8) that M(B') < n. (8.4)

On the other hand, M(B) < K M(B'), since w is K-quasiconformal. By (8.3) and (8.4) we thus have

d

< e"'K IZ2 -

zll .

Hence, if h = 2 e"'K, one of the arcs Ci lies inside a circle of diameter h IZI - z21, and the second part of the theorem is proved.

8.8. Turnin~ in the spherical metric. The property of a Jordan curve of being of bounded turning can also be expressed in the spherical metric. Of the arcs into which the curve C is divided by two (finite or infinite) points Zv Z2 we consider the one having the smaller spherical diameter. Then the following result is true: C is at bounded turning it and only it the quotient at this diameter and the spherical distance between ZI and Z2 is bounded tor all points Zv Z2 E C. This follows from Theorem 8.6: By a linear mapping which corresponds to a rotation of the Riemann sphere the curve C can always be transformed so as not to contain the point at infinity. Such a rotation preserves the spherical distances and the quasiconformality of a curve. Thus the curve C is quasiconformal and hence of bounded turning if and only if it satisfies the above spherical condition. For a Jordan arc it is convenient to give the definition in the spherical metric: We say that the arc C is of bounded turning if the ratio of the ~

.

spherical diameter of its subarc ZI Z2 to the spherical distance between ZI and Z2 is bounded for all pairs ZI' Z2 E C. If C is bounded, spherical distances can be replaced by euclidean ones,

and we obtain the following result:

A bounded Jordan arc is at bounded turning it and only it there exists a finite number h such that

(8.5) tor all points

Zv Z2' Z3

lying in this order on C.

§ 8. Quasiconformal Continuation

103

8.9. Local characterization of a quasiconformal curve. Let C be a curve or an arc of bounded turning and h 0 a number such that the inequality

>

.---..

Ll (zl'

Z2)

<

h

(8.6)

k(Z1' Z2)

.---..

holds for every pair of points

ZI' Z2

.---..

on C, where Ll (zl'

Z2)

denotes the

spherical diameter of a subarc Z1' Z2 of C bounded by Z1 and Z228. Since Ll < nJ2 in any case, (8.6) is automatically true if k(ZI' Z2) ::::: nJ2 h. To express more clearly the local character of the quasiconformality of a curve, we first note: Every closed subarc C1 of a quasiconformal curve C is of bounded turning. In fact, if

Z1

z~ of C. If

and

Z2

lie on CI , then (8.6) holds at least for one subarc

<

k(zl' Z2) Ll (C - C1)/h, this arc must belong to C1 . By the above remark, (8.6) then holds for subarcs of C1 for some h.

By 8.2 every closed quasiconformal arc is a subarc of a quasiconformal curve. We thus infer: A closed quasiconformal arc is of bounded turning. The converse is also true as has been proved by Rickman [1]. We restrict ourselves to the following result which is easier to establish: I! an open Jordan arc C is of bounded turning then its closed subarcs are quasiconformal. To prove this we may suppose that C lies in the finite plane. Every closed subarc C1 of C lies in a closed subarc C2 of C whose endpoints PI' P2 are at a positive distance 2 r from C1 • Since C2 does not separate the plane (1.1.3), we can choose a point qi in each of the discs Iz - Pil r, i = 1, 2, and join these points by a polygonal arc lying at a positive distance from C2. If we also join q1 and q2 with C2 by shortest possible line segments then (by discarding certain loops if necessary) we obtain a closed curve" C3 which contains C1 as a subarc. This curve C3 is of bounded turning. For if the points Z1' Z2 both lie on C2 n C3 or on C3 - C2 , then the required condition is obviously satisfied. In the other cases its validity follows from the fact that the qi are joined to C2 by the shortest possible line segments. Thus by Theorem 8.6, C3 is quasiconformal and hence so also is its subarc C1 •

<

The proposed local characterization of the quasiconformality of a Jordan curve runs as follows: Theorem 8.7. A Jordan curve C is quas~conformal if and only if every point of C belongs to an open subarc of C of bounded turning. 28

The smaller of two such arcs if C is a curve.

104

II. Distortion Theorems for Quasiconformal Mappings

ProOf: If C is quasiconformal, every closed subarc of C is of bounded turning as we have just seen. The same is true for open subarcs of C which are contained in some closed subarc of C. This means that the condition given in the theorem is necessary. We now suppose that every point of C belongs to an open subarc of bounded turning. Then C can be covered by finitely many sub arcs C1> ... , Cn of this type. Let Z1> Z2 be two points of C. If they lie on the same arc Ci then by hypothesis there exists an h satisfying (8.6). If Zl and Z2 do not belong to the same Ci then k(Zl' Z2) is greater than a positive bound r, and (8.6) holds for h = n/2 r. Hence C is of bounded turning and so quasiconformal by Theorem 8.6.

8.10. Examples of quasiconformal arcs. From condition (8.5) we can deduce simple sufficient criteria for the quasiconformality of a Jordan arc C. 29 If for example there is a homeomorphism cp of the unit interval 0 < t < 1 onto C, satisfying a Lipschitz condition m It1

-

t2 1 < Icp(t1 )

-

cp(t2) I :::;; M It1

-

t2 1

,

then all compact subarcs of Care quasiconformal. In particular, every smooth closed Jordan arc is quasiconformal. The same holds if C consists of finitely many smooth arcs which meet pairwise at non-zero angles. If the arc C has a zero angle at some point, it is not quasiconformal, for the condition (8.5) cannot hold in any neighbourhood of this point.

~/\'----

Fig. 11

On the other hand there exist quasiconformal arcs which are not rectifiable. Without giving a precise proof, we have shown two methods of construction in Fig. 11 which by unlimited repetition lead to non-rectifiable 29

See also Pfluger [3J, Tienari [1J, VaisaIa [2].

105

§ 9. Circular Dilatation

quasiconformal arcs. 30 The first of these does not even contain any rectifiable subarcs. Once it was shown that quasiconformal arcs can be non-rectifiable, the question arose about their Hausdorff dimension (see IH.1.l). This problem has recently been solved by Gehring and ViiisiiHi [2]. They proved that while the Hausdorff dimension of a quasiconformal arc is always < 2 it can take any value A, 1
§ 9. Circular Dilatation 9.1. Definition of the circular dilatation. In 1.9.4 we defined the dilatation quotient D of a homeomorphism w at every regular point of w. The dilatation quotient admits a differential geometric interpretation as the ratio of the axes of the image ellipse of an infinitesimal disc. At a (not necessarily regular) point z it is therefore natural to measure the local dilatation of w not only by the dilatation of quadrilaterals in the neighbourhood of z but also by the distortion of the images of small circles with centre z. For this purpose we write maxe< Iw (z + rei") - w(z) I · . ..~-H (z) = H(z) = 11m sup w r~ +0 mine< Iw (z + rei") - w(z) I

for Z =1= 00, w(z) =1= 00, and extend the definition by the formulae Hw(oo) = Hr;/O) where w(z) = w(1fz), and Hw(z) = H 1/ W (z) for w(z) = 00. The function H is called the circular dilatation of the topological mapping w. If z =1= 00 is a regular point and w(z) =1= 00, then

w (z

+ rei")

- w(z)

= r (wz(z) ei" + wz(z) e- i") =

r ~i" o",w(z)

+ oCr)

+ oCr) .

It follows that H(z) = D(z) in this case. Because of the invariance of H with respect to linear mappings this relation also holds if z = 00 or w(z) = 00. In other words the dilatation quotient D is the restriction ot the circular dilatation H to regular points.

9.2. Upper bounds for the circular dilatation. By Theorem 1.9.4, for a regular quasiconformal mapping we have F(z) = D(z) = H(z) at every point of its domain. We now examine the mutual relationship between F and H at non-regular points. 30

The idea of such direct constructions is due to G. Piranian.

106

II. Distortion Theorems for Quasiconformal Mappings

For the homeomorphism w, -~+iargz

w(z) = e

Izi

,

of the finite plane onto the unit disc we have H(O) = 1, F(O) = Thus we cannot infer that F(z) is finite if H(z) is.

00.

However, in the opposite direction the following result holds (Mori [2J) : Theorem 9.1. For a K-quasiconformal mapping, H zs bounded by a number depending only on K. Proof: If we do not attempt to find the best possible bound for H, the theorem can be proved as follows.

Let w: G ~ G' be a K-quasiconformal mapping; we may assume here that G and G' are domains in the finite plane. For Zo E G,

Dr = {z/ /z - zol

m1 (r) = .

< r}, Dr C G, we write m2 (r) =

max /w(z) - w(zo)/ , Iz-z,1 = r

min Iw(z) - w(zo) I Iz-z,1

~

r

and denote by Zl and Z2 the points at which the maximum and minimum are attained. Choose the number r so small that the disc Iw - w(zo) I < m1 (r) is contained in G'.

<

<

We consider the annulus B' = {w I m 2 (r} /w - w(zo) I m1(r)} C G'. Its preimage B is a ring domain which separates the points zo, Z2 from Zl' 00. Since IZI - zol = IZ2 - zol we have M(B) S; 2 It

(~ )

=

7l

by Teichmiiller's module theorem (see 1.3) and by the formula (2.8). On the other hand, M(B') It follows that

=

log (m 1(r)jm 2 (r)) and M(B') ~ K M(B).

(9.1) Hence also H(zo} ~ e

nK

•

9.3. Lower bound for the circular dilatation. For a K-quasiconformal mapping, H(z) can in fact be greater than K. We make use of the distortion function ),(K), introduced in 6.3, which has the asymptotic expansion 1

A(K) = - enK 16

1 ---,- -

2

and prove (Lehto-Virtanen-VaisaHi [1]):

+ O(e-

nK

)

§ 9. Circular Dilatation

107

Theorem 9.2. For every E lor which

> 0 there exists a K-quasiconlormal mapping H(z)

~

A(K) -

E

at some point z. Prool: By 2.2 and 6.3 the module of the quadrilateral consisting of the upper half-plane and the vertices - A(K), 0, 1, 00 is

~ p ( V).(K: +

1 )

=

K.

It follows from the remark made in 1.3.4 that there exists a K-quasi-

conformal self-mapping W of the upper half-plane with w(O) = 0, W(1) = 1, W (- 1) = - A(K), w(oo) = 00. Let this mapping be extended to the whole plane by reflection. We write B n = {z 11/n < Izi <.: n}, B: = w(B n ), n = 2, 3, ... , and map B: conformally onto an annulus An == g 1an < 1'1 < bn} by a function In with In(1) = 1. The composite K-quasiJ:onformal mapping w n = In 0 w: B n ->- An can then be extended to the punctured plane Izl < 00 by repeated reflection. Since z = and z = 00 are removable singularities by Theorem 1.8.1, w n admits a K-quasiconformal continuation to the whole plane and we have wn(O) = 0, wn(1) = 1,

°<

°

wn(oo) = 00. It follows from Theorem 5.1 that {w n} is a normal family. By considering a subsequence and applying Theorem 5.3 we see that w n converges to a quasiconformal mapping of the plane as n ->- 00. The mappings In = w n w- 1 are conformal in every compact subset of the punctured plane < Izi < 00 from some n onwards, and converge to a conformal mapping I of the plane. Since 1(0) = 0, 1(1) = 1, 1(00) = 00, I is the identity mapping. Hence to every E there corresponds an N such that

°

0

>°

Iw N(- 1)

+ A(K)I

= IwN(w- 1 ( - A(K)))

+ A(K)I < E.

Since w N (1) = 1 we have therefore on Izi = 1 max IWN(Z) I

. IWN ()I mIn Z

.

> A(K) -

E.

This inequality also holds on every circle Izi = N-2 k, k = 1, 2, ... , obtained as images of Izi = 1 under the above reflections. Hence (9.2) holds at z = for the K-quasiconformal mapping WN'

°

9.4. Supremum of the circular dilatation. By Theorem 9.2 the supremum of the values which the circular dilatation can assume for K-quasiconformal mappings is at least A(K). In fact, equality holds:

108

II. Distortion Theorems for Quasiconformal Mappings

Theorem 9-3. In the family W of K-quasiconformal mappings w we have sup Hw(z)

=

A(K) .

WE'/f!

The proof of this result requires function-theoretical tools different from the other methods we have used here. We shall therefore not prove the result but refer to the joint work of ViiisaJii and the authors [1].

III. Auxiliary Results from Real Analysis

Introduction to Chapter III In Chapters I and II we have developed the theory of quasiconformal mappings from the geometric definition by using topological and function-theoretical methods. However, it is clear that some fundamental problems in the theory of quasiconformal mappings have a natural connection with real analysis. As examples we mention the questions concerning differentiability and preservation of null sets. For the further development of the theory it is therefore necessary to apply methods belonging to measure and integration theory. While the auxiliary results from topology and complex function theory were only briefly dealt with in the first two sections of Chapter I, we have here preferred to devote a whole chapter to results from real analysis. This is not only because of the difficulty of giving precise references, but because some of the theorems are proved in a form in which they are directly applicable to quasiconformal mappings. For example, we deduce the results on differentiability and preservation of null sets in Chapter IV as special cases of theorems proved in the present chapter. In § 1 we collect some notations and basic results on the concepts of measure and integral. For further particulars we refer the reader to the text-books of Munroe [1J or Saks [1]. Also § 2, in which we assign to every homeomorphism a completely additive set function, is of a general nature. In § 3 we prove that a homeomorphism with almost everywhere finite partial derivatives is almost everywhere differentiable. This result will often find application later. In § 4 we investigate the module of a curve family by applying methods developed in §§ 1-3. This concept was introduced in Chapter I in special cases. Having presented in § 5 some methods of approximating a measurable function, we consider in § 6 functions with LP-derivatives. Many theorems proved here will be applied directly to quasiconformal

110

III. Auxiliary Results from Real Analysis

mappings in Chapter IV, for example, the result saying that a homeomorphism with V-derivatives is absolutely continuous with respect to the area measure. Finally in § 7 we deal with the two-dimensional Hilbert transformation. As an application we derive a relation between the complex derivatives of a function possessing L2-derivatives.

§

I.

Measure and Integral

1.1. Outer measure. We begin with some introductory remarks on the basic concepts of measure and integration theory. We assume that the reader is acquainted with these; our purpose is to establish the terminology to be used later. In what follows a real set function is a mapping into the extended line; thus the values 00 and 00 are not excluded. A real set function p, * defined for all subsets of a space Q is called an outer measure3l if it satisfies the following axioms: 1. A C B implies that p, *(A) ~ P, *(B). 2. p,*(fI) = O. 3. For every sequence of sets An we have p,* (U An) :::;; }; p,*(A n )· We consider only the case when Q is a metric space (usually a point set in the plane). All outer measures which occur will satisfy not only 1- 3 but also the following two axioms: 4. If the sets A and B are at a positive distance from one another, then p,* (A U B) = p,*(A) + p,*(B). 5. For every set A C Q there exists a sequence of open sets Q) Gl ) G2 ) ••• such that n Gn ) A and p,* (n Gn ) = p,*(A).

+

A set function which satisfies axioms 1-4 is called an outer measure in the sense ot Caratheodory. It follows from axiom 5 that p,* is regular. Of the outer measures to be used we mention the Lebesgue outer linear and area measures. The latter is defined for an arbitrary subset A of the finite plane 32 as follows: We consider all sequences R l , R 2 , ••• , where the elements R n are open rectangles whose sides are parallel to the coordinate axes and whose union covers A. The Lebesgue outer measure is defined as inf }; m(R n ) = m*(A). The Lebesgue outer linear measure l* can be analogously defined in the case when Q is a finite line. For a set A C Q we define l*(A) as the 31 Although f! * is defined as a function we speak of the outer measure of a set A instead of the value of f! * on.A.

We restrict ourselves here to the finite plane, since this is the union of CQuntably many open sets of finite area measure (d. 1.2). In 4.1 the area measure will be defined for sets of the whole plane.

32

111

§ 1. Measure and Integral

greatest lower bound of the sum E l(I,,), where II' 12 , • • • are open intervals whose union covers A, and l(In ) denotes the length of I,.. It follows from the above definitions that m* and l* satisfy axioms

1-5. The Lebesgue outer volume measure in an n-dimensional euclidean space R n , n 2, can be defined in an analogous way. We shall use this measure in the case of R4.

>

1.2. Measurable sets. Let ft* be an outer measure, defined.-Oll...~e subsets of the spa ce Q. and satisfying axioms 1 - 5. A set A C £2 is ~ measurable with respect to ft * ~very set 4:( £2 the relation ft*(X) = ft* (X n A) +ft* (X - A) holds. If A is measurabf~,7(A) is called the measure of A. To express the measurability of A we omit the star and write ft(A) instead of ft*(A). Measurable sets form a completely additive class Jfil, i.e. 0 E Jfil, A E Jfil implies that - A E Jfil and An E Jfil, n = 1, 2, ... , implies that U An E JJl. The significance of this class lies in the fact that equality ft CQjA n)

=n~t(An),

holds in axiom 3 if the sets An belong to Jfil and are nonintersecting. Thus the measure ft is completely additive in Jfil. If ft*(A)

= 0, then A is called a null set; such.lLsetis.always measurable" andJw-axiom:Lthe.s.am~.is trlleof _a~n~l!!>~~.!...Qf4. As usual we shall say that a statement holds almost everywhere if it holds everywhere with the possible exception of a null set.

It follows from axioms 1-4 that every open and every closed set is measurable. There exists a uniquely determined smallest completely additive class 3J which contains all closed (and consequently also all open) sets of the space £2. The elements of 3J are called Borel sets 33 ; it follows from the definition that they are always measurable. By axiom 5 there corresponds to every set A a Borel set B, such that A C Band ft*(A) = ft(B).

Since the outer measure ft* is regular it possesses the following continuity property (Munroe [1J, p. 95): If Al C A 2 C ... is a non-decreasing sequence, thenft* (U An) = limft*(A n). Foranon-increasingsequence of measurable sets Al ) A 2 ) ••• the corresponding relation ft (n An) = lim ft(A n) holds under the assumption that ft(A I ) is finite. Because We make the following remark on the dependence of the class $ on the space D: If a subspace Do of D is a Borel set when considered as a subset of D, then the subsets of Do are simultaneously Borel sets with respect to D and Do. 33

112

III. Auxiliary Results from Real Analysis

of this supplementary requirement it is not always possible, despite axiom 5, to associate with every measurable set A a sequence of open sets Gn ) A such that ,u(A) = lim ,u(Gn). A set which is the union of countably many sets An with ,u*(A n) < 00 is called a-linite (with respect to the outer measure ,u*). It follows from axiom 5 that we can associate with every a-finite measurable set A C fJ two Borel sets B l and B 2 such that B l cAe B 2 and,u (B 2 - B l ) = O. Furthermore, B l can be chosen as a countable union of closed sets. Thus there exists a sequence of closed sets F n CA such that lim,u(Fn) = ,u(A) (d. Munroe [1J, pp. 108-110). If the space fJ is the union of countably many open sets with finite measure (as for example in the case of the Lebesgue measure), then the following condition is necessary and sufficient for the measurability of a set A C fJ: To every f 0 there corresponds a closed set F C A and an oJl.flL~tLc;2A such that,u (G - F) f (d. Munroe [1J, p. 111f" The supremum of the measures ,u(F) of closed sets F C A is often called the inner measure of A and denoted by ,u* (A). If A is contained in a space fJ with finite measure then the above characterization of measurability coincides with the validity of the equation ,u*(A) = ,u*(A). The relation ,u(fJ) = ,u*(A) +,u* (fJ - A) is also equivalent to this.

>

<

1.3. Measurable functions. A function I which is defined in a measurable set and maps this into a topological space is called measurable if the preimage l-l(G) of every open set G is measurable. The measurability of I depends on the choice of the outer measure ,u*; if the preimages of the open sets are Borel sets then I is measurable with respect to every ,u *. In this case I is called a Borel lunction. All functions I considered are either real (i.e. mappings into the extended line) or complex-valued (mappings into the whole plane). In both cases there are various criteria for the measurability of I. A real function I, for example, is measurable if the set {z I I(z) r} is measurable for every rational number r. For the measurability of an almost everywhere finite complex-valued function I = u + i v it is necessary and sufficient that the real functions u and v are measurable.

>

The characteristic function cA of the set A is the real function which has the value unity at every point of A and is zero elsewhere. The function cA is measurable if and only if A is measurable. For measurable functions we have the following convergence theorem which is formulated here in the special case of the Lebesgue area measure:

113

§ 1. Measure and Integral

Egoroff's Theorem. Let In be a sequence 01 real or complex measurable lunctions which converges almost everywhere in a set A 01 linite measure to a linite lunction. Then lor every 8 0 there exists a closed set F C A, with m (A - F) 8, where the convergence is unilorm.

>

<

Egoroff's theorem in this form follows almost immediately from the definition of a measurable function and from the characterization mentioned in 1.2 of the measure of A as the supremum of the measures of the closed sets Fe A (ct. Munroe [1J, p. 157).

1.4. Integrable functions. Let A be a measurable set and I a real nonnegative measurable function defined in A. We define

J I dfl =

n

sup E fl(A k) inf I(z) ,

A

ZEAk

k~1

where AI> ... , An are disjoint measurable subsets of A and where the product fl(A k) inf I(z) = 0 if one of the factors vanishes even if the other is infinite. If the integral is finite we say that I is integrable in A. As was remarked in 1.2, a measurable O'-finite set can be represented as the union of a Borel set and a null set. In the definition of the integral over a O'-finite set A we may therefore consider only Borel sets A k instead of all measurable subsets of A. A real or complex measurable function I is defined to be integrable if III is integrable. If I is real, the non-negative functions 1+ = (III 1)/2, 1- = (III - 1)/2 are integrable simultaneously with I, and the integral of I can be defined as the difference

+

J I dfl = J 1+ dfl

A

-

A

J 1- dfl .

A

+

A complex integrable function I = u i v is finite almost everywhere. The real functions u, v are then integrable, n~ matter how they are defined at points where I is infinite. The integral of I is the sum

J I dfl A

=

J u dfl + i J v dfl .

A

A

1.5. Lebesgue integrals. The integrals defined with respect to the Lebesgue linear and area measures are called Lebesgue integrals. For these we shall use the notation introduced in Chapter I. Thus, for example, we write JJ I dO' instead of J I dm (see also 2.6). The following result on the mutual dependence between the two Lebesgue integrals will be used frequently (ct. Munroe [1J, p. 207). Fubini's Theorem. Let I be a real or complex valued measurable lunction delined in a rectangle R = {x i y I Xl X x 2, Yl Y Y2}'

+

< <

< <

114

III. Auxiliary Results from Real Analysis

< < Y2'

Then I is l-measurable as a lunction 01 x lor almost all y, YI Y and as a lunction 01 Y lor almost all x, Xl X x2. The integrals

< <

y,

x,

J I/(x, Y)I

J I/(x, Y)I dx

dy,

y.

x.

are also measurable as lunctions X2

and y, respectively, and we have

Y2

JJ III da = I dx I R

01 X

Xl

Y2

I

I/(x, Y)I dy =

X2

Yl

Yl

I

dy

lI(x, Y)I dx.

(1.1)

Xl

II I is integrable in R, then the lunctions delined by the integrals y,

I I(x, y) dy, y.

are integrable in

Xl

x,

J I(x, y) dx 34 x.

< X < X 2 and Yt < y < Y2' %2

Y2

Yll

respectively, and X2

JJ I da = I dx I I(x, y) dy = J dy I I(x, y) dx . R

Xl

Yl

Yl

(1.2)

Xl

Fubini's theorem can be analogously stated for arbitrary euclidean spaces R m, R n, Rm+n. We shall use the theorem also in the case m = n = 2. It then reads as follows: R z = {x + i Y I Xl < X < x 2 ' YI < Y < h}, R e = {~ + i'Y) I ~l < ~ < ~2' 'YJI < 'YJ < 'YJ2} be rectangles in the z = x + i Y and' = ~ + i 'YJ

Let

planes and let R = R z X R e = {(z, ,) 1 z E R z' 'E R c} be their Cartesian product in R4. If I is defined and measurable in R (with respect to the four-dimensional Lebesgue volume measure), then

II daz JJ I/(z, ')1 dae = JJ dadi Rz

If

Re

I is integrable in

Re

I/(z,

')1 da z •

R then

JJ da z II I(z, ,) dae = JJ dae II I(z, ,) da z • Rz

(1.1)'

Rz

RC

RC

(1.2)'

Rz

As a second result on Lebesgue integrals we mention the following theorem, which can easily be proved with the help of Egoroff's theorem (see 1-3) (Munroe [1J, p. 234). Le besgue con vergence theorem. Let In be a sequence 01 measurable lunctions which satislY the lollowing two conditions almost everywhere in a set A 01 linite Lebesgue area measure: 1. I/n(z) I ~ g(z), n = 1,2, ... , where g is integrable in A. 2. lim In(z) = o. Then lim

JJ In da =

0.

n-oo A 34 Yll

These functions exist for almost all x and y, respectively, since the integrals X2

J I/(x, y) Idy and J //(x, y) I dx are finite for almost all x 'YI

Xl

and y on account of (1.1).

115

§ 1. Measure and Integral

1.6. Points of density of a set. Let A be a point set on the line. For a finite point x we consider all intervals 1:< which contain x and set S(X)

.

.

= lim lilf

1* (A n Is) 1(1) .

1(1:<)-0

:<

If S(x) = 1, then x is called a point at (linear) density of the set A. The lower limit can then be replaced by the ordinary limit.

Analogously we define points of density for a plane set A. Fer a finite A we consider all closed squares Qz which contain z and have sides parallel to the coordinate axes. If

Z E

n Q,) =

lim inf m*

(A

m(Qz)-O

m(Qz)

1,

then z is called a point of density of A. In both cases we have the following result on the existence of points of density (Munroe [1J, p. 290, Saks [1J, p. 129): Almost every point at A is a point at density at A. For plane sets we need another more special concept of density. A finite point Zo = Xo i Yo of the z = x i y-plane is called an x-point. at density of the set A, if X o is a point of linear density of the onedimensional set {x I x i Yo E A}. Similarly, Zo is defined to be a y-point of density if Yo is a point of linear density of {y I Xo + i YEA}. A point which is both an x- and a y-point of density is called an x ypoint at density. In § 3 we shall need the following result on x y-points of density (Saks [1J, p. 298).

+

+

+

Lemma 1.1. Almost all points at a measurable set A in the tinite plane are x y-points at density at A.

1.7. Hausdorff measure. The Lebesgue linear measu.re is defined only for point sets which lie on a line. To determine the length of a more general plane set another type of measure must be introduced. Such a measure is obtained as a special case of the concept of iX-dimensional measure, iX 0, defined as follows:

>

--~.-

Let A be a~ the finite pl~~e. Consider alLcoveriugs of A bYSQunt.: abbr_J11.J1llY opens.ets of diameter less .tQaILapositive number d. We associate with every such covering {G v G2 , •• -TTfie·rii.lmberE d:, where d n denotes the diameter of the set Gn . The infimum of these numbers increases as d ~ 0; thus the limit ,u!(A)

=

lim (inf

B

d-o

n

d~) ,

116 exists. of A.

III. Auxiliary Results from Real Analysis

It is called the IX-dimensional Hausdorff outer measure

>

It follows immediately from the definition that for every IX 0, P; is an o,uter measure satisfying the axioms 1 - 5. The restriction PI!< of to sets measurable with respect to is called the IX-dimensional Hausdorff measure.

P:

P:

We conclude from the definition that for a fixed A, p;(A) does not increase with increasing IX and is positive and finite for at most one value of IX. We define the dimension of A as the least upper bound dim A = sup {IX I p;(A) = oo}; then we have p;(A)

=

°for IX > dim A and p:(A)

=

00

for IX

< dim A.

In some later examples we shall use the following result on the dimension of the Cartesian product of two sets: Lemma 1.2. Let A and B be compact sets on the finite x- and y-axes. Then dim (A X B) = dim A + dim B. The inequality dim (A X B) < dim A + dim B is an almost immediate consequence of the definition of dimension. To prove the converse relation dim (A X B) ::2: dim A dim B we refer to Marstrand [1J,

+

p.198. 1.8. Linear measure. Among all the above measures Pex the onedimensional Hausdorff measure PI will most often be used here. If the set A lies on a line then this measure agrees with the Lebesgue linear measure. We shall therefore call PI(A) the linear measure of A and denote it by l(A) also in the general case.

For a set A of finite linear measure the Lebesgue area measure vanishes. By Fubini's theorem the one-dimensional set A(yo) = {z = x + i Yo I z E A} is therefore of zero linear measure for almost all values Yo. In fact, a much sharper result is true: On every horizontal line y = Yo there is only a finite number of points of A, with the possible exception of a null set of values Yo (Gross [1J). This can be deduced from the following lemma, which will find an application in V.3. Lemma 1-3. Let A be a plane set, N a positive integer, and E the set of values Yo for which A (Yo) contains at least N points. Then l*(E) < l*(A)fN. Proof: We denote by En the set of values Yo with the following property: On the line y = Yo there exist N points of A whose mutual distances

117

§ 2. Absolute Continuity

are greater than 1/n. Then we have E I C E 2 C ... and U En = E. Thusl*(E) = lim l*(E n), and we must nowprovethatl*(En) < l*(A)IN for every positive integer n. For a fixed n we consider an arbitrary covering of A by open sets Gl> G2 , ••• , with diameters dI , d2 , ••• less than 1In. In view of the definition of the linear measure we have to prove that}; d k ~ N l*(E n). For Yo E En the line y = Yo contains N points which lie in different sets Gk • If P k denotes the projection of Gk onto the y-axis, then every point of En belongs to at least N sets P k • Let P k = U 1 jk , where 1 jk , j = 1, 2, ... , are disjoint open segments. Then every point of En belongs to at least N segments 1 jk • If the number of segments I jk is finite, the relation N l*(En) S }; }; l(Ij k) < }; d k obviously follows. For a countably infinite covering we obtain the same result by a limiting process. In fact, if Om denotes the set of points which belong to at least N of the first m segments in the sequence Ill> 112 , 121 , . . • , then ON C ON+l C ... and En C U Om' It follows that

l*(E n) < lim l(Om) S ~ }; dk m

,

~oo

and the lemma is proved.

§

2.

Absolute Continuity

2.1. Absolutely continuous additive set functions. Let "t" be a completely additive set function defined in the class of Borel sets of the finite plane, and fl a Hausdorff or Lebesgue measure. The function "t" is called absolutely continuous in a set E (with respect to the measure fl) if the following condition is satisfied: For every ~~ts CJ suchth<.l-Ur(B)I 8 for.~veryBQrelset B (E \\TUh p,(I3J CJ. 'fei is absolutely continuous in every compact ~ubset of E, we -say that "t" is locally absolutely continuous in E.

>°

<

8_>

<

If "t" is absolutely continuous and B is a Borel set with fl(B) = 0, then obviously "t"(B) = 0. Conversely it is easy to prove (Munroe [n p. 191): A bounded completely additive set function "t" is absolutely continuous if it vanishes for every Borel null set.

As examples of completely additive absolutely continuous set functions we exhibit the integrals of functions integrable with respect to fl. They are bounded and vanish for every null set. On the other hand, the

118

III. Auxiliary Results from Real Analysis

.. Radon:[email protected]~~}p (Munroe [1J, p. 196) states that a completely additive absolutely continuous set function 7: is always the integral of an integrable function,provided that 7: is bounded and vanishes outside a a-finite set.

2.2. Absolutely continuous homeomorphisms. Suppose that E and E' are Borel sets of the finite plane, f: E -+ E' is a homeomorphism, and f-l is a Hausdorff or Lebesgue measure. Since a homeomorphism maps closed sets onto closed sets, Borel sets remain Borel sets (d. the footnote in 1.2). The image of a Borel set lying in E is therefore measurable with respect to f-l. We associate with the mapping f a set function f-lt by the formula

f-lt( B)

=

f-l

{f (B n E)) ,

where B runs through all Borel sets. The function f-lt clearly has all the properties of a measure. We define the homeomorphism f to be absolutely continuous or locally absolutely continuous (with respect to f-l) if the set function f-lt is absolutely continuous or locally absolutely continuous in E. We shall consider only two measures here, the (Lebesgue) area measure and the (Hausdorff or Lebesgue) linear measure. In the first case E and E' will be chosen to be~ns, in the ~ s . Before dealing with these special cases we shall, however, make some general remarks on absolutely continuolls homeomorphisms. Suppose that f is absolutely continuous and B is a Borel set with f-l(B) = 0. According to the above remarks f-lt(B) = 0, i.e. Borel null sets are mapped onto null sets by f. Actually the following stronger result is true:

A n absolutely continuous homeomorphism maps all null sets onto null sets. In fact, every null set A is contained in a Borel null set B by 1.2, and f(A) is therefore a subset of the null set feB). If ACE is a measurable a-finite set, there exists by 1.2 a Borel set B, A C BeE s~h that B - 4J_s a null set. If f is an absolutely continuous homeomorphism, then we have f-l{f(B) - f(A)) = f-l{f(B - A)) = 0, and hence feB) - f(A) is measurable. Since feB), being a Borel set, is measurable, we infer that f(A) is also measurable. Thus we have shown:

An absolutely continuous homeomorphism maps every a-finite measurable set onto a measurable set.

§ 2. Absolute Continuity

119

As a consequence of this result an absolutely continuous mapping is often called measurable in the case when E is a domain. We have preferred the term "absolutely continuous" to avoid confusion with the concept of measurable function and to point out the analogy with the one-dimensional case. If E is a-finite and f is an absolutely continuous homeomorphism, then fli is defined for all measurable sets and satisfies the following condition: For every I> 0 there exists (J 0 such that fli(A) I> for every measurable set A with fl(A) (J.

>

>

<

<

2.3. Derivative of an additive set function. Retaining the above notation we now introduce the following special assumptions which are to remain in force throughout the rest of this section: E is either a bounded or unbounded segment or a domain of the finite plane, and fl is to be the linear measure in the first case and the area measure in the second. In both cases E is the union of countably many compact sets. It follows that the results of 2.2 on the preservation of null sets and measurability remain true if the homeomorphism in question is only loc,:l~solutelycontinuous. --Let r be a non-negative completely additive set function which is defined for all Borel sets; at present we do not require it to be absolutely continuous. The derivative of r can be defined in various ways. The following procedure is analogous to the definition of a point of density (d. 1.6). Consider a fixed point Z E E. If E is a domain, let Qz denote a square containing the point z and having sides parallel to the coordinate axes. If E is a segment, Q.~lo den<2t!L
= l'1m p(Qz) - 0

T

(Qz n E) p,(Qz)

exists, we call it the derivative of the set function r at z. In 1.6 we have already mentioned a special result on the existence of the derivative. In fact, if cB is the characteristic function of a measurable set Band r(A)

= J cB dfl ' A

then we conclude from the discussion of 1.6 that the set function r has the derivative r'(z) = cB(z) almost everywhere.

120

III. Auxiliary Results from Real Analysis

In the general case we have the following result (Saks [1J, pp. 115 and 119): Theorem of Le besgue: A non-negative, completely additive set junction T, bounded in E, has al;;:U;St-~v-ery-;fierein E a jinite derivative T'(Z) , which is measurable as a junction oj z. For every Borel set BeE we have (2.1) T(B) ~ JT' dfl, B

where equality holds for every B if and only if continuous in E.

T

is locally absolutely

2.4. Derivative of the set function associated with a homeomorphism. We now consider the special case when T is the set function fll associated with a homeomorphism f: E ---+ E'. If fll is bounded in every compact subset of E the finite derivative fl; exists almost everywhere in E by the above theorem of Lebesgue. This condition is always satisfied in the two-dimensional case, when E and E' are domains in the finite plane and fl is the Lebesgue area measure. In this case we call fl; the areal derivative of the mapping f.35 Lebesgue's theorem further states that in this case fl; is measurable and locally integrable in E (i.e. integrable over every compact subset of E) and that (2.2) fl(t(B)) ~ J fl; dfl B

for every Borel set BeE. The same holds in the one-dimensional case provided that fll is locally bounded. If f is locally absolutely continuous, Lebesgue's theorem can be sharpened. By 2.2, fll is then defined for every measurable set, since E is a-finite. Every a-finite measurable set A is contained in a Borel set B with the property f-l (B - A) = 0 and we see that equality holds in (2.2) not only for Borel sets but for all measurable sets:

If j is a locally absolutely continuous homeomorphism, then fl(t(A)) = J fl; df-l

(2·3)

A

for every measurable set A.

We need nQt assu_IJ1ellereih~tjtl is bouIJ,ded-: for a compact set ACE a locally absolutely continuous f-ll is automatically bounded and for an unbounded A we obtain (2.3) by a limiting process. It should be noted that the areal derivative coincides almost everywhere with the Jacobian if the homeomorphism has partial derivatives almost everywhere (see 3.3).

31i

121

§ 2. Absolute Continuity

2.5. Transformation of variables in integrals. Let I: E ~ E' be a locally absolutely continuous homeomorphism. We remind the reader that E is a segment on a finite line or a domain of the finit.e.j)lane. In the latter case E' is trivially a-finit'e;JJy the local absOlute continuity of 1 the set E' has this property also in the first case. Let g be a complex-valued function in E' and h = g 0 I. We conclude from the relation g-l(A) = l(h-1(A)), in view of 2.2, that the measurability of h implies that of g. Furthermore it follows from h-1tA) = l-l(g-l(A)) that hand g are Borel functions simultaneously, and that ~_~ ~~asurable if g is m~asllrab*-andf=-l:Js-.local1y~olutell con1!J!..!!QUs. If g is measurable and either integrable or non-negative, then the (finite or infinite) integral J g dft A'

exists for every measurable set A' C E. We prove the following generalization of (2.3): Lemma 2.1. Let j; E ~ E' be a locally absolutely continuous homeomorphism and g an integrable or non-negative junction in E', such that h = g 01 is measurable. Ij ACE is a measurable set and A' = I(A), then (2.4) where we set

n ft; =

0 at the points

01 A

where h ~

00

and ft;

= o.

Prooj: Without loss of generality we may assume that g 2: O. Since A' is a-finite we may suppose that in the definition gk

= inf g , A'k

the sets A~ are Borel, by the remark made in 1.4. Then A k = j-l(A~) is measurable, and by (2.3)

I; gk ft(A~) = I; g" J ft; dft < Ak

J h ft; dft < J h ft; dft .

UAk

A

Hence

(2.5) To prove the opposite inequality we consider next the set h-1(00) = A oo · Since g = 00 in A:x, = j(A oo ) we may suppose that

122

III. Auxiliary Results from Real Analysis

Then P; vanishes almost everywhere in A oo and, in view of our convention, so does h The set A oo can therefore be disregarded in the integration. It is thus enough to consider the case when h is everywhere finite. We el ~ h(z) choose an arbitrary e 0 and set A k = {z I Z E A, (1 (1 e)k+ 1} , A~ = f(A k). Then

p;.

>

< +

+

00

Jh p; dp < A

~ (1

00

+ e)k+l p(A~)

S (1

+ e)

k=-oo

~ gkP(A~) S (1

k=-oo

+ e) J g dp. A'

This, together with (2.5), yields (2.4).

2.6. Rectifiable arcs. In 2.2 we considered a homeomorphism f: E ~ E' in the general case when E and E' were arbitrary Borel sets of the finite plane, and in 2.3 - 2. 5 we made the special assumption that E was either a segment or a domain. We now look only at the case when E is a bounded or unbounded segment Ion - 00 < x < ()() and E' is a Jordan arc C of the finite plane. The measure p is the Hausdorff linear measure I and the set function PI will now be denoted by V. If V(I) = I(C) is finite, then the arc C is called rectifiable and I(C) is the, length of C. The arc C is called locally rectifiable if every compact subarc of C is rectifiable. A Jordan curve is called rectifiable if all its subarcs are. Since the Hausdorff linear measure has the continuity properties mentioned in 1.2, we have lim I(C,,) = I (n C,,) for every monotone sequence C1 ) C2 ) ••• of subarcs of C, provided that I(CI ) is finite. If we choose the sequence such that n C" is a point, then lim I(C n ) = O. In the case of a locally rectifiable arcJ:.-..tbg~~..!iL!!!.nctiol1Jl:i~JhITefore continuou~ !..I!....!Pe Jollowing §en~: If x is an arbitrary point of I, then f~very e 0 there exists a neighbourhood L1 of x such that V(L1)

<e.

>

From now on, we suppose that C is locally rectifiable. To the set function V there corresponds a function s of x, determined up to an additive constant, such that the growth of s on every subinterval [a, b] = {x I a S x < b} of I is equal to V([a, bJ). T_he function~~_called the arc Zength,j..s c:ontinuous at everY.. .P 9int of continuity of V, and hence on the whole interval I. . If V has a finite derivative at a point x E I, then s is evidently differentiable at x and s'(x) = V'(x). It follows from Lebesgue's theorem that this derivative exists almost everywhere on I and satisfies the inequality

J s'(x) dx < I

V(I) = I(C) ,

§ 2. Absolute Continuity

123

<

where equality holds for l(C) 00 if and only if the mapping I is locally absolutely continuous. Since the function s is continuous and strictly increasing it maps 1 topologically onto a segment 10 , By the definition of arc length, we have l(Io) = V(I) = l(C) and so 10 is bounded or unbounded according as C is rectifiable or not. We note that in the latter case C cannot be compact. Denote by' : 10 ->- C the homeomorphism composed of I and the inverse of s: 1 ->- 10 , We call' the length preserving parametrization of C. Among all parametrizations of C this one is distinguished by the fact that it maps every interval Ll C 10 onto an arc of length l(Ll). It follows that' is absolutely continuous and satisfies l(,(A)) = leA) for every measurable set A. The inverse is also absolutely continuous.

'-1

Since a mapping composed of two absolutely continuous homeomorphisms is obviously absolutely continuous, it follows from I = , s, s= I, that I and s are simultaneously locally absolutely continuous. In this case and only then the equation

'-1

0

0

J s' dx =

l(t(A))

(2.6)

A

holds for every measurable set A C I (d. (2.3)). Under the same assumptions Lemma 2.1 yields the more general formula

J (g

0

J g dl

f) s' dx =

A

(2.7)

I(A}

for every non-negative or integrable Borel function g defined on C. In the integral on the right we write instead of dl also jdzl, jdwl, ... , according to the variable used. In an obvious way the above considerations can be carried over to the case when the Jordan arc C consists olJiI!itely~any)ocallyrectifiable arcs. and their endpoints. The same is true if C is-a JQrdancufve; we onlyneedtodivide-Cl};to two s u b a r c s . - - -------

2.7. Functions of bounded variation on an interval. The definition of the set function III given in 2.2 assumes that I : E ->- E' is a homeomorphism. In the case when E is a segment, III admits another characterization which can be applied to non-topological mappings of 1. This generalization depends on the fact that the length of an arc I(Ll), Ll C I, can also be defined in the following way (see Saks E1J, IV. § 8): Consider all finite sequences of non-intersecting intervals ai < x < bi which, together w_iih-their__~dpoi~.!~!_'!!~<':QJ!ta.i!1.e.§.~~~: Then lUlLl))

=

V(Ll)

=

sup

L: i

I!(bi ) - !(ai)1 .

(2.8)

124

III. Auxiliary Results from Real Analysis

This formula yields the required new characterization for the restriction of the set function V to the family of all segments Ll c f. We use it as a definition in the case when I is an arbitrary continuous complexvalued function defined on f; the expression (2.8) is called the variation of I on Ll. If V(f) is finite we say that I is 01 bounded variation on f. If V(LI) < 00 for every compact segment LI C f, then I is of bounded local variation; in the case of a homeomorphism this is equivalent to the IQcal rectifi~bilit~f

Im- - -- ---

_ n

-

_0.

-

----

I

is defined by (2.8) for segments only. If ~ however, be extended to a completely additive set function defined in the class of Borel subsets of f. To do this we first associate with every set A C f an outer measure V*(A) by considering all coverings of A by sequences of open segments Ll 1 , Ll 2 , . . . and setting The variation V of

~ndedjtcan,

V*(A) = inf I

V (f

n Ll k) •

Then V* satisfies the axioms 1- 5 of 1.1 and so the Borel sets are measurable with respect to V*. Since V*(LI) = V(LI) for every compact segment LI C f we obtain the required extension of V by setting V(B) = V*(B) for every Borel set B. I!!is definiti9J}gL V~grees with the origiI1 al one when morphism. 0.

I

is a homeo-

-

As a generalization of the arc length of 2.6, we associate with every continuous function I of bounded local variation a function s on f, defined by the formula s(b) - sea) = V([a, bJ). The function s is continuous and determined up to an additive constant. If I is real we write Mx) = sex) + X, 12(x) = Mx) - I(x). Then

1=11 - 12' where 11 and 12 are strictly increasing and thus homeomorphisms of f. A c().!!!£~ex.:yalued funcjion.L=u +i vis said to be of bounded variation or of bounded local variation if and only if the real functions u and v have the property in question. Hence, if t is of bounded local variation on f, it admits a representation t = U 1 - U 2 i (VI - v2), where Uk' vk are homeomorphisms of f.

+

By Lebesgue's theorem a monotone function on f has a finite derivative almost everywhere on J. It follows from the above that a function I of bounded local variation on f also has this property. The derivative t' is integrable over every compact subset of f.

125

§ 2. Absolute Continuity

The derivatives of a function I of bounded local variation and of the function s associated with I satisfy the equation

It'(x) I = s'(x) for almost all x

E

I (Saks [lJ, p. 123).

2.8. Absolutely continuous functions on an interval. As in the case of a homeomorphism, an arbitrary continuous function Ion the interval I will be called absolutely continuous if its variation V is absolutely continuous on I. We see from the definition of V given in 2.7 that absolute continuity can also be characterized in the following way: The function I is absolutely continuous on I if to every B 0 there corresponds C> 0 such that I; I/(b;) - I(a;) I B for every finitll x bi which together with sequen'ce of non-intersecting intervals ai their endpoints lie in I and have a total length C>.

>

< < < <

>

Since an absolutely continuous function I on I is of bounded local variation it possesses a derivative which is integrable over compact subintervals of I. It follows from Lebesgue's theorem that b

I(b) - I(a) =

J t'(x) dx

a

for every pair of points a, b E I.

2.9. An example of a singular function. A function I of bounded variation on the interval I is called singular if the derivative t' vanishes almost everywhere on I. Since the increment of an absolutely continuous function equals the integral of its derivative, J!..().~ant_ function can be simllltaneously singular anCl~~()lui~!y_cg_l1_1JE~~~~ For later applications we need an example of a non-constant continuous singular function. This is constructed with the help of a Cantor set, which will also appear later in connection with certain other questions. A Cantor set is defined in the following way: Let a sequence of numbers 0 Pv 1, be given. We first remove from the middle of the closed interval I = [0, 1J an open interval In of length Pl' From the middle of the two remaining closed segments we take away open intervals 121 , 122 , each of length P2 (1 - PI)' At the nth step, we remove from the middle of the 2n - 1 remaining closed segments open intervals Ink, each of length 21 - n Pn (1 - PI) ... (1 - Pn-l)' The Cantor set E = E(Pv P2' •• ,) is defined as the complement of the union of all the removed intervals:

P.,

< <

+

E = 1 - U U Ink' n=1 k=1

III. Auxiliary Results from Real Analysis

126

The set E is perfect, that is it coincides with the set of all its limit points, and it has the linear measure 00

I(E)

=

II (1 -

P.) .

• =1

The proposed construction of a singular function can now be carried out as follows: We first set g(x) = 2- n (2 k - 1) for x E 1 n k' Then g is defined as a non-decreasing function in 1 - E whose values are everywhere dense in [0, 1]. Therefore the limit

I(x) = lim g(t) t ..... x

exists for every x E 1 and defines a continuous functionLQ!1 1 which decreases nowhere and grows from to 1.

°

°

By definition 1 is constant on every interval 1 n k' Hence I' (x) = for x E 1 - E. If the numbers P. are so chosen that I(E) = 0, say P. = P = const., then I'(x) vanishes almost everywhere on 1. Thus the function 1is singular and gives the required example.

2.10. Variation and absolute continuity on a Jordan arc. Let 1 be a continuous, complex-valued function defined on a locally rectifiable arc C. F o~__all-p_;g:CI:.!ilet~!:1:~tioIl?.t~L~E, t!:e_\T- C, the absolute continuity of f t depends on the homeomorphism t. 0

2.11. Integrals over oriented arcs. Let C be a closed rectifiable Jordan arc and 1;: 10 -'>- C, 1 0 = {s I a S;; s S;; b}its length-preserving parametrization .L~!L?-pd_g]:>_~~~l'ltimlolls_ complex-valued functions on C, g being of boundedvari
J 1 dg = ± J 1(I;(s)) dg(I; (s) ) .

C'

(2.10)

a

On the right hand side we have a classical Stieltjes integral, and the sign is chosen to be or - according as I; belongs to the orientation ex or not (see 1.1.4). A homeomorphism w of C onto an arc C' induces a mapping of the orientation ex of C onto an orientation ex' of C' (see 1.1.5), i.e. w carries

+

127

§ 3. Differentiability of Mappings of Plane Domains

C+ onto the oriented arc w(C+) = (C', (X'). The definition (2.10) thus implies the following invariance property: If C' is rectifiable, then

J I dg' = J (f

c+

0

w- 1) d (g

0

w- 1 )

•

(2.11)

w(C+)

§}. Differentiability of Mappings of Plane DomainJ' 3.1. Existence of partial derivatives. As we have seen in 2.4 it follows from Lebesgue's theorem o~ the derivative of a set function that ~meQmorQhj§illQtaplane dom
< <

< <

In every rectangle R = {x + i y I a x b, c y d}, ReG, 1 is of bounded variation (is absolutely continuous) as a function of x on almost all segments I y = {x + i y I a < x < b}, and as a function of yon almost all segments Ix = {x + i y [ c < y < d}.36 Since absolute continuity on a segment implies bounded variation, I is of bounded variation on lines in G if it is absolutely continuous on lines in G. By 2.7 a function of bounded local variation on a segment I has a finite derivative almost everywhere in I. The corresponding result for plane domains reads as follows: 36

We could just as well use closed segments I yo Ix here.

128

III. Auxiliary Results from Real Analysis

Lemma 3.1. A lunction I 01 bounded variation on lines in G possesses linite partial derivatives almost everywhere in G.

< <

< <

Prool: We consider a rectangle R = {x + i Y I a x b, c y d}, ReG. Let E denote the set of points of R at which f x exists and is finite. Since E is a Borel set its characteristic function cE is measurable. Thus by Fubini's theorem, d

m(E)

b

d

= fJ cE da = J dy f cE dx = f I (E Rca

c

where I y denotes the horizontal segment {x

nI

y)

dy ,

(3.1)

+ i y I a < x < b}.

f is of bounded variation on I y, then Ix exists almost everywhere on I y , and I (E n I y ) thus equals the length b - a of I y • This holds for almost all y E (c, d), by hypothesis, and so (3.1) yields If

m(E) = (b - a) (d - c) = m(R) .

The function I thus has a finite partial derivative Ix almost everywhere in R. In the same way we can show that Iv exists and is finite almost everywhere in R. Since G is a union of countably many rectangles with sides parallel to the coordinate axes we see that f has partial derivatives almost everywhere in G.

3.2. Differentiability of a homeomorphism. The existence of the partial derivatives of a function I at a point Zo = X o + i Yo does not necessarily imply the differentiability of f at zo, i.e. the validity of an expansion f(z) = I(zo) + IAzo) (x - xo) + Iy(zo) (y - Yo) + 0 (z - zo)' However, in the case when I is a homeomorphism the following result is true (Gehring-Lehto [1J): Theorem 3.1. Let G and G' be domains in the linite plane and f: G --+ G' a homeomorphism having linite partial derivatives almost everywhere in G. Then I is dilferentiable almost everywhere in G. Prool: It is clearly enough to prove that I is almost everywhere differentiable in an arbitrary compact subset Go of G.

We consider the function F h'

Fh(z) =

Ij(z + ~ -

j(z) _

Ix(z) I + Ij(z

+ i~)

- j(z) -

Iy(z) I'

for every point z E Go at which I has finite derivatives Ix and I y and for real numbers h =1= 0 such that z hand z i h lie in G. For sufficiently

+

+

§ 3. Differentiability of Mappings of Plane Domains

129

smalllhl, F h is a Borel function defined almost everywhere in Go. It follows that the functions gn'

gn(z) =

n = 1,2, ... ,

sup Fh(z) , O
are also Borel functions from some n = no onwards, since we can let h run through only the rational values in (0, 1/n). From the assumption on the partial derivatives 'of I we conclude that --+ 0 as n --+ 00 almost everywhere in Go. Hence, by Egoroff's theorem (see 1.3), there exists for every 'YJ 0 a closed set E C Go with m (Go - E) 'YJ such that

gn(z)

>

<

Ix (z)

= l'

1m

f(z

h~o

+ h)

- f(z)

h

'

I y (Z)

l'

=

1m

f(z

+ i h) h

- f(z)

h~O

uniformly for Z E E. The restrictions of Ix and Iy to E are then continuous. If e E (0,1) is given, we can thus find 15 0 such that for z, Zo E E the inequalities

>

I/x(z) - Ix(zo) I < e ,

I/y(z) - Iy(zo) I < e for

and

<

<

Iz - zol

< 15

(3.2)

<

IFh(Z) I e for 0 Ihl 15 (3·3) hold. Suppose that Zo = X o i Yo E E is an x y-point of density of E. By Lemma 1.1 almost every point of E has this property. According to the above the theorem will therefore be proved if we can show that I is differentiable at Zoo For this purpose we shall prove that I satisfies the inequality

+

I/(z) - I(zo) - Ix(zo) (x - xo) --- Iy(zo) (y - Yo) I < Me Iz - zol

(3.4)

in a neighbourhood of zo, where M is finite and depends neither on z nor on e. To simplify the notation we suppose that Zo = 0 and write L(z) = 1(0) I,.{o) x Iy(o) y. Using the triangle inequality we obtain

+

+

I/(z) - L(z)1 < I/(x

+ i y)

+ I/y(x) -

- I(x) - Iy(x) YI

+ I/(x)

- 1(0) - Ix(o) xl

Iy(o) I IYI .

From (3.2) and (3.3) it then follows that

I/(z) - L(z) I

<

for Izi 15, x i y E E.

E

< 3 e Izl

E. Analogously we see that (3.5) also holds for

(3.5)

Izi

< 15,

To prove that (3.4) (for Zo = 0) also holds without the special hypotheses x E E and i y E E, we first use our assumption that z = 0 is an

III. Auxiliary Results from Real Analysis

130

x y-point of density of E. It follows that there exists a square Q = {x + i Y Ilxl < d, IYI < d} whose length of side is at most <5 and which satisfies the following condition: For every closed segment J C Q lying on either coordinate axis and containing the origin we have

I (J

n E) > 3ll. 1.+6

Now take an arbitrary point z = x + i Y =1= 0 of the square Qo = {x + i Y Ilxl < d/2, IY[ < d/2}. By 0.6) each of the open segments ((1 - 8) x, x), (x, (1 + 6) x), ((1 - 6) i Y, i y), (i Y, (1 + 6) i Y) contains at least one point of E. Thus we can choose four points Xl' x2, i YI' i Y2 of E which satisfy the conditions Xl < X < x 2, YI < Y < Y2' X 2 - Xl < 2 6Ix[, Y2 - YI < 28 IYI· Let R denote the rectangle with corners Xl + i YI' X 2 + i YI' X 2 + i Y2, Xl + i Y2. Then every boundary point z* = x* + i Y* of R has the property that either x* or i Y* belongs to E. Since Iz*1 < <5, (3.5) then holds everywhere on the boundary of R. We now make use of the assumption that 1 is a homeomorphism. This implies that 1 satisfies the maximum principle in R. The expression I/(C) - L(z)l, considered as a function of C, therefore takes its maximum at a boundary point z* of R. But (3.5} holds at z* and so

I/(z) - L(z)1 < If(z*) - L(z)1 :::;; If(z*) - L(z*)1

+ IL(z*) -

L(z)!

+ [fx(O) [ Ix* - xl + Ify(O)I/Y* - YI . < 26 Iz[ < 2/zl. If we now replace the origin

< 36 Iz*1

By the above Iz* - zl by a general point zo, we obtain 0.4) for z - Zo 2 (Ifx(zo) I Ify(zo)I), and the theorem is proved.

+

+

E

Qo with M = 9

Remark. The above proof is based on a method due to Stepanoff (Saks [1J, p.300-303) as well as on the maximum principle. Since the maximum principle holds not only for h()meomorphisms but also for_.'l:ll continuous open mappings,•.. Theorem 3.1 is also true for these mappings. .. _.. _ "

_.-~

~

~

3.3. Areal derivative and Jacobian of a homeomorphism. Let f be a homeomorphism between the domains G and G' of the finite plane; henceforth we shall assume that f is sense-preserving. If f has finite partial derivatives at a point of G, then the Jacobian i

-

-

J = 2 (Ix fy - Ix f y ) = /1./ 2

-

1/.1 2 ,

also exists at that point. It is known that for sufficiently regular homeomorphisms J equals the areal derivative /hI defined in 2.4. The following result of this type will be needed later: Lemma 3.2. Let the homeomorphism 1 be diflerentiable at a point Zo Then 1 has a finite areal derivative at Zo and /hl(zo) = J(zo).

E

G.

131

§ 3. Differentiability of Mappings of Plane Domains

Prool: For the sake of simplicity we suppose that Zo = 0 and 1(0) = O. Let Q c G be a closed square with length of side r and let 0 E Q. We write L(z) = Ix(o) x + ly(O) y and

s (r) =

sup 0
f(Z) - L(z)

I

I.

z

Since I is differentiable at the origin we have s(r) """* 0 as r """*

o.

The linear function L maps the square Q onto either a parallelogram, or a segment, or a point. In any case, we have for the image L(Q) the formulae m(L(Q)) = r 2 ](0) for the area and d(L(Q)) S M r for the diameter, where M = 2 (1/Ao)1 + I/y(O)I).

<

In Q we have I/(z) - L(z)1 < Iz/ s(r) 2 r s(r), and so every boundary point of I(Q) lies within this distance of the boundary of L(Q). An elementary geometrical argument then yields r 2 ](0) - 8 M r 2 s(r) < m(t(Q)) S r 2 J(o) + 8 M r 2 s(r) + 4 n r 2(s(r))2. Hence .

m(f(Q))

')

hm -;2-- = fJ-t(O = J(O) , r~o

as required. 3.4. Integration of the Jacobian. If the homeomorphism I: G """* G' has finite partial derivatives almost everywhere in G then by Theorem 3.1 it is differentiable at almost all points of G. By Lemma 3.2 we then have J(z) = fJ-;(z) almost everywhere in G. Applying Lebesgue's theorem to the areal derivative of a topological mapping (see 2.4) we infer: Lemma ).3. II the homeomorphism I: G"""* G' possesses linite partial derivatives almost everywhere in G, then

JJ J da S

m(t(B))

B

lor every Borel set BeG. Equality holds lor every B il and only il I is locally absolutely continuous in G. Then

JJ J da =

m(t(A))

A

even lor every measurable set A C G.

By 2.1 and 2.2 the inverse 1-1 of I is absolutely continuous in every compact subset of G' if and only if I maps no set of positive measure onto a null set. Thus from the second part of Lemma 3.3 we obtain: L e m m a 3.4. Let the homeomorphism I be locally absolutely continuous in G and have linite partial derivatives almost everywhere in G. The

III. Auxiliary Results from Real Analysis

132

Jacobian J of f is positive almost everywhere in G if and only if the inverse f- 1 : G' ~ G is locally absolutely continuous in G'.

We remark that the transformation formula (2.4) of 2.5 can in the case considered here be expressed in the following form: Lemma 3.5. Let the homeomorphism f : G ~ G' and its inverse f- 1 be locally absolutely continuous in their respective domains and let f have finite partial derivatives almost everywhere in G. Let g be a complexvalued, measurable function in G' which is non-negative or integrable. Then JJ g(f(z)) J(z) da = JJ g da , A

I(A)

where A eGis an arbitrary measurable set.

§ 4. Module

of a Family of Arcs or Curves.

4.1. Generalization of the notion of module. The module introduced in Chapter I for quadrilaterals and ring domains can be characterized by means of a family of Jordan arcs or curves. In fact the concept of module can be generalized to arbitrary families of arcs or curves as we shall now show (d. AhHors-Beurling [1J, Fuglede [1J). As in Chapter I we consider in this section point sets of the whole compact plane. All St;t functions -r (measures and integrals) must therefore be extended to sets containing the point at infinity. This we do simply by writing -r(A) = -r (A - {=}). In the case when -r is the Lebesgue area measure the extended concept of measure differs from the usual one in that the measure of a set need no longer be the infimum of the measures of its open coverings. An arc C containing = is called locally rectifiable if it can be transformed into a locally rectifiable arc by a linear conformal mapping. If we remove the point at infinity from such an arc we obtain two locally rectifiable subarcs and the integration methods developed in § 2 can be applied to these. Let ~ be a family of Jordan arcs or curves in the plane Q. A nonnegative Borel function e defined in Q is called admissible for the family t if the relation (4.1) J e IdzJ > 1 c

holds i{)r.~~~~)' l()~a]y.-reetifiable C E...t. We write

me(Q) =

JJ e

2

Q

da

(4.2)

133

§ 4. Module of a Family of Arcs or Curves

(d. 1.4.2) for every admissible

e and call the infinum

M(t)

=

inf me(Q) e

the module of the family t. The number 1(M(t) is called the extremal length of t. Not~ that condition (4.1) does not concern arcs and curves which are not locally rectifiable so that these do not affect the module of t.

Let B be a Borel set of the plane such that all curves C E t lie in B. Then clearly the infimum (4.3) does not change if, instead of all admissible functions e, we consider only those which vanish outside B. Thus for sets B of this type we have M(t)

=

inf me(B) . e

(4.4)

4.2. Properties of the generalized module. We infer from the above definition that the generalized module M(t) is monotonic (d. 1.4.6). It follows first from

that every function

e admissible for t 1 is also admissible for t 2 • Hence

In order to obtain a more general form of monotonicity we consider families tv t 2 with the following property: To every C2 E t 2 there corresponds a C1 E t 1 which is a subarc of C2 • Then ~very function admissible for t 1 is admissible for t 2 , and so we have again 00

The module M(t) is also subadditive: If t = U tn' then n=1 00

M(t) :S }; M(t n) . n=!

(4.5)

To prove this we remark that (4.5) always holds if the right hand side is infinite. If it is finite, then given e 0, we can for every n choose a function en admissible for t n such that

>

men(Q) < M(tn)

+ z-n e.

The function e = (}; e~)!/2 is admissible for 00

M(t)

and hence (4.5).

< me(Q)

= }; men(Q)

n=!

t. 00

Consequently, we have

< }; M(t n) + e , n=!

134

III. Auxiliary Results from Real Analysis

From the first monotonicity property and the subadditivity it follows in particular that

(4.6) 4.3. Families of zero module. The next theorem contains a useful criterion for the module to vanish (d. Fuglede [1]). Lemma 4.1. A necessary and sufficient condition for M(t) to vanish is the existence of a non-negative Borel function eo which is square integrable over the plane and satisfies

J eoldzl

c

for every locally rectifiable C

E

=00

e.

Proof: If there exists a function eo with the above properties then all the functions eo/n, n = 1,2, ... , a!"~!~_!()r_~! and we have

,M(t)

< lim n-oo

:2 ff e~da = o. n

Conversely, if M(t') = 0 then there exists a sequence functions which are admissible for t' and satisfy

JJ e; da < 4 -n ,

n

el' e2' . .. of

=

1, 2, ...

<

JJn~ rnn~

!)

Then eo =

1: en is a Borel function, and

JJe~ = JJC~ da

r

n2 n2 / 2 / enf da

! ) ! )

n

2 e; da

!)

Since each en is admissible for t', we have further J eo Idzl c

= n~J en Idzl =

00

c

for every locally rectifiable C E t, and the lemma is proved. From this lemma we deduce the following result: The module of a family t' vanishes if all arcs and curves of t possess a common point zoo If Zo is finite, we see this by setting 1

eo(Z)

= -

Iz - zollog jz - zol

135

§ 4. Module of a Family of Arcs or Curves

= 0 elsewhere. In the case Zo =

in a neighbourhood of Zo and eo(z) we choose

00

1

eo(z) =

1

+ Izllog Izi •

The latter function eo can also be used to prove the following result: The family of all non-rectifiable arcs and curves has module zero.

4.4. Modules and L2-convergence. A measurable function f belongs to the class LP, 1 < P 00, in the measurable set A if Ifl P is integrable over A; we then write f E LP(A). In what follows we use the Lebesgue area measure. The class LP(A) becomes a metric space if we define

<

Ilf - gllp = (~f If - glP dafP as the distance between f and g, and identify functions which are equal almost everywhere in A. The metric is called the LP metric and the expression 1I/II p the LP norm of f. We shall later need the following theorem of Fuglede [1J on convergence in the L2 metric. Lemma 4.2. Let fn, n = 1,2, ... , be a sequence of Borel- functions belonging to the class L2 in a Borel set B and converging to a Borel lunction I in the L2 metric. Then there exists a subsequence Ink 01 the sequence In such that (4.7) lim J Ifnk - flldzl = 0 k ..... 00 C

for all locally rectifiable arcs C C B, with the exception of a family of module zero. Proof: We choose that subsequence fnk such that

If Ifnk


- fl 2 da

B

3k

(4.8)

and set Ifnk - fl = gk' Let to be the family of those locally rectifiable arcs C C B for which (4.7) does not hold. If C E to, there exist arbitrarily large values of k such that

J gkldzl

C

Writing 'til =

{C lee B.j

gk

>r

Idzl

>r

to

C

}

Utk

k!";

for every n.

k

k.

n

we therefore have

(4.9)

136

III. Auxiliary Results from Real Analysis ,

On the other hand, since the function 2 k gk is admissible for t' k it follows from the definition of module and from (4.8) that

M(t'J < 22k

ff g%da < r

k.

B

Since the module is subadditive we have

MC~nt'k) -:;'k~ MWk) < 2- n+

1 ,

and M(t'o) = 0 follows from (4.9).

§

J. Approximation

of Measurable Functions

5.1. Preparatory remarks. In this section we discuss two types of approximation theorems. In 5.2 and 5.3, we approximate. pointwise a given measurable function by continuous functions or step functions. The second type of problem is dealt with in 5,4- 5.8, where the approximation is measured in the LP metric. We use the Lebesgue,.area measure, and the functions to be approximated are defined and measurable in the whole finite plane. If necessary we then have to extend functions which are measurable in a set A by assigning them the value zero outside A. Let f be a complex-valued finite measurable function. We first construct a sequence of measurable functions gn' n = 1, 2, ... , converging to f, each gn taking only finitely many values. To this end for every n we divide the square {u + i v I - n u ;":;; n, - n v
<

Iqkl -:;, inf If(z) I , zEAk

<

V2

sup If(z) - qkl < ---.;;- .

(5.1 )

tEAk

We now write 4n'

gn=};qkcAk' k~1

where CA is the characteristic function of A. By the measurability of f, gn is measurable, and by (5.1) Ign(z)1 -:;, If(z) I and lim gn(z) = fez) for every z. If f is bounded, the convergence is uniform.

5.2. Pointwise approximation by continuous functions. By a small modification of the functions gn we obtain for f an approximating sequence of continuous functions fn' However, we can only conclude that f n

§ 5. Approximation of Measurable Functions

137

converges to I almost everywhere. On the other hand, be only almost everywhere finite.

I then

needs to

To construct an approximating sequence of continuous functions we again consider a function gn having the constant value qk in the set A k , k = 1,2, ... ,4 n 4 , and vanishing in A o = - U A k • By 1.2, every set A k n D n , k = 0, 1, 2, ... ,4 n 4 , where D n = {z I Izi n}, contains a compact subset F k such that

<

k~'m (A k n Dn -

F k)

=

m(Dn -

~: F k) < r

n

•

We denote by d(z, F k) the distance between the point z and the set F k and by d the smallest of the mutual distances between the sets F k , k = 0, 1, 2, ... ,4 n 4 • If we write qo = 0,

hk(z) = ( 1 -

2 d(z, P k ) )

d

k = 0, 1, ... , 4 n 4

qk'

°

,

>

for d(z, F k) ~ d/2, and hk(z) = for d(z, F k) d/2, then every function hk is uniformly continuous in the finite plane. The sum 4n'

In = }; hk k=o

coincides with gn in the set U F k. Thus it differs from gn inside D n at most in a set of measure less than 2-n . Since the discs D n exhaust the finite plane as n --->- 00 we conclude that for almost every z the equation In(z) = gn(z) holds from some h onwards. Hence lim In(z) = lim gn(z) = I(z) almost everywhere, and we have constructed the required approximation to f by continuous functions. .. The values IIn(z) I can be greater than I/(z) I at certain points z, but they cannot exceed sup I/(z)l. z

Remark. Consider the restriction of the above function f to a set A of finite measure. By Egoroff's theorem (see 1.3), for every s there is a closed set F C A with m (A - F) s, such that the approximating sequence fn converges uniformly in F. Since the limit function of a uniformly convergent sequence of continuous functions is continuous we have proved the following result on the relationship between a measurable and a continuous function.

>°

<

Lusin's Theorem. Let I be a complex-valued, almost everywhere finite, measurable function in a set A of linite measure. Then for every S there exists a closed set F C A with m (A - F) s such that the restriction of I to F is linite and continuous.

>°

<

138

III. Auxiliary Results from Real Analysis

5.3. Special approximating sequences. We call the set of open squares Qhk = {x + i y I (h - 1) (J x h (J, (k - 1) (J Y k (J}, h, k = 0, ± 1, ± 2, ... , anetands~CQIllesji~~~.eIl_Q. is made smaller. A function which takes a constant value in every square of some net is called a step lunction.

< <

< <

As above suppose that I is an almost everywhere finite, measurable function in the finite plane. To approximate I by step functions we consider the sequence In constructed in 5.2; this converges to I almost everywhere. Since every In is uniformly continuous, we can construct a net N n for every n, so fine that I/n(zl) - In (Z2) I 1jn if Zl and z2lie in the same square Qhk E N n' For every Qhk we denote by Zh k the point of Qhk where Ilnl assumes its minimal value. We then set tpn(z) = In(zhk) if Z E Qhk and tpn(z) = In(z) if Z is a boundary point of Qhk' The function tpn is then a step function which satisfies the inequalities Itpnl ::;; Ilnl, Itpn - Inl < 1jn everywhere in the finite plane. We thus infer:

<

For any almost everywhere finite, measurable lunction I we can construct a sequence 01 step lunctions tpn such that sup Itpn(z) I ::;; sup I/(z)1 and I(z) = lim tpn(z) almost everywhere.· Z Z Finally we remark that an almost everywhere finite, measurable function I can also be approximated by polynomials. To prove this we again consider the above approximating sequence of continuous functions In' By Weierstrass's approximation theorem, there exists for every Ina sequence of polynomials Pnk(z, z) such that Pnk(z, z) -->- In(z) as k -->- 00, uniformly in every bounded set. Such polynomials are, for instance,

If. [ If [ - (_ICI

1 Pnk(z, Z) = -;;;

In(')

1 -

(IClog- kZI)2Jk de; ,

1'1 < log k

where ck

=

1

log k

)2]k d = e;

2

nlog k k

+1

.

1'1
For sufficiently large k = k n we have for P nkn = P n the inequality /Pn(z) - In(z) I 1jn in [zi n. Then lim Pn(z) = lim In(z) = I(z) almost everywhere. In what follows it is significant that from some n onwards the absolute value of the polynomials P n in an arbitrary given bounded set is not larger than the supremum of III in the plane. Since IPn(z) I - 1jn ::;; Iln(z) I < sup I/(z)1 for Izi n. this can be achieved by replacing P n by (1 - 1j(n sup Iii)) P n. We remark that Pnisapolynomial in Z and z (or x and y) and should not be confused with an analytic function of Z = x i y.

<

<

<

+

139

§ 5. Approximation of Measurable Functions

5.4. Smoothing of integrable functions. Let I be a measurable function which is integrable in every compact subset of the finite plane Q. Such a function is called locally integrable in Q. Since I is finite except perhaps on a null set it can be approximated by the methods set out in 5.2 and 5.3. We now study the approximation of I relative to the LP metric. Let fJ be a finite continuous function which vanishes outside a disc. The function I * fJ defined by

* fJ(z)

I

fJ I(z

=

- C) fJ(C) da =

Q

fJ I(C)

fJ (z - C) da

(5.2)

Q

is then finite in the finite plane. The convolution process (5.2) regularizes j': the lunction continuous. This follows immediately from

1* U(z) - I

* fJ(zo) = ff I(C)

I * fJ

1S

n- U(zo - mda,

[fJ(z -

Q

since I is locally integrable and fJ(z - C) - fJ(zo - C) vanishes outside a disc and tends to zero uniformly as z -+ zoo If fJ is continuously differentiable, then the mean value theorem, applied to the real and imaginary parts of fJ, implies that the expression 1J(z

+ h) h

fJ ( ) x z ,

1J(z) _

where h #- 0 is real, tends to zero uniformly as h

r

1m

h~O

= !~

f * 1J(z

JJ

+ h)h - f * 1J(z)

I(C)

[1J(Z

+ h - ~~ -

I

_

->-

0; It follows that

* fJx (Z)

1J(z -

~)

-

fJ x(z - C) ] da

=

0.

Q

Thus

I * fJ

has the partial derivative

(f * fJ) x = I * fJ x ' which is continuous everywhere in the finite plane since fJ x is continuous. In the same way we can show that the partial derivative (f * U)y exists and equals I * fJy' We say that a function belongs to the class en if it is n times (in the case n = 00 infinitely many times) continuously differentiable. By repeating the above argument we deduce in general that I * fJ E en if fJ E en. The support of a function I is the closure of the set of all points z at which I(z) #- O. The class of functions belonging to en in the finite plane Q and having bounded support will be denoted by e~. If I has

140

III. Auxiliary Results from Real Analysis

bounded support in Q then it follows from the definition (5.2) that

I * {} also has bounded support. 5.5. A lemma on convergence in the LV metric. The importance of the above regularization process lies in the fact that for suitably chosen {} the function I * {} can be used to approximate I in the LP metric. First we prove the following lemma:

11 I belongs to LP in the linite plane Q, then lim ;,~o

If I/(z + A)

- l(z)JP da

Prool: For every given

8

+ A)

(5.4)

> 0 we can find a number R such that

JJ IfIP da < Izi

Since I/(z

0.

=

Q

- l(z)JP ::;; 2PI;,(A)

2pc+Z •

;";R-I

=

1

ff

(1/(z I/(z

+ A)JP + II (z)IP) , the

+ A) -

integral

l(z)IP da,

A

then satisfies

I;, ({zllzl ~ R})

< 4c

(5.5)

for IAI < 1. The integral of III P over A is absolutely continuous as a function of the set A (see 2.1). Thus we can find an'Yj 0 such that this integral is less than 812Hz for every A with m(A) 'Yj, and consequently

<

>

(5.6)

< +

By the theorem of Lusin proved in 5.2thedisc D R + 1 = {zllzl R 1} contains a closed subset F with m (D R + 1 - F) 'Yj, such that the restriction of I to F is continuous, and since F is compact, even uniformly continuous. Thus there exists 15 E (0, 1) such that

<

+ A)

- l(z)IP

provided that z E F, z

+ A E F.

I/(z

< 4 n:(F)

IAI

for

< 15 ,

(5.7)

We shall prove that I;,(Q) < 8 for IAI < 15. To this end we consider the following three sets: A I = {ZIZEF,z+AEF}, A 2 = {ZIZ+A E D R + 1 - F}, As = D R + 1 - F. Since IAI 1, z A belongs to D R + 1 if Izi R. The set Al U A 2 U As therefore covers the disc DR and so

<

<

3

I;,(Q) < l: I;,(A k ) k=1

+ I;, ({z Ilzl

+

~ R}).

141

§ 5. Approximation of Measurable Functions

The last term on the right was estimated in (5.5). From (5.7) we see that I",~AI) 13/4. Since the measures of A 2 and A 3 are less than 1] we deduce from (5.6) that I",(A k ) 13/4, for k = 2,3 also. Thus I",(Q) < 13 and the assertion (5.4) is proved.

<

<

5.6. Approximation in theLP metric. By applying the above result (5.4) we can carry out the required approximation to I in the LP metric. Denote by {}n a finite non-negative continuous function which vanishes outside the square Qn = {x + i Y I - 1/n x 1/n, -1/n Y 1/n} and satisfies the condition (5.8)

< <

< <

The following lemma states that the functions I * {}I' I * {}2' .. " constructed from the sequence {}t> {}2' ... by (5.2), approximate I in the LP metric. Lemma 5.1. Let I E LP in the linite plane Q. Then

If II - 1* {}nl P da =

lim

(5.9)

0.

n-oo{J

Prool: Because

I(z) - f

* {} n(z) = ff

[fez) - I(z - C)] {}n(C) da

Q

it follows from Holder's inequality 37 (or immediately if p = 1) that

I/(z) - I * {}n(z)IP =

'ff I [fez) Q

~ II/I(Z)

I(z -

m({) n(C))P ({}n(C)) 1

1

1

- I(z - C)IP {}n(C) da.

-P da

IP

(5.10)

Q

Now integrate both sides with respect to z. In order to be able to apply Fubini's theorem as quoted in 1.5 to the right hand side we must know that the integrand is measurable with respect to the Lebesgue volume measure in the four-dimensional z C-space. This follows from the fact that I(z - C) can be transformed into a function of z by a rotation of the z C-space and that the Cartesian product A X Q = {(z, C) I z E A, CEQ} is measurable in the four-dimensional volume measure if A is measurable with respect to the area measure. We can thus apply Fubini's theorem and obtain from (5.10)

fJ Q

37

I/(z) - 1* {}n(z)IP da z :;:;;

Holder's inequality

holds for

F,

G

~

0,

fJ (fJ Q

If F G da:;:;;

p>

1,

1/p

Q

I/(z) - I(z - C)IP daz) {},g) da, .

(fJ FP da)l/P (fJ Gq da)l/q

+ 1/q =

1.

142

III. Auxiliary Results from Real Analysis

Since {}n vanishes outside Qn it is sufficient to take the outer integral on the right hand side over Qn' Then lei V2/n. Since the inner integral tends to zero as C-+ 0 by (5.4) and {}n is normalized by (5.8), it follows that the whole double integral has limit zero as n -+ 00. This proves the lemma.

<

5.7. V-approximation by

C~-functions.

Write

1

{}n( Z) -_

an e

Izl' - n-'

for

1

1 <~,

Iz/

for Izl2:-;-, (5.11)

n

where an is to be chosen so that (5.8) is satisfied. Then {}n E Coo, and by 5.4 the functions 1* {}n also belong to Coo. A function I E LP can thus be approximated in the LP metric by infinitely differentiable functions. In fact, a stronger result holds: A lunction IE LP(D) can be approximated in the LP metric by lunctions 01 the class C~.

<

To see this we write Ifk(Z) = I(z) for z E D k = g I lei k}, and Ifk(Z) = 0 for Z E - D k , k = 1,2, .... If {}n is the function defined by (5.11), then by Lemma 5.1 we have

JJ Ilfk -Ifk * {}nl P da < k- P [)

for a sufficiently large n = n k • Every approximating function I k = Ifk * {}nk belongs to Coo and vanishes outside the disc Dk+l/nk' The triangle inequality now yields

(£J II -

P Ikl dar'P <

(!L'/' P dafP + ~ ,

and the desired result follows. 5.8. Pointwise convergence of an LP-approximation. For a continuous function I E LP the sequence I * {}n constructed above converges not only in the LP metric but also pointwise. Lemma 5.2. Let the integrable lunction I be linite and continuous in an open set G. Then (5.12) I(z) = lim 1* {}n(z) , the convergence being unilorm in every compact subset F 01 G. Prool: We choose a compact set F o such that F C F o C G and F lies at a positive distance d from the complement of Fo. It follows from the definition of 1* {}n that

/f(z) - I

* {}n(Z) I < JJ I/(z) Qn

- I(z - C)/ {}n(C) da.

(5.13)

§ 6. Functions with LP-derivatives

143

>

V2Jd and z belongs to F, then the point z - Clies in F o if CE Qn' Since I is uniformly continuous in the compact set F o, the assertion (5.12) follows immediately from (5.13).

If n

-Iei'

§ 6. Functions with LP-derivatives 6.1. Definition of a function with LP-derivatives. We now introduce a class of functions which will play an important role in Chapters IV to VI. In what follows G will always denote a domain of the finite plane. We say that a finite complex-valued function I defined in G has LP-derivatives, p > 1, if it satisfies the following two conditions: 1.

I is

absolutely continuous on lines in G (d. 3.1).

2. The partial derivatives

Ix' I y belong to

LP in every compact subset

of G. A function with V-derivatives is also called absolutely continuous in the sense 01 Tonelli. It follows from Holder's inequality that a function with LP-derivatives also has U-derivatives for q < p.

6.2. Integral transformations for functions with LP-derivatives. We shall give two more characterizations of the functions with LPderivatives, both expressing important properties of this class of functions. The first of these characterizations is based on the fact that a transformation formula from classical analysis between curve and surface integrals holds for functions with LP-derivatives and only for these. We remaind the reader that aD denotes the boundary of the Jordan domain D, positively oriented with respect to D (d. 1.1.4). Lemma 6.1. A lunction I which is linite and continuous in G possesses LP-derivatives in G il and only il there exist two lunctions g and h belonging to LP in every compact subset of G such that the following condition holds: II R, ReG, is an arbitrary rectangle with sides parallel to the axes, then (6.1) J I dx = - JJ g da , J I dy = JJ h da . oR

In this case

R

Ix =

h,

I = y

oR

R

g almost everywhere in G.

We first suppose that f has LP-derivatives In G. Let R = {x + i y I a x b, c y d}, ReG. Since the restridion of I to Proof:

< <

< <

III. Auxiliary Results from Real Analysis

144

the vertical segments {x + i Y I e < Y < d} is an absolutely continuous function of y for almost all x, a x b, we have

< <

b

J I dx = J (I(x + i

oR

e) - I(x

a

+ i d)) dx =

b

d

- J dx J Iy(x a

c

+ i y) dy .

Here I y belongs to LP C V in R, and the double integral can therefore be transformed into a surface integral over R by Fubini's theorem. We thus obtain the first of the relations (6.1) with g = Iy' The second relation is proved analogously and the necessity of (6.1) follows. To prove that the condition is also sufficient we start with two functions g and h locally in LP in G and satisfying (6.1). We apply the first relation (6.1) to the rectangle R xy C R with vertices a + ie, x + ie, x + i y, a + i y. This gives x

J (1(; + i

e) -

1(; + i

y)) d;

= -

JJ g dO' =

R xy

a

x

y

- J d; J g(; a

c

+ i 1]) dO'.

(6.2)

< <

Let Yn' n = 1, 2, ... , be a sequence everywhere dense on e y d. If we differentiate (6.2) with respect to x for a fixed Yn and remember that the derivative of an integral is almost everywhere equal to the integrand, we obtain I(x

+ i Yn ) -

I(x

+ i e)

Yn

J g(x + i 1]) d1]

=

(6·3)

c

< <

for every x, a x b, with the exception of a set En of linear measure zero. If x does not belong to the null set E = U En' then (6.3) holds for every Yn . Since both sides of (6-3) depend continuously on Yn , we have I(x

+ i y)

- I(x

+ i e)

y

= J g(x

+ i 1]) d1]

everywhere in R except for the vertical segments x E E. J3eing an fujegral. t is aD _ab~Q.lutely ~9nj:ill_gous function of y for almost every x E (a, b). Furthermore: we see th~t Iy(x y) = g(x i y) for almost all y provided that x (! E. Since I y and g are measurable with respect to the area measure it follows from Fubini's theorem that Iy(z) and g(z) coincide except on a set of zero area measure.

+--i

+

If we exchange the roles of x and y we see that I is also absolutely continuous on almost all horizontal segments and has the partial derivative Ix = h almost everywhere. The proof is now complete.

6.3. Approximation of functions with LP-d~rivatives. A further characterization of the functions with LP-derivatives depends on the

§ 6. Functions with LP-derivatives

145

fact that they admit a particular Coo-approximation. The approximating sequence can be chosen such that the functions converge pointwise and their derivatives converge in the LP metric. Lemma 6.2. Let f, g, h be three functions defined in the domain G and suppose that g and h belong to LP in every compact subset of G. Further let fn be a sequence of continuously differentiable functions in G such that the following conditions hold for every compact subset F of G: 1. lim (In -

fl

=

° uniformly in F,

n~OO

2. }~~

IP,

afn

ffl ax - hi

da

=

0,

F

Then f has LP-derivatives in G and f x

= h, f y =

g almost everywhere in G.

Conversely, if f has LP-derivatives inG there exists a sequence of functions fn E C;;" satisfying conditions 1 and 2 with h = f x ' g = f y • If the support of f is a compact subset of G, then the functions fn can be chosen so that the conditions 1 and 2 hold with F replaced by G. Proof: We start by proving the first assertion. It follows from condi-

tion 1 that f is continuous. If R, ReG, denotes a rectangle with sides parallel to the axes, then J

U

f dx

ff~n da.

= lim Jfn dX = -lim n-oo

n-oo

U

R

Y

From the second equation in 2 and Holder's inequality we deduce that lim n-oo

ff~nY da = R

ffg da . R

.

Hence

J f dx = - JJ g da . oR

R

In the same way we can prove that

J f dy = JJ h da .

oR

R

Lemma 6.1 now shows that f has LP -derivatives in G and that = g almost everywhere in G.

fx =

h,

fy

Conversely, let us assume that f has LP-derivatives in G. In order to construct a sequence of C;;"-functions with properties 1 and 2 we make use of the regularization process introduced in 5.4.

146

III. Auxiliary Results from Real Analysis

It is enough to carry out the construction so that conditions 1 and 2 are satisfied with h = lx' g = Iy for a fixed compact Fe G. The required approximation of I in every compact subset of G is then obtained by· exhausting G by an increasing sequence of compact sets, constructing the corresponding sequence of ego-functions and choosing the diagonal sequence.

We choose a compact set F o, F C F o C G, such that F is at a positive distance d from the complement of F o. The function I defined by the formulae fez) = I(z) for z EF o, fez) = 0 for z ({ Fo, is integrable in the plane. We can choose F o such that its boundary h~s zero area measure (d. n.S.1). Then the derivatives x and belong to LPin the finite plane. Using the functions (5.11) introduced in 5.7 we construct the sequence

f

In=I*U n ,

I:

(6.4)

n=1,2, ... ,

where every In then belongs to ego. By Lemma 5.2 we have lim In = I = I uniformly in F, and so condition 1 is fulfilled. If the support of I lies in F, then In ~ I uniformly even in G. To prove that condition 2 holds we use Lemma 5.1. Since LP in the finite plane, we obtain lim

JJ I/x- Ix * UnlP da =

Ix belongs to (6.5)

0 .

n-oo F

If the support of I lies in F, then this relation holds with G replacing F. The derivative Iy is approximated in the same way by Iy * Un'

-

-

-

To c~mplete the proof we must show that Ix * Un = (f * Un)x and Iy * Un = (f * Un)y from some n onwards, everywhere in F. It will appear from our proof that these relations are valid everywhere in G if the support of I lies in F. Choose n

> V2jd and, for brevity, write U instead of Un'

Since

ix (z -

C)

= f:.(z - C) for z E F, 1; E Qn (with Qn as defined in 5.6) we first obtain

_ Ix

* U(z)

=

JJ fx(z -1;) U(1;) da =

Qn by Fubini's theorem.

lin lin J d'YJ J Ix(z - ; - i 'YJ) U(; -lin -lin

+ i 'YJ) d;, (6.6)

By the hypothesis, the function f{J defined bYf{J(1;) = I(z -1;) is absolutely continuous on almost all horizontal segments 1'] = {1; = ; i 'YJI - 1jn < ; < 1jn}, 1'] C Qn' For every such value of 'YJ the inner integral on the right hand side of (6.6) can be transformed by integrating by parts. Since U vanishes at the endpoints of 1'] we obtain

+

l~

l~

JlxCz-;-i'YJ) 8(;+il'YJ)d;= JI(z-;-i'fj)U;(;+i'fj)d;. -l~

-l~

147

§ 6. Functions with LP-derivatives

This equation holds for almost all 'YJ and it follows from (6.6) that _ lin lin * {}(Z) = f d'YJ f I(z - ~ - i 'YJ) {}x(~ i 'YJ) d~ -lin -lin

«

+

= f f I(z - C) {}x(C) da =

-

Qn

- * {})x'

By (5.3) we have 1* {}x = (I = (f * {})x (z) as required. The proof that iy * {}(z) = proved.

i * {}x(Z) .

and consequently Ix

* {}(z)

(F * {})y (z) is similar, and the lemma is now

6.4. Approximation in the finite plane. We now consider the special case when p = 2 and G is the finite plane Q. If we supp'Ose that J.,l!e partiaLg~rivatives of I are notm{lrely locally_sqlla.t~ iriifgrable_.b1!t belong to L2(Q), then Lemma 6.2 can be complemented as follows. Lemma 6.3. Let I be absolutely continuous on lines in the linite plane Q and suppose that its partial derivatives belong to V(Q). Then there exists a sequence 01 lunctions In' n = 1,2, ... ,in the class C;{' lor which the lollowing conditions hold:. 1. lim In(z)

= I(z) unilormly in every bounded set,

n~OO

2. lim

f f Il x - (fn)xI 2 da = lim f f Il y - (In)yI2 da = O. n-oo D

n-oO D

Prool: If the support of I is bounded, then the assertion follows immediately from Lemma 6.2. Hence it is sufficient to approximate I in the above way by functions In with V-derivatives and bounded support. We set inez) = VJn(lzl) I(z), n = 2,3, ... , where the function VJn is continuous for alllzi = r ~ 0, takes the values VJn(r) = 1 for 0 S r < n, VJn(r) = 0 for r ~ en, and has derivative VJ~(r) = cnf(r log r) for n r en. Then

<

<

en

f ~dr= - 1 , r logr

(6.7)

n

and so C

I:

n

=

1·

log log n - log n

(6.8)

Each function is absolutely continuous on lines in Q and has bounded support. We assert that the above co~ditions 1 and 2 hold with In replacing In' It then also follows that In has V-derivatives.

148

III. Auxiliary Results from Real Analysis

Condition 1 clearly holds. We denote by Ix 1f!n the function with the value IAz) 1f!n(lzl) at z. Since Ix 1f!n -+ Ix in the L2 metric, the first relation in 2 will follow from the triangle inequality if we show that

fJ

lim

I/ x 1f!n - (l;.)xI 2 da

n-oo[J

(6.9)

0.

=

The corresponding relation for derivatives with respect to y can then be proved in the same way. We have

JJ Ilx 1f!n -

en

2

(l;.)xI da

JJ

~ dr 11f!~(r) I(r ei'P) 12 r drp n

D

= If

I is

21l

0

J~ JI/(r ei'P) 12 drp en

( 2"

n r log2 r

)

dr.

0

(6.10) •

continuously differentiable in the disc Izi :::;; r, then .

.

2

I/(r e''P) - l(e''P)1 =

e''P) J-ar- dr :::;; J-;-JI-ar-I r dr , r af(r

. e''P)

2

r

1

I1

1

r

dr

af(r .

2

1

and consequently 2"

f

o

I/(r ei'f) - l(ei'P)12 drp :::;; log r ff (1/ x12 Izi <

+ I/ y l2) da .

r

From Lemma 6.2 we see that this also holds without the assumption of differentiability. For all sufficiently large values of r we thus have 2"

f

If(r ei 'f)1 2 drp :::;; (1

o

+ 211/xll; + 211/y l@ log r =

A log r.

From (6.10), using (6.7) and (6.8), we deduce that

ff D

ll x 1jJn -

(j)n x12 da < =

A cnf~dr r log r = -A cn =

---~~_._- log log n

log n

n

from some n onwards. This implies (6.9), and so Tn satisfies conditions 1 and 2.

6.5. Generalized Green's formula. By aid of Lemma 6.2 the classical Green's formula can be generalized: Green's formula. Let I and g be lunctions having respectively LP- and U-derivatives in the domain G, where 1/P + 1/q = 1. II D, D c G, is a Jordan domain with a rectiliable boundary on which g is 01 bounded variation, then (6.11)

§ 6. Functions with LP-derivatives

149

The same relation holds il I has V-derivatives and

g

belongs to the class

C2.38

Prool: If I and g have LP- and U-derivatives, Lemma 6.2 implies the existence of functions In' gn E Coo, n = 1, 2, .. " such that lim lin - II n-oo = lim Ign - gl = 0 uniformly in D and n-oo

}~~JJI OUno~!) IP da = !~ooJJlo Uno;!) IP da D

D

=}~~JJI O(gno: g) Iq da = !~~JJI O(gno; g)-!qda = D

If g E C2 and

above (with

O. (6.12)

D

I has V-derivatives, then we choose the sequence In as 1) and set gn = g for n = 1,2, ....

p=

We assume here the validity of Green's formula (6.11) for C2-functions (see for example Apostol [1J, p. 289). Then

I d = ff(Oin ogm _ oin Ogm) da n gm ox oy oy ox J iJD D

(6.13 )

for every pair In' gm in both cases above. Now let m -+ 00. Then gm converges uniformly to g on the boundary of D. Since In' gm and g are continuous and of bounded variation, we obtain by integrating by parts lim m-oo

J Indg m =

iJD

-lim m-oo

J gmdln = - J gdln = J Indg. iJD

To carry out the limiting process m (6.13) we note first that the integrals

iJD

-+ 00

(6.14)

iJD

on the right hand side of

JJI~:IPda, D

are uniformly bounded, by (6.12). Since we only need to deal with the case p 1, Holder's inequality yields

>

38

The requirement g

E

C 2 can clearly be replaced by a much weaker one.

III. Auxiliary Results from Real Analysis

150

By (6.12) the right hand side tends to zero as m ~ 00. It follows from this and (6.14) that (6.13) remains valid if g;" is replaced by g. The uniform convergence of I.. on the boundary of D implies that lim J .. _00

In the case p

I.. dg =

cD

J

I dg .

(6.15)

cD

> 1 we apply Holder's inequality as above and infer that

lim n-oo

11((Of,,-- Of) og _ (ofn _ Of) {}g) da ax ax oy oy oy ax

=

o.

(6.16)

D

For p = 1, g E C2 , we have

Of.. - of) og _ (Of.. _ of) Og) dal ax ax oy oy oy ax If(( I If I o(fn-oy < If Io(fnax- I + D

f)

rna! Ig,,(z)l zeD

da

D

rna! Ig..(z)1 zeD

f)

I da,

D

and we see that (6.16) holds in this case too. By (6.15) and (6.16), we can replace I.. by I in (6.13), as we have replaced gm by g. The generalized Green's formula is now proved. Example: In the two simple cases g(z} = z and g(z} = k, we obtain the following result from Green's formula: III has V-derivatives in G, then J I dz

cD

=

2 i fJ D

I. da ,

Jldz= -2ifJl z da

cD

(6.17)

D

lor every Jordan domain D, D C G, with a rectiliable boundary.

6.6. Absolute continuity of homeomorphisms with L2-derivatives. Of the functions with LP-derivatives, the case p = 2 is the most important to us. In fact, we show in IV.1 tb.J!..:t_~~e!:Y-9.11~siconformal mapping ha?_ V-derivative,? The following result can therefore later be-appueddirectly to quasiconformal mappings. Theorem 6.1. A homeomorphism w 01 the domain G, having V-derivatives in G, is locally absolutely continuous in G. Prool: Without loss of generality we may assume that w is sensepreserving. By Lemma 3.3, the Jacobian] of w satisfies the inequality

JJ] B

da::;; m(w(B))

(6.18)

§ 6. Functions with LP-derivatives

151

for every Borel set BeG, 'YAffe equ
J u dv = f f J da ,

oR

R

J u dv oR'

=

ff

da ,

R'

where in the left hand equation u and v denote the real and imaginary parts of w, and in the right they are the coordinates of the w = u + i vplane. ~-,jjlt~~hand sides of the aQQY~ equations ar~_~qual. Thus equality holds in (6.18) if B is the closure of a rectangle R with sides parallel to the axes. The same is then true for every open BeG, since an open set of the finite plane consists of countably many non-intersecting half-open rectangles with sides parallel to the axes. To demonstrate equality in (6.18) for general Borel sets B, we need 0 consider only sets B with BeG. If B is such a set, then for every . there exists an open set DB' B C DB' DB C G, with m (DB - B) 8. Since the integral (6.18) is locally absolutely continuous in G as a set function, we then have

8> <

ff J da =

lim

B

8-0 De

ff J du =

lim m(w(D B)) ;;::: m(w(B)) .

8-0

By (6.18) equality holds, and Theorem 6.1 is proved.

6.7. Composition of functions with LP-derivatives. As a second application of Green's formula we establish the following result:

+

+

Lemma 6.4. Let w = u i v: G -+ G' and its inverse w- 1 = x i Y be homeomorphisms which have LP-derivatives with p ;;::: 2 in G and G'. Further, suppose that f is a function with U-derivatives in G', where 1JP 1Jq = 1. Then the composite function f 0 w has V-derivatives in G and its partial derivatives satisfy the chain rule

+

(I ~w)x = (lu ow) u x + (Iv ow) vx' almost everywhere in G.

(I Owly = (lu ow) U y + (Iv ow) vy (6.19)

152

III. Auxiliary Results from Real Analysis

Proal: By Theorem 6.1, wand w- l are locally absolutely continuous in G and G', respectively. Furthermore it follows from Theorem 3.1 and Lemma 3.4 that wand w- l are differentiable and have a positive Jacobian except on certain null sets Eo C G, E~ C G'. Since a locally absolutely continuous homeomorphism maps null sets onto null sets, the set E = Eo U w-I(E~) is also of zero area measure. If the point CE G does not belong to E, then w is differentiable at Cand w- l at w(C) = A simple calculation then shows that

r.

Uy(C) = - xv(C') !(C) ,

Vy(C) = xu(C') !(C) ,

(6.20)

where! denotes the Jaco1:;>ian of w. Thus these equations hold almost everywhere in G. Having this preliminary result, we want to show that low satisfies the condition (6.1) given in Lemma 6.1. To this end we consider a rectangle R, ReG, with sides parallel to the axes. First we suppose that w is absolutely continuous on the boundary of R. Then w(Fr R) is rectifiable. Green's formula (6.11) can now be applied to the functions I and x in the domain R' = w(R). This gives (6.21)

Furthermore Lemma 3.5 tells us that

Hence, in view of (6.20) and (6.21),

J (low) 3R

dx

= -

JJ R

((tu

0

w) u y + (Iv

0

w) v y) da.

(6.22)

In the same way we prove the formula. (6.23)

These relations are proved under the assumption that w is absolutely continuous on the boundary of R. However, since w)s absolutely continuous on lines in G by hypothesis, it follows from the continuity of I and the surface integrals appearing in (6.22) and (6.23) that (6.22) and (6.23) are valid for every rectangle R, ReG, with sides parallel to the axes. Lemma 6.1 now gives the required result that low has V-derivatives in G and that the relations (6.19) are true almost everywhere in G.

Remark. If the mappings wand w- l are continuously differentiablein G and G', respectively, it follows from (6.19) that low has U-derivatives simultaneously with f.

153

§ 6. Functions with LP-derivatives

6.8. Absolute continuity on arbitrary arcs. It follows from the above that~fun~withLP-derivativ~§~ely.sontinuous not_()~ OJ! horizontal anQ.y~ti~.~Jl'~gments,but on much J!!.QI P general am;.

To-~xpress this fact in an exact form-we -~~cept of the module of a family of arcs introduced in § 4. In the case p = 2 one can prove the following (d. Vaisala [1J):

Theorem 6.2. Let I be a lunction with V-derivatives in G. Then I is absolutely continuous on all arcs C, C C G, with the exception 01 a lamily 01 module zero. Prool: Since the family of all non-rectifiable arcs has zero module by 4.3 we may restrict ourselves to rectifiable arcs.

Let F be a compact subset of G. By Lemma 6.2 there exists a sequence of functions In E Coo, n = 1, 2, ... , such that In -+ I uniformly in F and the partial derivatives of In converge to I" and Iy in V(F). Being derivatiYe£_of a ~Il.t1Qus ftl:r:t~tion,_ I"" ~nAI y. are Borel functiolls. Hence, by Lemma 4.2, there exists a subsequence of In> also -(fenated by In' such that the relations

. JI ax ofn

hm

n'+oo

of OX I Idzl

Ja - a I

. = hm

n-oo

c

10fn Y

c

of Idz[ Y

= 0

(6.24)

hold for all rectifiable arcs C C F, apart from a family of module zero. We show that I is absolutely continuous on every arc C for which (6.24) holds. Let 1,; = x i y: I -+ C be a length-preserving parametrization of C, choose two points SI and S2' SI S2' on I, and set ZI = 1,;(SI)' Z2 = 1,;(S2)' If the partial derivatives of In are considered as functions of s, then

+

<

J[~; 5,

I(Z2) - I(ZI) = lim (tn(Z2) - In(ZI)) = lim n-OO

n-oo

x'

+ ~nY y'] ds,

(6.25)

5,

where 52

J[(

Ix'i < 1, Iy'l <

1. Using the inequality

I . Ofn - Of) x' + (ofn _ Of) ,] dsl s ofn _ of + ofn _ of ox ox oy oy y OX ox oy oy

J( I

II

I

c

~

and the limit relations (6.24), we then obtain from (6.25)

I(Z2) - I(ZI) =

5'(Of

Of) a; x' + oy y' ds. J

5,

Being an integral

I is thus

absolutely continuous on C.

I) Idzl

154

III. Auxiliary Results from Real Analysis

Until now we have considered only arcs C in the compact set F. Let F v F 2 , ••• , be compact sets which exhaust G. By the above M(t',,) = 0 for every n, where 't" is the family of those arcs C C F" on which j is not absolutely continuous. Since the module is subadditive by 4.2, we have and the theorem is proved.

§ 7. Hilbert Transformation 7.1. Generalization of Cauchy's integral formula. As an application of Green's formula we shall give an integral representation for functions with V-derivatives, which coincides with Cauchy's formula in. the case of an analytic function. This result in the differentiated form and combined with certain general properties of LP-spaces leads to farreaching consequences. Let j be a function having V-derivatives in a domain G and D, D C G, a Jordan domain with rectifiable boundary. Further, let z be a point of D and denote by Dr the disc {CI IC - zl < r} , Dr CD. The function 'IjJ defined by the formulae

'IjJ(C)

=

for CE G - Dr'

j(C) (C - Z)-1

'IjJ(C) = r- 2 j(C) (C - z)

CE Dr'

for

then has V-derivatives in G. Applying the first formula (6.17) to in the domains D and Dr and subtracting gives

J

f(?:')

?:,-z

J

dC -

aD·

f(?:')

?:,-z

dC = 2 i

aD r

If

fe(?:')

?:,-z

'IjJ

da.

D-D r

Now let r --+ O. Since f is continuous, the second integral on the left hand side has the limit 2 n i j(z), and it follows that

j(z) =

_1_. 2nz

J

dC - ..!-lim

f(?:')

?:,-z

n

r-O

aD

The first integral

W j(z) = D

If

f,(?:')

?:,-z

da.

(7.1)

D-D r

_1_.

2nz

J

f(?:')

?:,-z

dC

(7.2)

aD

defines an analytic function WD j in D. For the second integral we write 1. - -hm n

If

r-O D-D

r

oo(?:,) r--da .,,-z

= - -n1

J

D

oo(?:,) r--da .,,-z

= TDw(z) ,

155

§ 7. Hilbert Transformation

even if the integral exists only as a Cauchy principal value. Using thi,s notation we can express (7.1) as follows: Lemma 7.1. A function f having V-derivatives in G has the representation f=WDf+TDIz in D where W D f is analytic.

f is identically zero on the boundary of D, then W D and the above formula reduces to ,the simpler form

If

f vanishes in D (7.3)

-Cd _C- R 2

JC-

z -

JC(C -

dC

=

C, the analytic

R2J(C-1 1)C dC-

--

z) -

----

z

i!D

iJD

< Rand f(C)

lei

In the special case when D is the disc part of f also vanishes. For then

z

-0

aD

for ZED and so

Z=

-~If~. n C- z

(7.4)

ICI
7.2. Definition of the Hilbert transformation. The question of differentiating (7.3) leads to the so-called Hilbert transformation which will now be defined. We restrict ourselves first to C~-functions (see 5.4) in the finite plane Q. For W E C;;" we write 1 T w(z) = - -;;

If

w(C)z da. C_

D

Fix a point Zo

E

Q and define a function 'lfJ by the formula

'lfJ(z) = T w(z) - w(zo) Z.

(7.5)

We show that 'lfJ has at Zo a derivative 'lfJ'(zo) ='lfJz(zo) which is independent of direction, and that 'lfJ,(zo) therefore vanishes. This leads to the Hilbert transformation of wand simultaneously yields rules for the differentiation of T w. In view of the definition of T wand (7.4) we have 1JI(zo

+ h)

-1JI(zo)

h

= _

~

ff

n.

(C -

w(C) - w(z_o)_ da Zo - 11) (C - z o ) '

(7.6)

D

<

where D = {CIICI R} is an arbitrary disc containing the point Zo and the support of w. We let h -+ 0 and show that this limiting process commutes with the integration.

156

III. Auxiliary Results from Real Analysis

To do this we note that there exists a finite number M such that Iw(C) - w(zo)1 :::;:; M IC - zol in D. Thus the integral

ff

£0 (zo) I da IC - ZOl2

IW(C) -

D

exists and

iff = ff I

w(C) - w(zo) da (C - Zo - h) (C - zo)

ff

D

rotC) -

w(zo) da I (C - ZO)2 I

D , £0 (C) - w(zo) (C - ZO)2 (C - Zo - h)

D

h da ~ Mlhl

ff

da

IC - zollC - Zo - hi'

Ie - zol < Ihl/2

For sufficiently small Ihl, the discs Ihl/2 lie in D. Outside these discs

<

(7.7)

D

and

Ie - Zo -

hi

and we have

+ 31hl I

21hl

~

II

da

IC-

Zol

2

= 4n Ihl

4R

+ 6n Ihllog-1hl .

le-zol <2R

This expression tends to zero with h, and so by (7.6) and (7.7) the derivative (7.8)

exists. The term w(zo) can here be omitted. In fact, in view of the second formula (6.17), we have (d. 7.1) 2

ill D-D r

-

(1;

~zo)2 = I c· ~z~ -

for every disc Dr cipal value

iJD

=

{CI IC - zol 5 w(z)

I

-

C

iJD r

~ Zo =

< r},

1 = - --;;

ff

- R2

I

iJD

C2

/~ zo)

=

0

Dr CD, and the Cauchy prin(C w(C) _ Z)2 da

(7.9)

Q

thus exists. The linear operator 5 in (7.9) is called the Hilbert transformation, and 5 w is the Hilbert transform of w.

157

§ 7. Hilbert Transformation

7.3. Differentiation of the Hilbert transform. By (7.5) and (7.8) we have (7.10) (Tw)z = 5w, (Twh = w. Applying the second formula (6.17) we further obtain (d. 7.1) 5 w(z) - T wz(z)

If aca = J

(W(C) C_ z ) da

1 =;;-

D·

_1_.

2:nt

cD

w(C)

C-z

it + _1_. lim 2:nt

0

r-

J cDr

w(C)

C-z

if"

<

where Dr = {CI IC - zl r}. On the right hand side the first integral vanishes since w vanishes on Fr D, and it is .clear that the second term on the right is also zero. Consequently, (7.11)

5w = Tw z '

and we deduce from (7.10) that (5 wh

= wz '

(7.12)

It follows from (7.5) that T w is everywhere continuous. Thus lty (7.11) 5 w is also continuous. From (7.10) we then infer that T w is

continuously differentiable and (7.11) shows that 5 w has the same property. Repeating this argument we see that T wand 5 w belong to the class Coo in Q. Furthermore, it follows from (7.10) and (7.12) that T wand 5 ware analytic outside the support of w.

7.4. V-norm of the Hilbert transform of a C~-function. We wish to extend the definition of the Hilbert transformation to functions which belong to V(Q). This extension is based on the fact that the Hilbert transformation preserves the V-norms of

C~ -functions:

JJ IS wl 2 da = JJ [w1 2 da .

D

(7.13)

D

To prove this we use the identities (7.14) and obtain from (7.10)

JJ Iwl 2 da = JJ w (T w)z da = JJ (OJ D

D

~

T w)z da -

JJ W z T w da

~

(7.15)

158

. III. Auxiliary Results from Real Analysis

<

for every disc DR = {zllzl R} containing the support of w. On the other hand, it follows from (7.10), (7.12) and the second identity (7.14) that ff 15 wl 2 dCT = lim ff 5 w(T w}-z dCT fJ

R-oo DR

= lim (ff (5 w T w}z dCT - ff w. T w dCT) . R-oo

DR

(7.16)

DR

Applying (6.17) gives

II(w Tw).dCT = -

1 2i

Da

I w Twiz, WR

II (5 w T w}-z dCT =

1 2i

I 5 w T w dz . ~DR

DR

The integrals in the first equation are equal to zero since w vanishes Rowe obtain on the boundary of DR' If the support of w lies in Izi for T wand 5 w at a boundary point z of DR) DR, the estimates

<

15 w(z)1 ::;; n (izi

~ R O)2 If Iwl dCT. [J

Thus lim fI(5 w T w}-z dCT =

1 2i

R-ooDR

lim I5 w Tw dz = 0,

R-oo

~DR

and the required equation (7.13) follows from (7.15) and (7.16).

7.5. Completeness of V-spaces. With the help of (7.13) the Hilbert transformation 5 can now be extended to the class V(Q). To this end we first make some general remarks on the spaces LP, P ::2: 1. A sequence In' n = 1, 2, ... ,in LP (ef. 4.4) is called a Cauchy sequence 0 we can find an n. such that 111m - Inll p E for m, if for every f n n•. A convergent sequence in LP is certainly a Cauchy sequence. The converse is also true: II In is a Cauchy sequence in LP then there exists IE LP such that lim Il/n - Ill p = 0 (Munroe [1J, p.243). This property of LP is called completeness.

>

>

<

In general it does not follow from the convergence of a sequence In in the LP-metric that the functions In converge pointwise. However, we have (Munroe [1J, pp. 225 and 229): II the 11mctions In converge to I in the space LP, then there is a subsequence Ink 01 In such that lim k-oo

lor almost alZ z.

Ink(z)

=

I(z)

159

§ 7. Hilbert Transformation

7.6. Extension of the Hilbert transformation to V. Using (7.13) we can now extend the Hilbert transformation to the space LZ in the following way. Let W be an arbitrary function which belongs to V in the finite plane D. By 5.7 there exists a sequence of functions w n E Cgo such that lim Ilwn - OJllz = 0 . n~oo

By (7.13) we have

115 OJ".

-

5 OJnll z = I[OJ In

-

OJnll z

for every m and n. Thus 5 w n ' n = 1, 2, ... , is a Cauchy sequence. In view of the completeness of V there exists an V-function, which we denote by 5 w, with the property lim 1[5 OJ n

-

5 OJllz = 0 .

This function 5 OJ, which is a uniquely determined element of LZ, is called the Hilbert transform of w. It defines the Hilbert transformation 5 as a continuous linear operator in the whole space V(D), having the integral representation (7.9) in the subclass CO.39 The equation holds for every

OJ

E V(D).

7.7. Application to functions with LZ-derivatives. We now return to formula (7.3) of 7.1. Let t be a function which is absolutely continuous on lines in the finite plane D and whose partial derivatives are L2_ integrable not merely locally but also over the whole of D. By Lemma 6.3 there then exists a sequence of functions tn E Cgo such that (7.17) n~oo

n~oo

The following remark can be made on the integral representation of the Hilbert transform of an arbitrary function WE L2(Q).

39

If we write 1

SeW(Z) = - -;;

If

I("-zl

w(~)

(~_ Z)2 da,

>e

then lim

liSe W

-

SwI12

=

O.

e~O

(The result is due to Beuding; a proof can be found in Ahlfors [2].) Considered as an element of L2, the integral defined by (7.9) exists therefore as a Cauchy principal value (with respect to the L2- metric) for every W E L2(Q) and coincides with the above defined Hilbert transform of w. In fact, Se w(z) even converges pointwise almost everywhere as e --->- 0 (Calder6n-Zygmund [1]). We shall not make use of these results in the sequel.

160

III. Auxiliary Results from Real Analysis

Applying formulae (7.3) and (7.10) we deduce that

(In)z = S(ln)z

(7.18)

for every n. Since

IIS(lnh - S 1z112

=

/I(lnh -1z112

Iz

it follows from (7.17) that S(lnh ->- S in V. Thus by 7.5 the sequence a subsequence for which both terms in (7.18) converge pointwise almost everywhere. We have proved the following result:

In has

Lemma 7.2. If the lunction I is absolutely continuous on lines in the finite plane Q and has square integrable derivatives over Q, then

almost everywhere.

IV. Analytic Characterization of a Quasiconformal Mapping Introduction to Chapter IV The methods developed in the previous chapter will now be applied to quasiconformal mappings. To bridge the gap between the theorems of Chapter III and the theory of quasiconformal mappings, we prove first of all that a quasiconformal mapping is absolutely continuous on lines. Once we have this result we obtain immediately a series of properties of quasiconformal mappings. Thus it follows from Lemma lII.3.1 and Theorem III.3.1 that a quasiconformal mapping is differentiable almost everywhere. This and Theorem 1.9.3 imply that the dilatation condition max", IO",wl :::;; K min", IO",wl holds almost everywhere for a Kquasiconformal mapping, a result which sheds new light on the connection between the general and the classical regular quasiconformal mappings. It follows from the dilatation condition and Lemma III.3.3 that a quasiconformal mapping has V-derivatives. Consequently all the results proved in III.6 can be carried over to quasiconformal mappings. For example, we see that a quasiconformal mapping preserves null sets and possesses a positive Jacobian almost everywhere. In § 2 we investigate the converse problem of how far the properties established in § 1 are sufficient to characterize a quasiconformal mapping. With the help of a counterexample we show first that the quasiconformality of a homeomorphism w cannot be characterized by prescribing the local behaviour of w only almost everywhere. We also deduce from this example that not only the dilatation condition but also the absolute continuity on lines must be required. We therefore obtain the Analytic definition, fundamental in our theory, under minimal conditions, when we prove the following result: A sense-preserving homeomorphism w, which is absolutely continuous on lines and satisfies the dilatation condition max", IO",wl ;;; K min", [o",w/ almost everywhere, is K-quasiconformal. Using the analytic definition we obtain in § 3 various modifications of the original geometric definition. For example, we show that a K-

162

IV. Analytic Characterization of a Quasiconformal Mapping

quasiconformal mapping w of the domain G satisfies the module condition M(w(t)) < K M(t) for every family t of ares lying in G. In § 4 we shall prove, again by the aid of the analytic definition, that a sense-preserving homeomorphism is K-quasiconformal precisely when its circular dilatation is everywhere finite and exceeds K in at most a null set. Finally in § 5, as a preparation for Chapter V, we introduce the complex dilatation of a quasicohformal mapping w; its domain of definition consists of the regular points of w, where it agrees with w-z1wz• We prove the uniqueness theorem which states that the complex dilatation determines a quasiconfornial mapping up to a conformal transformation. Finally the convergence properties of the complex dilatation of quasiconformal mappings wn of a domain G are examined, under the hypothesis that the sequence w n converges uniformly in compact subsets of G.

§

I.

Analytic Properties

~f aQuasiconformal Mapping

1.1. Absolute continuity on lines. In this section we give a series ot theorems on the continuity and differentiability properties of quasiconformal mappings. As was mentioned in the introduction to this chapter, most of these theorems follow immediately from the results of the previous chapter once we have proved that a quasiconformal mapping is absolutely continuous on lines (d. III.3.1). This result is due to Strebel [1J and 11ori[2]' In our proof we follow the method of Pfluger [2J who obtains the result directly from the definition of absolute continuity using Rengel's inequality.

The definition of absolute continuity on lines in III.3.1 refers to a domain of the finite plane and to a finite-valued function. Here we extend the definition as follows: A homeomorphism I is called absolutely continuous on lines in a domain G if it is absolutely continuous on lines in the former sense in the domain G - {oo} - {f-l( oo)}. Analogously we say that I has LP-derivatives in G if it has them in G - {oo} - {f-l(OO)}. Lemma 1.1. A quasiconlormal mapping w 01 a domain G is absolutely continuous on lines in G.

+

Prool: Let R = {x i y I a < x < b, c < y < d} be a rectangle such that ReG - {f-l( oo)}, and I y the borizontal segment a < x < b with ordinate y. We have to prove that w is absolutely continuous on almost y d. all I y for c

< <

§ 1. Analytic Properties of a Quasiconformal Mapping

163

For this purpose we consider the subrectangle Ry of R lying below 1y , and its image w(R y)' The area A(y) of w(R y) is an increasing function of y and thus has a finite derivative A'(y) for all y, C Y d, except for a set of zero linear measure. We show that w is absolutely continuous on 1 yo ' C < Yo < d, if A'(yo) exists and is finite.

< <

Let (x k ' x;), k = 1, ... , n, be an arbitrary system of non-intersecting open subintervals of the segment a x b. To prove that w is absolutely continuous on 1yo ' it is enough to show that the sum L; Iw: - wkl, where w: = w (x"t + i Yo), wk = w (x k + i Yo), has an upper bound which tends to zero with L; (x: - x k ).

< <

<

We first choose a positive r5 such that Yo + r5 d. Let R k , k = 1, ... , n, denote the quadrilateral consisting of the rectangle {x + i y I x k X Yo Y Yo + r5} and its corners. The horizontal sides will be regarded as a-sides for R k (d. 1.2.3). Then

< x:,

< <

M(R k)

1 (Xk* = 6"

x k) .

<

(1.1)

For the module of the image of R k , Rengel's inequality (see 1,4.3) gives the estimate (1.2) where d k (r5) denotes the euclidean distance between the b-sides of w(R k), measured in the plane. These sides converge to the points wk and as r5 -+ 0, and so

w:

lim d k (r5) = Iw: - wkl .

(1.3)

6~o

From (1.1) and (1.2) we get the inequalities k

= 1, ... , n,

where K denotes the maximal dilatation of w. ,If we add these over k and use Schwarz's inequality we obtain n

(

i

d k (fJ))2

K L; (x: - x k ) ~ r5 _~_=~1 - - ' - k~l

(1.4)

1: m(w(Rk )

k=l

Now we make use of the assumption that the function A introduced above has a finite derivative at y - Yo' Since

A(yo

+ (5)

n

- A(yo) ~ L; m(w(Rk ») k=l

164

IV. Analytic Characterization of a Quasiconformal Mapping

it follows from (1.4) that

C~ dk(~)r~ K If we let

~ -'>-

A (Yo

+ o~ -

A (Yo)_

k~

(x: - x k) .

0 this becomes by (i.})

C~ Iw: '- wk1r < K A'(YO)k~ (x: -

x k),

and the absolute continuity of w on [Yo is proved. In the same way we can show that w is absolutely continuous on almost all vertical segments c y d, x = constant, lying in R.

< <

Remark. A quadrilateral consisting of a rectangle and its comens is called a horizontal rectangle if its a-sides are parallel to the x-axis. From the above proof it follow that Ltmma 1.1 holds for every homeomorphism w which satisfies the dilatation condition M(R)JK ~M(w(R)) ~ K M(R) for all horizontal rectangles. On the other hand, a K-quasiconformal mapping w of G need not be absolutely continuous on every closed segment in G. By II.8.10 the image of a segment [y can indeed be non-rectifiable and then w is not even of bounded variation on [y. 1.2. Differentiability and local dilatation conditions. It follows from the above Lemma 1.1 and from Lemma II1.}.1 that a quasiconformal mapping w of the domain G has finite partial derivatives almost everywhere in G. Thus by Theorem II1.}.1 w is differentiable almost everywhere in G. From this result, important in itself, we can deduce more about the IGcal behaviour of a K-quasiconformal mapping. According to Theorem 1.9.} the inequality max", lo",w(z) I < K min", Io",w(z) I holds at every point where w is differentiable. By the above this is true for almost every z E G, and we obtain the following res~lt: Theorem 1.1. A K-quasiconformal mapping w of the domain G is differentiable and satisfies the dilatation condition (1.5)

at almost all points z of G. We remind the reader that for a regular K-quasiconformal mapping (1.5) holds at all points of G by definition. 40 A general quasiconformal mapping naturally need not be differentiable at all points of G.

&0

§ 1. Analytic Properties of a Quasiconforrnal Mapping

165

Theorem 1.1 is due to Mori [2J. He proved the differentiability almost everywhere using the distortion theorem 11.9.1 and the RademacherStepanoff theorem (Saks [1], pp. 310-311). We prefer to replace this method by our Theorem III.3.1, which we shall also need later.

1.3. V-derivatives of a quasiconformal mapping. The Jacobian of the mapping w satisfies J(z) = max 10 ",w(z) [ min 10",w(z) /

'"

'"

at every point z where w is differentiable. The inequality (1.5) is thus equivalent to (1.6)

'" From this and Theorem 1.1 it follows that the partial derivatives of a K-quasiconformal mapping w of G satisfy the inequalities JwAz)1 2 ~ K J(z) ,

(1.7)

almost everywhere in G. We deduce from Lemma II1.3.3 that J is integrable over every compact subset of G - {w-1(00)}. By (1.7) the same is true of the functions IWxl2 and IW y / 2. By Lemma 1.1 w is absolutely continuous on lines in G, and we obtain the following result (Bers [1J): Theorem 1.2. A quasiconformal mapping of a domain G has Vderivatives in G. This theorem will playa central role in the sequel, for it enables us to transfer the results of II1.6 to quasiconformal mappings.

1.4. Absolute continuity with respect to the area measure. By combining the above Theorem 1.2 with Theorem II1.6.1 we obtain the following result (d. Morrey [1J): Theorem 1.3. A quasiconformal mapping w of the domain G is locally absolutely continuous in G - {oo} - {w- 1 ( oo)} . In view of II1.2.2 we deduce from this theorem: A quasiconformal mapping of G transforms every area-measurable. subset of G into an areameasurable set, and preserves null sets. Further, by Lemma III.3-3, we have

JJ J da = A

for every measurable set.A. C G.

m(w(A))

(1.8)

166

IV. Analytic Characterization of a Quasiconformal Mapping

Remark. From the examples constructed in 11.8.10 we see that a quasiconformal mapping need not be absolutely continuous with respect to the linear measure. As far as sets of zero linear measure are concerned, Beurling and AhHors [1J have shown that under a quasiconformal mapping such a set can have an image with positive linear measure. As an example we can take a quasiconformal mapping of the plane, whose restriction to the real axis is a real-valued singular function. The existence of such a mapping follows from the fact that there exist singular functions satisfying the boundary condition (6.11) of Theorem II.6·3·

1.5. Regular points of a quasiconformal mapping. If a mapping w: G ~ G' is K-quasiconformal, then its inverse w- 1 is K-quasiconformal, and by Theorem 1. 3 locally absolutely continuous in G' - {(x)} - {w(oo)}. Hence by Lemma II1.3.4 the Jacobian J of w is positive almost everywhere in G. By the definition given in 1.1.6 a sense-preserving homeomorphism w of G is regular at a point Z E G - {(x)} - {w-1 ( oo)} if w is differentiable at z and J(z) o. From the aboye and Theorem 1.1 we obtain:

>

Theorem 1.4. A quasiconformal mapping w of the domain G is regular at almost every point of G. We refer to the result of 1.9.6 which states that at a regular point the conditions min lo"w(z)1

> 0,

wz(z)

=1= 0 ,

(1.9)

" are satisfied. Thus these relations hold for a quasiconformal mapping w almost everywhere in G.

It further follows from Theorem 1.4 that the dilatation quotient D (d. 1.9.4) which we introduced at regular points, is defined almost everywhere in G. The inequality (1.5) can be written in the form

D(z)

~K

for almost all z E G.

§

2.

Analytic Definition ofQuasiconformality

2.1. Statement of the problem. The results of the previous section show that the definition of quasiconformality given in 1.3.2 strongly restricts the local behaviour of the mapping. Conversely, we know that the regular quasiconformal mappings can be characterized by local

167

§ 2. Analytic Definition of Quasiconformality

analytic properties. We shall now give a similar analytic characterization for general quasiconformal mappings. We assume that w is a sense-preserving homeomorphism which satisfies the dilatation condition (1.5) at almost every point. This condition makes sense only if w is differentiable almost everywhere. By Theorem 1.1 these conditions are necessary for w to be quasiconformal. On the other hand, they are not sufficient. In fact the following counterexample shows that a sense-preserving homeomorphism need not be quasiconformal even if it is conformal except in a null set.

2.2. A counterexample. To construct the example we consider a Cantor set E(Pl> P2' ...) with linear measure zero and the corresponding singular function I (see IiI.2.9). The formula

w(z) = z

+ i I(x) <

_defines a homeomorphism w of the square Q = {z = x + i Y I 0 x 1,0 y 1} which leaves the x-coordinate of every point invariant and adds the increment I(x) to the y-coordinate. If, as in III.2.9, Ink denote the open intervals on which I is constant, then the restriction of w to each rectangle R nk = {x + i Y I x E Ink, 0 y 1} is a translation. Since U R nk contains almost every point of Q, the dilatation condition (1.5) holds with K = 1 almost everywhere in Q.

<

< <

< <

In spite of this, the mapping w is not quasiconformal in Q. This follows from Lemma 1.1, since the singular function I is not constant on {x I15::;; x ~ 1 - 15}, 15 = (1 - P1)!4, and consequently the mapping w cannot be absolutely continuous on any of the segments {x + i Yo I 15 < x ::;; 1 - 15}, 0 Yo 1.

< <

We note that, with respect to the area measure, the mapping w is not only absolutely continuous, but is in fact measure-preserving: m(w(A)) = meA) for every measurable set A C Q. Thus this property, even if combined with conformality outside a null set, is not enough to ensure the quasiconformality of the mapping. In particular, if we consider the Cantor set E(Pl> P2' ...) with 1 - 2- n, n = 1, 2, ... , then an elementary calculation shows that dim E = 0 (d. III.1.7). The above exceptional set Q - U R nk is then so small that it is not only of zero area but of dimension 1, as follows from Lemma III.1.2. 41

Pn =

2.3. Analytic definition. It follows from the above example that we cannot characterize the quasiconformality of a homeomorphism by the 41

By Theorem V.3.2 an exceptional set of this type cannot be essentially smaller.

168

IV. Analytic Characterization of a Quasiconformal Mapping

properties in § 1 without taking into account the concept of absolute continuity on lines (Lemma 1.1). If a homeomorphism w: G ->- G' possesses this property, then finite partial derivatives W x and wy exist almost everywhere in G, and by Theorem III. 3.1 w is differentiable almost everywhere in G. The hypothesis that w is absolutely continuous on lines in G can therefore be combined with the dilatation condition (1.5) without any assumptions about the differentiability of w. We shall show that these properties are sufficient to characterize quasiconformality. Although we present this result as a theorem, we call it the Analytic definition of quasiconformality. Indeed, since the converse result is also true we could use the statement of the theorem as a new definition equivalent to our geometric definition. Analytic definition. Let a sense-preserving homeomorphism w of the domain G satisfy the following two conditions: 1. w is absolutely continuous on lines in G. 2. The dilatation condition max lo",w(z)[

'"

~

K min lo",w(z)1 '"

holds almost everywhere in G. Then w is a K-quasiconformal mapping of G. Proof: Since an isolated point is a removable singularity of a quasiconformal mapping (Theorem 1.8.1), there is no loss of generality in assuming that G and its image G' are domains of the finite plane. We show first that w has V-derivatives in G. By Lemma III.3.1 and Theorem II1.3.1, a homeomorphism with property 1 is differentiable almost everywhere in G. By Lemma III.3.3 the Jacobian J of w is then locally integrable in G. From condition 2 it follows that the functions '[w x [2 and [Wy12 are majorized by K J almost everywhere in G (d. 1.3). Thus w possesses V-derivatives in G. To prove that w is K-quasiconfOlmal we consider a quadrilateral Q, Q c G, and its image w(Q). We have to show that the modules M = M(Q) and M' = M(w(Q)) satisfy the inequality M'~KM.

Let f2 be the canonical mapping of w(Q) onto the rectangle R 2 = {u i v I 0 < u < M', 0 < v < 1}, .and fl the inverse of the canonical mapping of Q onto the rectangle R 1 = {~+ i rj [ 0 < ~ <: M, o rj 1}. The composed mapping w* = f2 0 W 0 fl is a sense-

+

< <

169

§ 2. Analytic Definition of Quasiconformality

preserving homeomorphism of R 1 onto R 2 , which can be extended topologically to the boundary. Our next step will be to show that w* satisfies conditions 1 and 2 in R 1 . Since w has L2-derivatives in G and 11 is conformal in R v we can use Lemma III.6.4. This tells us that the mapping w 0 11 is absolutely continuous on lines in R 1 , and that the equation O~(w 011) (,)

=

1;(') O~+fJW(t1(m '

where {3 = arg 1;('), holds almost everywhere in R 1 for every direction iX. Since 12 is conformal, w* = 12 0 W 011 is also absolutely continuous on lines in R 1 • Further we have o~w*(')

= 1~(w(f1(m) 1;(') O",+fJW(t1(O)

for almost all CE R 1 • Since the derivatives I~ and I; are independent of the direction iX, and w satisfies condition 2, this condition is also satisfied by w*. The inequality M' S K M can now be proved as follows. Since w* satisfies condition 2, we have Iwtl2 S K J* almost everywhere in R v where J* is the Jacobian of w*. It follows from Lemma IIL3.3 that

JJ Iwtl

R,

2

da ~ K

JJ J* da <

K m(R 2) = K M' .

R,

(2.1)

By Fubini's theorem (see IIL1.5)

ff

R,

Iwtl 2 da

M

1

=

f 0

d'YJ

f 0

Iwtl 2 d~ ,

(2.2)

and by Schwarz's inequality

!MIwtl2d~ ~ 1(M)2 ! Iwtl d~ . M

(2·3)

For further estimation, condition 1 is needed. Since w* is continuous on the boundary of R v we have M

f

o

Iwt(~

+ i 'YJ)! d~ :? Iw* (M + i 'YJ)

- w*(i 'YJ)I :? M'

(2.4)

for every 'YJ for which w* is absolutely continuous on every closed subsegment of I = {~ i'YJ I 0 ~ M}. Since w* satisfies condition 1 in R 1, (2.4) holds for almost all 'YJ, 0 'YJ 1. Our inequality M' < K M now follows from (2.1)-(2.4).

+

< <

< <

2.4. Earlier fo\'ms of the analytic definition. As was mentioned in § 1, Mori [2J proved that a K-quasiconformal mapping satisfies the

170

IV. Analytic Characterization of a Quasiconformal Mapping

conditions of the Analytic definition. The first versions of the converse are due to Yuj6bO and Bers. Yuj6bO [1J proved that a homeomorphism w is K-quasiconformal if it has V-derivatives, is differentiable almost everywhere, and satisfies the conditions 1 and 2. Bers [1J replaced the first two of these conditions by the assumption that w has V-derivatives (d. also Bers [2J). Pfluger [2J noted that it is enough to assume the existence of the V-derivatives. In the above formulation the Analytic definition is to be found in the paper of Gehring and Lehto [1]. The counterexample in 2.2 shows that conditions 1 and 2 are in a certain sense the weakest possible. The disadvantage of these minimal conditions is that condition 1 is stated in a form which depends on the coordinate system.

§ 3. Variants of the Geometric Definition 3.1. Quasiconformality and module conditions. By definition a Kquasiconformal mapping magnifies the module of quad:r;ilaterals by at most a factor K, and in 1.7.1 we proved that the same holds for ring domains. Since these modules are special cases of the general concept of module introduced in IlIA, it is natural to ask whether the same dilatation condition holds for the module of every family of arcs. The answer is affirmative, as we shall show in this section. On the other hand, one can seek the weakest possible module conditions which imply quasiconformality. In fact it was shown in 1.5.} and 1.9.2 that if the dilatations of analytic quadrilaterals or of small quadrilaterals are bounded, then the homeomorphism is quasiconformal. Further results of this type are given in }. 5 and}.6 of this section.

3.2. Absolute continuity on arcs. We begin with the following result, which will be applied in deriving the above-mentioned general module condition. Theorem }.1. A quasiconformal mapping w: G -->- G' is absolutely continuous on all closed arcs C C G ,vith the exception of a family of module zero. Proof: By Theorem 1.2, w has V-derivatives in G. If G and G' lie in the finite plane then the assertion follows from Theorem III.6.2. In the case when G or G' contains the point at infinity we only need remark that the family of all arcs passing through a fixed point has module zero by II1.4.}.

171

§ 3. Variants of the Geometric Definition

Theorem 3.1 contains Lemma 1.1 on absolute continuity on lines as a special case. To see -.!his we consider a rectangle R = {x + i y I a x < b, c < y < d}, ReG, and denote by I y the horizontal segment a < x < b with ordinate y lying in R. It then follows immediately from Lemma IIlA.1 that a family of segments I y is of module zero if and only if the corresponding y-values in c < y < d form a set of zero linear measure. We do not know whether it is possible to find a more precise estimate for a family on whose arcs a quasiconformal mapping is not absolutely continuous or in what manner the exceptional family depends on the maximal dilatation.

<

3.3. General module condition. We now prove the above-mentioned general result on the magnification of the module of a family of arcs (ViiisiiHi [1J). Theorem 3.2. Let w: G --+ G' be a K-quasiconformal mapping, family of arcs C C G and ~' the image family of ~ under w. Then M(~)IK < M(~')

<

K M(t) .

~

a

(3.1 )

Proof: Since the inverse of w is also K-quasiconformal it is enough to prove one of the above inequalities, for example that M(~) < K M(~').

Because the module is monotonic and subadditive we may delete from ~ an arbitrary family of module zero without changing the value of the module of ~ (d. formula (4.6) in IlI.4.2). In view of Theorem 3.1 and the monotonicity of the module (IlI.4.2) we may therefore assume that no arc C E ~ contains a closed subarc on which w is not absolutely continuous. Further it follows from the remark at the end of IlIA.3 that we need only consider rectifiable arcs C E ~. The image arcs C' E e' are then locally rectifiable. By Theorem 104 there exists a Borel set E of zero area such that G - E consists of regular points of w only. The partial derivatives of wand so also the derivatives o"w are Borel functions in G - E. The same holds for the function max" IO"wl since O"W is continuous as a function of iX, and so max" jo"w! = sUP/1 IO/1wl, where fJ assumes only rational values. Let e* be a non-negative Borel function defined in the w-plane and admissible with respect to ~', i.e.

J e* Idwl

C'

for every arc C'

E ~'.

21 .

(3·2)

172

IV. Analytic Characterization of a Quasiconformal Mapping

If

e*(w(z)) max" lo"w(z)1 e(z) =

{

(X)

o

for z E G - E, for z E E , for z E - G,

then e is also a Borel function. We shall show that e is admissible with respect to e. Let C: I --->- C be a parametrization of C with the arc length s as parameter. The composed mapping w 0 C: I --->- C is a parametrization of C. Let s* be the arc length of C' increasing with s. It then follows from equation (2.9) of III.2.7 that

Id(Wd;~) 1= ~s*

(3·3)

almost everywhere (In I. On the other hand, for every So E I for which the derivatives in (3.3) exist and Zo = C(so) is a regular point of w, we have d(W o~) I = lim 1~~iS)L= w(~(SO!21 < lim inf IW(~(S)) - w(~(so))1 [ ds ,S=So S-So S - So S-So ~(s) - ~(so) I

< max" lo"w(zo)/ .

(3.4)

Since the function e defined above assumes the value (X) at every nonregular point of w, we deduce from (3.3) and (3.4) that ds*

e(C(s)) ~ e*(w(C(s))) d;for almost all s

E

(3·5)

I.

The absolute continuity of w on every compact subset of C is by definition equivalent to s* being absolutely continuous as a function of s on compact subsets of I. It follows therefore from (3.5), in view of the transformation formula (2.7) in III.2.6, that

J

J

C

I

e Idzl =

e(C(s)) ds

~

J

e*(w(C(s)))

I

d;s*

ds =

J

e* Idwl .

C'

Hence by (3.2)

Ie Idzl

2: 1 ,

C

e is admissible with respect to e. It follows that the module of e satisfies the inequality M(e) :::;:: II e2 da.

i.e.

By the definition of

e we have

(3.6)

G

(e(z))2 = (e*(w(z)))2 max" lo"w(z)12 = (e*(w(z)))2 Dw(z) J(z)

(3.7)

173

§ 3. Variants of the Geometric Definition

almost everywhere in G, where D w denotes the dilatation quotient and J the Jacobian of w. At every regular point z E G we have Dw{z) = Dw-.(w{z)). Therefore, by Theorem 1.3 and Lemma III.3.5,

JJ (e*(w{z)))2 Dw{z) J{z) da = JJ (e*)2 Dw-' da .

G

q

Consequently, it follows from (3.6) and (3.7) that M(t)

< JJ (e*)2 D w -' da .

(3. 8)

G'

Since this inequality holds for every we finally obtain M(t) ~ inf e*

JJ (e*)2 Dw-' da <

e*

admissible with respect to t'

K inf e*

G'

JJ (e*)2 da =

K M(t') ,

G'

as required.

3.4. Conformal invariance of the module. For K = 1, Theorem 3.2 yields: The module of a family of arcs is invariant under conformal mappings. From this we deduce that the general module condition (3.1) contains the conditions for quadrilaterals and ring domains as special cases. Indeed, if Q is a quadrilateral and t denotes the family of all Jordan arcs which join the a-sides of Q inside Q, then M(t)

=

M(Q).

If Q is a rectangle this follows immediately from Fubini's theorem. For general quadrilaterals it is a consequence of the conformal invariance of the module. 42

Similarly for a ring domain B we have M(t') =

M(B) 2:n

'

M(t") =

~(~) ,

where t' is the family of Jordan curves separating the components of the complement of B, while til consists of the arcs which join them.

3.5. Characterization of quasiconformality by means of rectangles. As we have already mentioned, the quasiconformality of a homeomorphism w: G ~ G' is guaranteed as soon as the dilatation condition M(~(Q)) :::;:; K M(Q) is valid for a suitable subclass of the quadrilaterals Q, Q c G. We shall consider some classes of quadrilaterals which are sufficient for this purpose. 42

Cf. the footnote 7 on p. 21.

174

IV. Analytic Characterization of a Quasiconformal Mapping

As a first result we prove the following: Theorem 3.3. If a sense-preserving homeomorphism w : G -+ G' satisfies the module condition M(w(R)) ::;; K M(R) for every rectangle R, Ii c G, then w is a K-quasiconformal mapping of G.

Proof: We show that w satisfies conditions 1 and 2 of the Analytic definition. At the end of 1.1, as a supplement to Lemma 1.1, we remarked that a homeomorphism w: G -+ G' is absolutely continuous on lines if it satisfies the module condition M(R)jK < M(w(R)) < K M(R) for every horizontal rectangle 43 R. Thus condition 1 is satisfied. It further follows from Lemma II1.3.1 and Theorem II1.3.1 that w is differentiable almost everywhere in G. By the remark in 1.9.5 we have max /o",w(z)/ ::;; K min lo",w(z)1

'"

'"

at every point of G at which w is differentiable, assuming that the dilatation of every square R, ReG, is at most K. Hence condition 2 is also satisfied, and w is K-quasiconformal by the Analytic definiti~. . The above proof shows that Theorem 3.3 can be sharpened slightly. In fact, w is K-quasiconformal if the module condition M(R)jK ::;; M(w(R)) < K M(R) is satisfied for all horizontal rectangles and all squares.

3.6. Module condition for horizontal rectangles. It is natural to ask whether the validity of the module condition only for horizontal rectangles implies quasiconformality. This is in fact true, although the maximal dilatation of the mapping can be greater than the dilatation of horizontal rectangles (Gehring-Vaisala [1J). Theorem 3.4. A sense-preserving homeomorphism w: G -+ G' satisfying

the module condition M(R)jK::;; M(w(R)) < K M(R) for aU horizontal rectangles R, ReG, is (K + VK2 - 1)-quasiconformal. The bound is best possible. Proof: As in the proof of Theorem 3.3 we can deduce that w is absolutely continuous on lines and almost everywhere differentiable in G. Hence, Note that the double inequality is necessary here since the a-sides are horizontal by definition.

43

175

§ 3. Variants of the Geometric Definition

by the Analytic definition, w is (K condition max [o",w(zo) I ::;; (K +

+ YK2 yK2 -

1)-quasiconformal if the

1) min !o",w(zo)1

ex

ex

holds at every point Zo E G where w is differentiable. Using complex derivatives we write this inequality in the form (d. 1.9.4)

IW,(zo) I + Iw.(zo) [ <

(K + yK2 - 1) ([wz(zo) I -

jw.(zo)J)·

(3.9)

We may confine ourselves to the case when wz(zo) =F 0, since otherwise both sides of (3.9) vanish (d. 1.9.4). Further, by means of a translation of the z-plane and a similarity transformation of the w-plane, we can achieve the normalizations Zo = w(zo) = 0, wz(zo) = 1. Denoting w.(O) by x we then have w(z) = z + x Z + o(z) , with

°::;; [xl ::;; 1.

The required inequality (3.9) becomes 1

+ /x[

<

(K + yK2 -

1) (1 - [xl) .

(3.11)

Let R be the horizontal rectangle with corners 0, a, a + i b, i b, where a 0, b 0. If we denote the module ajb of R by M, then the module of the image quadrilateral R' = w(R) satisfies

>

>

MIK ::;; M(R')

< K M

(3.12)

by hypothesis. By Rengel's inequality (see 1.4.3) we have further s~

,m(R')

m(R')

~ M(R)::;;

T'

where sa and Sb denote the distances, measured in R', between the aand the b-sides, respectively, and m(R') is the area of R'. From (3.12) we deduce that S~ ::;;

S; :'S ~ m(R') .

K M m(R') ,

(3·13 )

By (3.10) we have on the vertical sides of R w (a

+ i y)

= a (1 + x) + i Y (1 (i y) = i Y (1 - x) + 0 (a

-

w

+ b) .

x)

+0

(a

+ b) ,

This gives the estimate Sb

~

a

11

+ xl

-

b 11

-

x[

+ o(a + b) .

Correspondingly, for sa we have

sa ;;::: b 11

-

xl -

a

11

+ xl + 0 (a + b) ,

176

IV. Analytic Characterization of a Quasiconformal Mapping

and a simple calculation gives m(R')

=

Ix1 2 )

a b (1 -

+ o(a + b 2

°

2

) •

If we now let a -+ with ajb = M fixed, then these estimates, together with (3.13), yield the following result: For M 11 x/ ~ 11 - xl, we have

(11 and for M 11

+

+ xl

+ xl < 11

- 11 ~ "I

r

< K (1 - 'xI 2)

,

- xl,

+ xl)2 <

(11 - x/ - M /1 It follows that the values x = M -+ 0, we obtain

±

K (1 -

'x/ 2 )

•

1 cannot occur. As M

-+ 00

or

+

Adding, we deduce that 1 Ixl 2 < K (1 - IxI 2), and the required inequality (3.11) follows by a simple calculation. The mapping w,

w(z) shows that the bound K

=

z

+ VK2

+ i V~ ~

:z,

- 1 is best possible.

3.7. Other special module conditions. In the proof of Theorem 3.4 we needed the module condition MjK < M(R') :::;; K M for large and small values of M. The validity of this condition for a fixed value of M is not enough to imply quasiconformality. For example it is easy to construct a homeomorphism of the plane which is not quasiconformal but leaves invariant the modules of all horizontal squares. If we only know that the dilatation of every square R, R C C, is at most K, the above methods cannot be applied. To our knowledge it is an open question whether this condition implies the quasiconformality of a homeomorphism w: C -+ C'. If it does then w must in fact be Kquasiconformal as can be seen from the proof of Theorem 3.3 (d. 1.9.5).

By Theorem 1.7.2 a sense-preserving homeomorphism w: C -+ C' is quasiconformal if it satis~es the module condition M(w(B)) < K M(B) for all ring domains B, B C C. This theorem too can be sharpened to

§ 4. Characterization with the Help of the Circular Dilatation

177

concern subclasses of the family of all ring domains. For results of this type we refer to Gehring-Vaisala [1J. Finally a result due to Renggli [1J should also be mentioned. It states that a sense-preserving homeomorphism is quasiconformal if it carries every curve family with module 00 into a family with module 00.

§ 4. Characterization ojQuasiconjormality with the Help oj the Circular Dilatation Besides using the geometric and analytic .definitions we can characterize a quasiconformal mapping by means of the behaviour of the circular dilatation introduced in 11.9.1. We then have the advantage that no assumptions about absolute continuity on lines are necessary.

4.1. Necessary conditions for the circular dilatation. Let w: G ~ G' be a homeomorphism and denote its circular dilatation by H as before. We shall first give necessary conditions on H for w to be Kquasiconformal. We proved in 11.9.1 that at every regular point H(z) equals the dilatation quotient D(z), which by 1.9.6 nowhere exceeds the maximal dilatation K(G). If K(G) is finite, Theorem 1.4 states that almost all points of G are regular. Thus H(z) < K(G) almost everywhere in G. On the other hand, Theorem 11.9.2 shows that the circular dilatation of a K-quasiconformal mapping can exceed K at certain points. By Theorem 11.9.1, however, it is always bounded by a number depending only on K. In view of the above remarks, this theorem can be extended as follows: Theorem 4.1. The circular dilatation of a K-quasiconformal mapping w : G ~ G' is bounded in G and at most equal to K almost everywhere in G.

4.2. Sufficient conditions for the circular dilatation. The validity of the condition H(z) < K almost everywhere in G is not sufficient to imply that a sense-preserving homeomorphism w: G ~ G' is quasiconformal. This follows from the example constructed in 2.2, in which a non-quasiconformal homeomorphism has circular dilatation 1 almost everywhere in its domain. However, if we add the condition that the circular dilatation is bounded, then we can prove that w is K-quasiconformal. Thus Theorem 4.1 has a converse. The result can even be

178

IV. Analytic Characterization of a Quasiconformal Mapping

slightly sharpened; in fact H need not be bounded in G, but only everywhere finite there. 44 Theorem 4.2. A sense-preserving homeomorphism w: G ->- G' is Kquasiconformal if its circular dilatation is finite at every point of G and satisfies H(z) :::::; K almost everywhere in G.

Proof: The following proof is based on the Analytic definition of quasiconformality. In view of Theorem 1.8.1 on removable singularities we can assume that neither G nor G' contains the point at infinity. First we show that a homeomorphism w whose circular dilatation satisfies the above conditions is absolutely continuous' on lines in G. The proof is similar to that of Lemma 1.1. We again take a horizontal

+

< < < <

< <

rectangle R = {x i y Ia x b, c y d}, R C G, and let !'y be the horizontal segment a x b with ordinate y. Denote by R y the part of R lying under I y. Then the area m(w(R y)) = A(y) of the image of R y is an increasing function of y. Hence, the finite derivative A'(y) exists for almost all y in c y d.

< <

Since H(z) :::::; K almost everywhere in R, it follows from Fubini's theorem that the same inequality holds almost everywhere on I y for almost all y, c y d. Hence it is enough to prove the absolute continuity of w on I y under the hypotheses that A'(y) exists and H (x i y) :::::; K holds for almost all x, a x b. We consider a fixed I Yo = I with these properties.

< <

< <

+

Let F C I be a compact set. We suppose first that H(z) does not exceed a finite bound N in F. As a first step in the proof we show that the linear measure of w(F) = F' has an upper bound which tends to zero with l(F). The result will then be extended to an arbitrary Borel subset of I, and it follows that w is absolutely continuous on I. We denote by M(z, r) and m(z, r) the maximum and minimum of Iw(C) - w(z)1 on the circle Ie - zl = r. If the positive number s is smaller than the distance betweenF and - R, we can construct the closed set F.

= { z I z E F, (r

M(z, r)

~ s) ~ m(z, r) :::::; N

+ 1} .

In what follows we shall restrict ourselves to numbers s = 1!q, where q is a natural number. In fact, it is enough to assume that H is finite outside a set of a-finite linear measure (ei. V.3.4). The result was formulated by Yl1jobo [1] and proved by Gehring [1]. Our proof of Theorem 4.2 follows the method of Gehring.

44

§ 4. Characterization with the Help of the Circular Dilatation

179

As B ~ 0, the sets F. form a non-decreasing sequence which converges to F, i.e. F = U F •. The images F~ of F. then satisfy U F~ = F', and consequently (d. III.1.2) lim l(F~) = l(F') . (4.1) Since F. is compact it can be covered by a finite collection of discs D.(Zh) = {Clle - zhl < B}, zIt E F., h = 1, ... ,n•. We may assume that these discs cover their union De = U De(z,,) at most twice, for if three discs D.(z,,) have a non-empty intersection we can dispense with one of them without reducing the set De n I. Since each of the segments D,(Zh) n I has length 2 B, we have

l (De

n I)

~

Bne .

(4.2)

If now 0, I) 0 ) F, is open in I, then D. n leO as soon as B is smaller than the distance between F and I - O. Since l (0 - F) can be chosen arbitrarily small, we therefore have lim sup l (De n I) :::;;: l(F). On the other hand, De n I ) F. for every B and, since lim l(F.) = l(F), we obtain lim l (De n I) = l(F) . (4·3) 8-0

As for the images D~(z,,) of De(z,,), h = 1, ... , n., we see immediately that the area of their union is at least n I: (m(z", B))2/2. Since all discs D.(Zh) lie in the rectangle {x + i y I a x < b, Yo - B < y < Yo B} we deduce that

+

<

2 (A (Yo

+ B) -

ne

A (Yo - B)) 2: n

I: (m(z", B))2 .

"=1

The expression on the right hand side can be further estimated. Since the points zIt belong to Fe' we have M(Zh' B)/m(z", B) < N 1, h = 1, ... , n., and it follows that

+

2 (A (Yo

+ B) -

A (Yo - B))

(N: 1)2,,~

n.

~

(M(z", B))2 .

By Schwarz's inequality,

2 (A (Yo

+ B) -

A (Yo - B)) ;;::: n

e

t~ M(z", B)Y.

(N:It+ 1)2

Thus by (4.2) we have l (D

•

Letting

n I) A (Yo + B) 2B ~

A (Yo - B) B

> =

:It

4 (N

+ 1)2

°we obtain from (4.3)

lim l (D. .~O

n I)

A (Yo

+ B) 2~ A (Yo

))2

(4.4)

l(F) A'(yo) .

(4.5)

( ;;

- B). =

"=1

M(

ZIt, B .

180

IV. Analytic Characterization of a Quasiconformal Mapping

To investigate the behaviour of the right hand side in (4.4) as E ---* 0 we first take a fixed Eo and note that the set F;. is contained in U D~(Zh) for E < EO' Since the set D~(Zh) lies inside a circle with radius M(zh' E), there exists for every E, Eo ~ E 0, a covering of F;. by discs the sum of whose diameters is

> n.

2 }; M(Zh' E) • h=l

Since w is uniformly continuous in R, the greatest of the numbers M(Zh' E) tends to zero as s ---* o. By the definition of the linear measure I(F;.) cannot therefore exceed n.

2 lim inf }; M(Zh' E) . • -0

This holds for every

EO'

h=l

and by (4.1) n.

1(F') ~ 2 lim inf }; M(Zh' s) . £-0

h=1

Consequently, in view of (4.4) and (4.5), we obtain the estimate (4.6) This inequality holds for every compact set F C I under the assumption that H(z) < N for Z E F. The next step is to show that (4.6) also holds for all Borel subsets of I in which H(z) ~N. To do this we construct an approximation of a Borel set by closed sets, and we start by proving that the image I' = w(I) of I is a-finite with respect to the linear measure (d. III.1.2). Since H(z) is finite for every

Z E

I, the same is true of the upper bound M(z, r)

ffi

'P(z) -- sup-r m(z, r) ,

where r runs through the positive numbers for which both Z + rand Z r belong to I. If E is a compact subinterval of I, then the set

IN

=

{z I f/>(z) ~ N} ,

is closed. For z E EN we have H(z) ~ N, and by (4.6) the image E'zv of E Iv has a finite linear measure. Since E = U EN' where N runs through the positive integers, E' = w(E) is a-finite. The same is then also true of I', since I is the union of countably many closed segments.

§ 4. Characterization with the Help of the Circular Dilatation

181

Let Bel be a Borel set in which H(z) < N. The image B' of B is also a Borel set, and since I' is a-finite, there exists by III.1.2 a sequence of closed sets F~ C B', k = 1,2, ... , such that lim l(F~) = l(B'). The preimages F k = w-l(F~) are also closed and so (4.6) holds with F = F k for every k. By passing to the limit we deduce that (4.6) also holds if F and F' are replaced by Band B', respectively. Finally we want to remove the restriction H(z) < N. To do this we first consider a Borel set B o C I of zero linear measure and its image B~. We have H(z) < N in the set B o n IN. Since l (B o n IN) = 0, the linear measure of the image (B o n N )' also vanishes. Since B~ = U (B o n IN)' we see that l(B~) = o.

r

Since I was chosen so that H(z) ~ K for almost all z E I, there exists a Borel set B o C I of zero linear measure such that H(z) does not exceed K in I - B o. Then l(B~) = 0 as proved above. If now Bel is an arbitrary Borel set, then its image satisfies l(B') = l ((B - B o)'). However, H(z) sKin B - B o' and (4.6) yields (l(B'))2

< ~ (K

+ 1)2 A'(yo) l(B) .

Since this inequality holds for every Borel set Bel and its image B', the homeomorphism w is absolutely continuous on I by the definition in nI.2.2. In the same way we can prove that w is absolutely continuous on almost all vertical segments joining the horizontal sides of R. Condition 1 of the Analytic definition is therefore satisfied. As regards condition 2, we remark that w is differentiable almost everywhere in G by Lemma III.3.1 and Theorem III.3.1. Let z be a point where this holds and where H(z) S K. We can distinguish two cases according as z is a regular point of w or min lo",w(z) I vanishes. In the first case it follows from the discussion of 4.1 that D(z) = H(z) < K. In the second case m(z, r) = o(r). Since H(z) < K we also have M(z, r) = o(r), which means that o",w(z) vanishes for every ix. Thus the dilatation condition max", lo",w(z)[ < K min", Io",w(z) I is trivial in this case. Hence condition 2 of the Analytic definition is satisfied and w is quasiconformal.

Remark. Methods similar to the above can be applied to other problems in the theory of quasiconformal mappings. As an example we mention the question of the characterization of a quasiconformal mapping by means of its distortion of angles; we refer here to the work of AgardGehring [1J and Taari [1]. By slightly altering the above proof it can also be shown that the class of mappings which Volkovyskij [1J called K-quasiconformal with a pair

182

IV. Analytic Characterization of a Quasiconformal Mapping

of characteristics is a subclass of our K-quasiconformal mappings (NaiWinen [1J).

§ J. Complex Dilatation 5.1. Definition of the complex dilatation. This section will serve as a preparation for Chapter V, in which we deal with the problem of the existence of a quasiconformal mapping with prescribed dilatation. In order to formulate this problem it is convenient to introduce a new measure of dilatation, which will express not only the size of the dilatation quotient, but also the direction of the maximal distortion. Let w: G -7 G' be a K-quasiconformal mapping and z EGa finite point at which w is differentiable. From the equation (5.1) we deduce by a simple calculation that the dilatation condition (1.5) can also be written in the form

(5.2) If z is a regular point, then w.(z) 7'= 0 (d. 1.5). The quotient x(z) =

wz(z) w.(z)

(5-3)

thus exists at every finite regular point z where w(z) 7'= 00; by Theorem 1.4 it therefore exists almost everywhere in G. We call the function x the complex dilatation of the mapping w. Being the quotient of two derivatives of a continuous function, x is a Borel function. The complex dilatation has a simple geometrical interpretation. We first see from equation (5.1) that lo"w(z)1 assumes its maximum Iw.(z) I (1 + Ix(z)l) when 1

(X

= (XM(Z) = 2 arg x(z) .

Next it follows from the definition of the dilatation quotient that D(z)

= 1 + l:>e(z) I 1 -

l:>e(z)j

.

(5.4)

Thus x(z) determines D(z), and in the case x(z) 7'= 0 also the direction (XM(z) of maximum distortion, the latter up to a multiple of n. Conversely, D(z) and (XM(Z) uniquely determine the complex dilatatiOll x(z).

183

§ 5. Complex Dilatation

By (5.4) (or (5.2)) we have D(z) - 1 K - 1 -D"":'(z":"')-+-1. < -K-+-1

I,,(z) I =

at every finite regular point z at which w(z) =F

(5.5)

00.

Since ,,(z) = 0 is equivalent to D(z) = 1, the complex dilatation of a conformal mapping vanishes identically. Conversely, if we suppose that the complex dilatation of a quasiconformal mapping w: G -+ G' vanishes almost everywhere in G, then w is 1-quasiconformal by the Analytic definition and so conformal by Theorem 1.5.1.

5.2. Transformation formulae for the complex dilatation. Let w : G -+ G' be quasiconformal, II quasiconformal in G', and low the composite mapping. Denote By "w' "I and "low the corresponding complex dilatations. Suppose that z and w(z) are finite regular points of wand I, respectively, and that I(w(z)) is finite. Then, by a formal calculation, we obtain

+ "/(w(z)) e- 2iargw.(z) + "w ()z "I (()) -2iargw-(z)' wz e z

"w(z)

"/ow(z) =

1

(5.6)

The set of non-regular points of I in G' is of zero area by Theorem 1.4. Th.e same is true of the preimage in G of this set, because the quasiconfotmal mapping w- 1 preserves null sets by Theorem 1.3. Since almost all points of G are also regular points of w, (5.6) holds almost everywhere in G. By 5.1 the mapping I is conformal if and only if "I vanishes almost everywhere in G'. Since wand w- 1 preserve null sets, this is equivalent to "/( w(z)) = 0 for almost all z E G. Thus by (5.6), I is conformal if and only if (5.7) almost everywhere in G.45 For later references we state the sufficiency part of this result in the following form: Uniqueness theorem. Let w: G -+ G' be a quasiconlormal mapping with the complex dilatation ". Then every quasiconlormal mapping 01 G whose complex dilatation equals" almost everywhere in G is 01 the lorm low, where I is a conlormal mapping 01 G'. If the mapping w in (5.6) is conformal, then in general same as "low' In fact, in this case

"/ow(z) = "/( w(z)) e- 2 i arg w'(z) . Note that the conformality of that f iio quasiconformal.

45

f

"I

is not the (5.8)

follows from (5.7) only under the assumption

IV. Analytic Characterization of a Quasiconformal Mapping

184

Finally, if f and ware quasiconformal mappings of the same domain, (S.6) yields

(S.9) where C= w(z). 5.3. Beltrami's differential equation. The definition

(S.1 0) w-• = "w • of the complex dilatation of a quasiconformal mapping w can be interpreted as a differential equation. An equation of type (S.1 0) is called a Beltrami equation. If w is conformal, so that" is identically zero, then (S.10) reduces to the Cauchy-Riemann equation W z -; o. Now we suppose that" is a. measurable function defined in G with suP. 1,,(z)1 1. In general there is no function w for which (S.10) holds at every point of G. We shall therefore explain what will be meant here by a solution of (S.10).

<

A function w is called a generalized solution of (S.1 0) in G if w is absolutely continuous on lines in G and the derivatives w.' W z (which by Lemma III.3.1 exist almost everywhere) satisfy (S.10) almost everywhere in G. Correspondingly, w will be called a generalized LP-solution in G if.it possesses LP-derivatives and (S.10) holds almost everywhere in G. Note that changing the values of " in a null set does not affect the generalized solutions of (S.10). Thus" need be defined only almost everywhere in G. By S.1 and Theorem 1.2, every K-quasiconformal mapping of G is a generalized V-solution of a Beltrami equation (S.10), where" is the complex dilatation of the mapping. Then

/,,(z)1 <

K-1 K 1

+

(S.11)

holds at every point where ,,(z) is defined. Conversely, every homeomorphic generalized solution of (S.10) is K-quasiconformal, provided that" satisfies (S.11). Since conditions 1 and 2 of the Analytic definition are then obviously satisfied, we need only show that w is sense-preserving. In any case w is either K-quasiconformal or the composition of a reflection and a K-quasiconformal mapping. By Theorem 1.4, the Jacobian] of w is therefore non-zero almost everywhere in G. On the other hand it follows from the equation J(z) = Iw.(z)1 2 - /w z(Z)!2 = Iw.(z)1 2 (1 - lu(z)1 2 ) ,

185

§ 5. Complex Dilatation

>

together with (5.11), that J (z) ;;:::: 0 almost everywhere. Hence J (z) 0 for almost all z, and it follows from 1.1.6 that w is sense-preserving. Theorem 5.1. A K-quasiconformal mapping w of the domain G is a generalized V-solution in G of the Beltrami equation (5.10) where x is the complex dilatation of w. If, conversely, x is a measurable function in G which satisfies (5.11), then every homeomorphic generalized solution of (5.10) in G is a K-quasiconformal mapping of G with complex dilatation x(z) for almost aU z E G.

This theorem is essentially contained in the Analytic definition of quasiconformality and in the theorems of § 1. Thus the characterization of quasiconformal mappings as homeomorphic solutions of Beltrami equations is here of a formal nature. These equations can however be transformed into integral equations and this leads to new methods of investigating quasiconformal mappings (see V.5).

5.4. Good approximation of a quasiconformal mapping. After introducing the complex dilatation we can complement the convergence theorems in II.5 by considering the convergence of the dilatations also. The rest of this section is devoted to problems of this kind. Let w n be a sequence ofK-quasiconformal mappings ofG with complex dilatations x..' n = 1,2, .... We say that the sequence w.. is a good approximation of a quasiconformal mapping w of G with complex dilatation x if the following two conditions are satisfied: 1. lim w..(z)

=

w(z), uniformly on compact subsets of G.

n~oo

2. lim x..(z) = x(z) almost everywhere in G. n~oo

If G contains the point at infinity or the point w- 1 ( 00) then the conver-

gence in condition 1 is to be with respect to the spherical metric. We illustrate the meaning of condition 2 by a few remarks. From condition 1 alone it follows by Theorem 1.5.2, that lim inf (sup Ix..(z) n-oo

. ZEG

I) ; : : sup Ix(z)j. ZEG

In the case when the mappings w.. are conformal and so each x.. is identically zero we therefore have x = 0, and condition 2 follows from condition 1. However, in the general case the uniform convergence of w.. to w does not imply the convergence of the complex dilatations x.. to x almost everywhere. In fact, the following example shows that the

186

IV. Analytic Characterization of a Quasiconformal Mapping

local distortion properties of the mappings wn can be entirely different from those of the limit mapping w:

For every e, 0 < e < 1, there exists a sequence of quasiconformal mappings W n of the square R = {x + i Y 10 < x < 1, 0 < Y < 1} with the following properties: lim wlI(z) =

Z ,

uniformly in R,

n-oo

IUn(Z) I = 1 - e almost everywhere in R for every n. To prove this we divide R into n 2 squares R hk = {x + i y I (h - 1)ln x hln, (k - 1)ln y kin}, h, k = 1, ... ,n. We construct a quasiconformal mapping fhk of each R hk onto itself as follows: First map R hk onto the unit disc lei 1 by a conformal mapping ({i such that the midpoint Zhk of R hk goes into the origin. Then map the unit disc (21e - 1)-quasiconformally onto itself by the function t,

< <

< <

<

and finally map the unit disc back onto R hk by means of ({i-I. The complex dilatation of the composite mapping Ihk = ({i-lot ({i has then the absolute value 1 - e everywhere in R hk except at the midpoint Zhk' This mapping can be extended to a homeomorphism of R h k onto itself and the extended mapping leaves every boundary point of R hk invariant. 0

Thus the formula wn(z) = Ihk(Z) for Z E R hk defines a homeomorphism wn of R onto itself. By Theorem 1.8.3 on the removability of an analytic arc it follows that w n is quasiconformal with maximal dilatation 21e - 1. Since wn maps each R hk onto itself, we have Iwn(z) - zi ::::::; {iln at every point Z E R. If we perform the above construction for every n = 1, 2, ... , we obtain a sequence w n with the required properties. Since the limit mapping has complex dilatation zero everywhere in R, the sequence w n is not a good approximation. We remark that this is so despite the fact that the absolute values Iun(z) I converge almost everywhere in R. 5.5. Weak convergence of the derivatives. In order to investigate the conditions under which a sequence of quasiconformal mappings is a good approximation we first make a few preliminary remarks. Let In' n = 1, 2, ... , be a sequence of functions which are locally integrable in the domain G. We say that the sequence In converges

187

§ 5. Complex Dilatation

weakly in G to a locally integrable function f if lim

JJ (fn -

f) da

=

(5.12)

0

n-oo R

for every horizontal rectangle R, ReG. We shall now prove that the uniform convergence of a sequence of quasiconformal mappings implies the weak convergence of the derivatives (Bers [1J). It follows from the example of 5.4 that the derivatives need not converge almost everywhere. Lemma 5.1. Let Ww n = 1,2, ... , be a sequence of K-quasiconformal mappings of G which converges uniformly to a finite function w on compact subsets of G. Then the derivatives (w,,). and (w"h converge weakly to w. and wz' respectively.

Proof: It follows from Theorem II.5. 3 that the limit function w is either quasiconformal or constant. Therefore wand all the mappings w" possess V-derivatives by Theorem 1.2. Let R, ReG, be an arbitrary horizontal rectangle. It follows from formulae (6.17) of UI.6.5 that

JJ ((w,,)z- wz) da = ~ J(W nR

w) dz,

oR

JJ ((Wn)z - wz)da = 21i J (w n- W) dz. R

oR

Since w" tends uniformly to w on the boundary of R, the above integrals converge to zero, as was asserted. . As an extension of this result we shall show in V.5.6 that for a good approximation, the derivatives converge in the L2 metric to the corresponding derivatives of the limit function.

5.6. A criterion for good approximation. We again consider the case when a sequence w" of K-quasiconformal mappings of G converges to a quasiconformal mapping w uniformly on compact subsets of G. The example constructed in 5.4 shows that 1",,(z)1 can converge to a limit which is greater than ,,,(z)1 for almost every z E G. If however the argument of ",,(z) also converges almost everywhere in G, then we can show that w" is a good approximation to w in the sense of the definition given in 5.4 (Bers [1J). Theorem 5.2. Let w", n = 1,2, ... , be a sequence of K-quasiconformal mappings of G which converges to a quasiconformal mapping w with

188

IV. Analytic Characterization of a Quasiconformal Mapping

complex dilatation u, uniformly on compact subsets of G. If the complex dilatations un(z) of W n tend to a limit uoo(z) almost everywhere in G, then wn is a good approximation to w, i.e. uoo(z) = u(z) almost everywhere in G. Proof: We may assume that G and its images wn(G) are finite domains. Since u W z = 0 and W z '1= 0 almost everywhere in G, we have to prove that C(z) = w.(z) - uoo(z) wz(z) = 0

w. -

for almost every z E G. We write

C= [w. - (wnhJ

+ [(wn). -

un(wJzJ

+ [un(wn)z -uoo(wn)zJ +

[uoo(wn)z -

U

oo wzJ ,

and integrate Cover an arbitrary horizontal rectangle R, ReG. Since the term in the second bracket vanishes almost everywhere in G, we obtain (5.13) fJ Cda = II,n Iz,n [3,n,

+

R

+

where

By Lemma 5.1 we have lim [I,n

= o.

(5.14)

Further, it follows from Schwarz's inequality and the dilatation condition(1.6)that •

II z,n1 2 ~

fJ

IUn - uoo l2 da fJ I(W n).l2 da ~ K m(wn(R)) R

fJ

IU n - uoo l2 da. (5.15) Since w is finite inG and wn converges uniformly to win R, m(wn(R)) is R

R

uniformly bounded. The integral on the right hand side of (5.15) tends to zero as n --+ 00 by Lebesgue's convergence theorem (see IlL1.5), and so (5.16) lim Iz,n = o. To estimate the third integral in (5.13) we refer to the result in IlL5.3, by which the function U oo can be approximated by uniformly bounded step functions f{Jk with constant values in horizontal squares so that lim f{Jk = U oo almost everywhere in R. We write

[3,n

=

If (u oo R

f{Jk)

((wn)z - wz) da

+ fJ f{Jk ((wn)z R

wz) da. (5.17)

§ 5. Complex Dilatation

189

A further application of Schwarz's inequality and Lebesgue's convergence theorem shows that to every s 0 there corresponds a ko 0 such that for k = ko and every n the absolute value of the first integral on the right in (5.17) is less than s. On the other hand, we deduce by applying Lemma 5.1 to the finitely many squares in which ({!k, has a constant value that lim fJ ({!k, ((wn)z - w z) da = O.

>

>

n-oo R

Thus l3,n also tends to zero, and (5.13), (5.14) and (5.16) give

fJ'da = 0 .

(5.18)

R

It follows that' vanishes almost everywhere in G. To prove this we first remark that every open set C G can be represented in the form (U R h ) U Eo where the sets R h , It = 1, 2, ... , are disjoint horizontal rectangles and Eo is a set of zero area. By (5.18), the integral of' over therefore vanishes. If A, A C G, is an arbitrary measurable set, we can construct a decreasing sequence of open sets Ok' A C Ok' Ok C G, k = 1,2, ... , such that lim m (Ok - A) = 0 (see III.1.2). Since the Lebesgue integral is locally absolutely continuous as a set function, we have (5.19) Jf'da=O

°

°

A

for every measurable A, A C G. Let ~ be the real part of ,. The set E = {z I Z E G, ~(z) =F O} is the union of the sets E k = {z I Z E G, ~(z) 11k}, E_ k = {z I Z E G, ~(z) 11k}, k = 1, 2, . . .. Since each of these sets has zero area by (5.19), we have m(E) = 0, and so ~ vanishes almost everywhere in G. In the same way we can show that the imaginary part of' vanishes almost everywhere, and the theorem is proved.

<-

>

V. Quasiconformal Mappings with Prescribed Complex Dilatation Introduction to Chapter V In the previous chapter we introduced the complex dilatation of a quasiconformal mapping. It is a Borel function defined on the set of the regular points of the mapping i.e. at almost all points. Its absolute value is bounded by a number less than one. We shall prove the following Existence theorem, which is fundamental in the theory of quasiconformal mappings: If " is any measurable function in a domain G with suP. 1,,(z)1 1, then there exists a quasiconformal mapping of G whose complex dilatation is equal to " almost everywhere. By the Uniqueness theorem proved in IV.5 this mapping is uniquely determined up to a conformal mapping.

<

The Existence theorem will be proved in § 1 by a method which rests on the function theoretical sewing theorem of II.l. At the end of § 1 we briefly refer to other methods of proof. In §§ 2- 5 we make repeated use of the Existence theorem. First, in § 2 we examine systematically the mutual relationships between the dilatation measures introduced earlier: the local maximal dilatation, the dilatation quotient and the circular dilatation. In § 3 three problems on the removability of sets are considered in detail. Such earlier results as the removability of a point (Theorem 1.8.1) and the invariance of the maximal dilatation under removal of analytic arcs (Theorem 1.8.3) turn out to be special cases of more general theorems. In § 4 we show that every quasiconformal mapping admits a good approximation by regular and even real-analytic quasiconformal mappings. This leads us to a characterization of quasiconformal mappings as nonconstant limits of regular quasiconformal mappings. In § 5 we apply the Hilbert transformation introduced in IILl to the study of the derivatives of a quasiconformal mapping. Some earlier results of this type can be extended and sharpened in this way.

191

§ 1. Existence Theorem

The last two sections, 6 and 7, are devoted to the study of the local behaviour of a mapping with a given complex dilatation. In particular, we give conditions under which a mapping is regular at a given point. Such questions can be attacked in various ways; our method is essentially based on module estimations with the help of extremal lengths.

§

I.

Existence Theorem

1.1. Formulation of the existence problem. We now pose the problem of constructing a quasiconformal mapping with given local dilatation properties. In view of the definition of quasiconformality, it would be natural to choose F or D as a measure of dilatation. Since, however, even the complex dilatation can be prescribed almost everywhere we shall prove the existence theorem in this formulation.

We draw attention to some restrictions which are necessary for the solvability of our problem. First of all we may not prescribe the value of the complex dilatation at every point where it is defined. This follows, for example, from the fact mentioned in IV.5.1 that the vanishing of the complex dilatation 'outside a null set already implies the conformalityof a quasiconformal mapping. Secondly, we knowthat the complex dilatation is a Borel function. Since it coincides with the prescribed function only almost everywhere it is sufficient to assume that this function is measurable with respect to the Lebesgue area measure. Our problem can thus be formulated as follows: Given an arbitrary measurable junction" in a domain G with

sup ,,,(z)!

<1,

ZEG

construct a quasiconjormal mapping oj G with complex dilatation equal to ,,(z) jor almost every z E G.

We remark that the problem can be transformed to the case when G is the whole plane. For an arbitrary domain G, we need only extend each measurable function" by writing ,,*(z) = ,,(z) for z E G, ,,*(z) = 0 for z ({ G; the function is then measurable in the whole plane. If the problem can be i30lved for ,,*, then the restriction of the solution to G solves the original problem.

,,*

1.2. Solution of a sewing problem: We first consider the special case when " is a step function (see III. 5.3). Let N be the net formed by the squares Qh k = {x i Y I (h - 1) (j x h (j, (k - 1) (j y k (j }, h, k = 0, ± 1, ± 2, ... , and" a step function with supz 1,,(z)1 1 which has a constant value "hk in each square Qhk' Since the complex dilatation of the required mapping is to coincide with " only almost

+

< <

< <

<

192

V. Quasiconformal Mappings with Prescribed Complex Dilatation

everywhere, the values of x on the sides of the squares and at infinity are without significance; we may set x(z) = 0 at such points. For each individual square Qhk our problem can be solved immediately; for example the affine mapping Tjhk defined by 'flhk(Z)

=

Z

+ Xhk Z

(1.1)

has the given complex dilatation Xu in Qhk' We shall construct the required quasiconformal mapping of the plane starting with the functions (1.1), and joining the squares Qhk together step by step. For this purpose we must first solve the following sewing problem.

Let R 1 and R 2 be two congruent non-intersecting rectangles having the open segment A as a common side. Let Wi' i = 1, 2, be a quasiconlormal mapping 01 R i with the (not necessarily constant) complex dilatation Xi' It is required to construct a quasiconlormal mapping 01 the rectangle R = R 1 U A U R 2 having the complex dilatation xi(z) lor almost all Z E R , i = 1, 2. i By IV.5.2 the mappings I 0 Wi and wi have the same complex dilatation if I is conformal. Hence there is no loss of generality in assuming that w1(R 1) = HI and w2(R2) = H 2 are the upper and lower half-planes, respectively. The mappings WI and W2 can then be extended topologically to R 1 and R 2 • By means of suitable conformal mappings of HI and H 2 onto themselves we may ensure further that WI and W2 map the segment A onto the same interval I of the real axis and that the points W;-l(OO) and W;l(OO) are symmetrically placed with respect to A. If the mappings WI and W2 coincide on A then the sewing problem is solved. In fact the mapping W defined by w(z) = w1(z) for Z E R 1 U A, w(z) = w2(z) for Z E R 2 is then a homeomorphism of R onto the domain HI U I U H 2, and it follows from Theorem 1.8.3 on the removability of an analytic arc that W is quasiconformal. In the general case when WI and W2 do not necessarily coincide on A we construct two conformal mappings II and 12 which map HI and H 2 onto disjoint Jordan domains such that 11 0 WI and 12 0 W2 have the same boundary values on A. If we then write w(z) = 11(wl(~)) for z E R 1 U A, w(z) = Mw2(z)) for z E R 2, then the mapping W is a homeomorphism. In the same way as above we deduce that W is quasiconformal. Since Ii 0 Wi' i = 1,2, has the same complex dilatation as Wi' the sewing problem is solved.

We still have to prove the existence of the auxiliary conformal mappings 11 and 12' The above boundary condition 11 WI = 12 W2 holds on A if, for every x E I,ll and 12 satisfy the equation 0

11(x) = 12(w2(w;-1(x))) .

0

§ 1. Existence Theorem

19}

In view of the sewing theorem in II.7.5 the existence of the required mappings 11 and 12 thus follows if we prove that the homeomorphism qJ: I ---+ I, defined by qJ(x) = W2(W;I(X)), is quasisymmetric. For this purpose we extend the quasiconformal mapping w2 to R 1 by reflection in A. The composite mapping w 2 W;I is then defined in the upper half-plane and maps it quasiconformally onto itself, keeping the point at infinity fixed. On I the boundary function of the mapping w2 0 W;I coincides with qJ which is therefore quasisymmetric. The above sewing problem is thus solved. 0

1.3. Proof of the Existence theorem. Using the solution of the above sewing problem we now construct a quasiconformal mapping with prescribed complex dilatation, first in the case of a step function. We again consider the net N introduced in 1.2 and the corresponding step function x. From the squares Qhk E N and their sides we can obviously construct a sequence of rectangles R n , n = 1,2, ... , satisfying the following conditions: 1. R1 is one of the squares

Qhk'

2. R n , n = 2, 3, ... , consists of R n _ l , a rectangle congruent to R n _ l , and their common side.

3. U R n is the finite plane. n~1

Starting with the affine mappings (1.1) we can then, by repeated solution of the sewing problem, construct a sequence of quasiconformal mappings w n : R n ---+ R~, such that the complex dilatation of w n coincides with x almost everywhere in R n . The mappings w n can be normalized by means of linear transformations in such a way that they do not assume the value 00 and have 0 and 1 as fixed points from some n onwards. By Theorem II.5.1 the family {w n I n ~ m} is then normal in R m • Consequently, for every R m we can find a subsequence wm,n' n = 1, 2, ... , of the sequence w n ' converging uniformly in compact subsets of R m . These sequences can be chosen so that wm+l,n is a subsequence of wm,no m = 1,2, .... The diagonal sequence WI,I' W 2 ,2' . . . then converges uniformly in every bounded domain. In view of the above normalization of the mappings it follows from the results of II.5 that lim wn,n = W is a quasiconformal mapping' of the finite plane onto itself. By Theorem IV.5.2 the complex dilatation of W coincides with x almost everywhere in the plane. The existence problem of 1.1 is thus solved in the case when x is a step function. By a limiting process we can now easily treat the general case in which 1. By IIL5.}, x is an arbitrary measurable function with suP. Ix(z)1

<

194

V. Quasiconformal Mappings with Prescribed Complex Dilatation

x can be approximated by a sequence of step functions x n such that suP. Ixn(z)/ ::;; sup.lx(z)l, and xn(z) --->- x(z) almost everywhere in the plane. For each x n we construct a quasiconformal mapping W n of the finite plane onto itself, with complex dilatation coinciding with x n almost everywhere. If each W n is normalized so that Wn(O) = 0, Wn (1) = 1, then the family {W n } is normal and we can choose a uniformly convergent subsequence of W n which has as limit function W a quasiconformal mapping of the finite plane. This can be extended to the whole plane by writing W( (0) = 00. By Theorem IV.5.2 the complex dilatation of W coincides with x almost everywhere. We have thus obtained the desired fundamental result: Existence theorem. If G is an arbitrary domain and x an arbitrary measurable function in G with sup Ix(z)1

< 1,

'EG

then there exists a quasiconformal mapping w of G whose complex dilatation coincides with x almost everywhere in G.

We remark that the set of all quasiconformal mappings of G having complex dilatation x almost everywhere coincides with the family {low} , where f runs through all conformal mappings of w(G). This follows from the Uniqueness theorem proved in IV.5.2. By the remark in 1.8.1 the conformal and quasiconformal equivalence classes coincide for simply connected domains. In view of the Riemann mapping theorem we deduce from the Existence theorem the following Mapping theorem. Let G and G' be conformally equivalent simply connected domains and x a measurable function in G with suP. Ix(z) I 1. Then there exists a quasiconformal mapping w: G --->- G' whose complex dilatation coincides with x almost everywhere. This mapping is uniquely determined up to a conformal mapping 01 G' onto itself.

<

1.4. Other proofs for the Existence theorem. The above method of proof resembles one used by Lavrentieff [1J. Here we indicate some others. It should first be mentioned that one is allowed to make very restrictive assumptions about the given measurable function x. Indeed, by Theorem IV.5.2 it is enough to consider a class of functions which approximate every bounded measurable function almost everywhere (d. III.5.2-3). In the above proof we have made use of this freedom by first choosing x to be a step function. Secondly, it should be emphasized that if x is sufficiently regular, then it is relatively easy to find locally homeomorphic solutions of the equa-

§ 2. Local Dilatation Measures

195

tion W z = ){ w z ' This had already been proved by Gauss under the assumption that ){ is real-analytic. In the special case when ){ is a polynomial the existence of a locally homeomorphic solution will be shown by direct calculation in 4.2, in connection with another problem. We can also obtain such local solutions by applying the integral transformation introduced in III.7 (see § 5, where such questions are discussed in more detail). The transition from local homeomorphisms to a global homeomorphic solution can be carried out in various ways. The classical method depends on the solvability of the boundary value problem for the real part of the solution (see Morrey [1J, where references to the older literature are also to be found, and Martio [1J). Another method is based on the application of uniformization theory. The plane is covered by discs U i , i = 1, 2, ... , n, in such a way that in each U i a homeomorphic solution of the equation W z = ){ W z can be constructed. By the Uniqueness theorem these solutions depend conformally on each other in common parts of the discs and define therefore a conformal structure for the plane. The Existence theorem then follows if we apply the uniformization theorem to the corresponding Riemann surface. This proof is less direct than the one given in 1.1 -1. 3 in that the uniformization theory must be brought in. On the other hand, it is possible to give a proof of the uniformization theorem which is similar to that of the sewing theorem. We also refer to Kuusalo [1J who shows how the use of the uniformization theorem can be replaced by fairly simple normal family considerations. Starting from the above-mentioned integral transformations we can also obtain solutions which are homeomorphic not only locally but globally. The proof was given in its final form by Bojarski [1J (d. 5.7); see also Ahlfors-Bers [1]. If){ has bounded support, this method yields . a representation for a normalized solution in terms of){. The formula is important as it can be used to study the dependence of the mapping on ){.

§

2.

Local Dilatation Measures

2.1. Connections between D, It and H. The Existence theorem and the results of Chapter IV give us new information about the mutual dependence of the local dilatation measures D, F and H. Our earlier results are contained in 1.9 and 11.9; we have shown that for any quasiconformal mapping F(z) 2: D(z) = H(z) (2.1)

196

V. Quasiconformal Mappings with Prescribed Complex Dilatation

at every regular point, while H(z) can exceed F(z) at a non-regular point. Let w be a quasiconformal mapping of the domain G. Since almost all points of G are regular for w, (2.1) holds almost everywhere in G. We shall now show that these relations can be sharpened. If U is a subdomain of G, then by Lemma 1.9.1 the maximal dilatation K(U) of w in U is equal to the least upper bound of F(z) in U. Since D(z) < F(z) at every point z where D(z) is defined, we have K(U) ~ supz D(z) for z E U. On the other hand, it follows from the Analytic definition that the restriction of w to U is K-quasiconformal with K = supz D(z). Hence K(U) cannot exceed this bound and so

= sup D(z) .

K(U)

ZEU

In view of the definition of F(z) it follows that

F(zo) = inf (sup D(Z)) U,ZoEU

for every Zo

E

(2.2)

ZEU

G.

Thus we obtain the following result: F is the least upper semicontinuous majorant 01 D.

In (2.2) z only runs through the regular points of w lying in U. In addition to the set of non-regular points one may omit an arbitrary null set without diminishing the least upper bound of D(z). This follows from the Analytic definition according to which the restriction of w to U is K-quasiconformal as soon as D(z) ::;: K holds almost everywhere in U. Thus if we write as usual sup ess D(z) = ZEU

inf E,m(E)=O

(sup D(Z)) , ZEU-E

lim sup ess D(z) = inf (sup ess D(Z)) , Z-Zo

U,ZoEU

ZEU

then K(U)

= sup ess D(z) Z E

U

and

F(zo)

=

lim sup ess D(z)

(2·3 )

z-zo

for every Zo

E

G. By (2.1), (2.3) and Theorem 1.9.2 we also have

F(zo) = lim sup F(z) = lim sup ess F(z) . z -Zo

z-zo

(2.4)

197

§ 2. Local Dilatation Measures

The circular dilatation H(z) coincides with D(z) almost everywhere in G (see (2.1)) and so

lim sup ess H(z) = lim sup ess D(z) z

z-zo

for every Zo ing result:

E

-Zo

G. If we combine this with (2.2-4) we obtain the follow-

Theorem 2.1. The dilatation measures D, F and H of a quasiconformal mapping w : G ~ G' satisfy the relations

F(zo) = lim sup F(z) = lim sup ess F(z) = lim sup D(z) z -Zo

z-zo

z -Zo

= lim sup ess D(z) = lim sup ess H(z) z-zo

z-zo

for every

Zo E

G.

2.2. Existence theorem for the local maximal dilatation. The Existence theorem of the previous section, in which the complex dilatation is prescribed, gives rise to the question of how far the real dilatation measures D, F and H can be prescribed. For the dilatation quotient D = (1 Ixl)((1 - fxl) this is already solved by the Existence theorem. Since H(z) = D(z) at all regular points z, the circular dilatation H can also be prescribed almost everywhere. Using Theorem 2.1 we can prove a corresponding existence theorem for F.

+

Theorem 2.2. Let ([J be a real-valued, upper semicontinuous, bounded function defined in G, with inf ([J(z) ~ 1. Then there exists a quasiconformal mapping w of G whose local maximal dilatation F(z) coincides with ([J(z) for almost all z E G and exceeds ([J(z) nowhere in G.

Proof: By the Existence theorem of § 1, there is a quasiconformal mapping w of G with dilatation quotient D(z) = ([J(z) almost everywhere in G. For the local maximal dilatation of w we then have F(z) ~ ([J(z) almost everywhere in G. On the other hand, Theorem 2.1 shows that

F(z) = lim sup ess D(C) C-z

=

lim sup ess ([J(C)

c-z

~

lim sup ([J(C) .

c-z

Since ([J is upper semicontinuous we thus have F(z) ~ ([J(z) at every point of G. Thus the mapping w possesses the required properties. Naturally there exist many different quasiconformal mappings with the same local maximal dilatation. We shall now show that not even the dilatation quotient D is determined by F.

v.

198

Quasiconformal Mappings with Prescribed Complex Dilatation

2.3. An example where F(z) is greater than D(z) almost everywhere. Theorem 2.1 does not imply that F(z) coincides with D(z) almost everywhere. We can indeed construct an example in which F(z) exceeds D(z) almost everywhere and differs from it by an arbitrarily large amount in a set of positive measure. Theorem 2.3. Let G be a domain and 8 and M two positive numbers. Then there exists a quasiconformal mapping w of G for which F(z) D(z) almost everywhere in G and F(z) = D(z) M except for a set of area

+

:s;;

>

8.

Proof: Let E be the set of points of G whose coordinates are rational numbers. Since E is countable it has zero area. Thus we can construct a decreasing sequence of open sets Gk , E C Gk , k = 1,2, ... , such that G1 = G, m(G2) :s;; 8 and lim m(Gk ) = o. We set D*(z) = 1 + M (1 - 11k) for z E Gk - Gk+l' k = 1,2, ... , which defines D*(z) for almost all Z E G. By the Existence theorem there exists a quasiconformal mapping w of G with dilatation quotient D(z) equal to D*(z) for almost all z E G. We shall determine the local maximal dilatation of w using Theorem 2.1. Let U be an arbitrary subdomain of G. Since U contains points of E, Gk n U has a positive area as a non-empty open set, for each k = 1,2, .... In Gk we have D*(z) ~ 1 M (1 - 11k) almost everywhere, and so sup ess D(z) = M 1.

+

+

ZEU

It follows from Theorem 2.1 that F(z) = M + 1 at every point z of G. Thus F(z) D(z) for almost every z E G. In the set G - G2 we have D(z) = 1 almost everywhere and consequently F(z) - D(z) = M. Since m(G 2 ) :s;; 8, the mapping w satisfies all the requirements of Theorem 2.3. We see that two quasiconformal mappings with the same local maximal dilatation F can display quite different behaviour. The mapping w constructed above and an affine mapping with F(z) = D(z) = M + 1 will serve as examples.

>

Remark. Since D(z) = H(z) almost everywhere in G, Theorem 2.3 will also hold if D(z) is replaced by H(z). However, a result of Gehring ([1J, p. 25) shows that the inequality F(z) H(z) cannot hold at every point of G. On the other hand, Gehring [1] has constructed a quasiconformal mapping for which H(z) F(z) in a set 01 Hausdorff dimension 2.

>

>

§ 3. Removable Point Sets

199

Since H(z) < F(z) at regular points we conclude: The set of non-regular points of a quasiconformal mapping can possess Hausdorff dimension 2. This set is, however, always of zero area, as has already been mentioned several times.

§ J. Removable Point Sets 3.1. Three problems of removability. In 1.8 and II.8 we investigated various continuation problems for quasiconformal mappings. These are related to a group of problems which will be treated here. Let G be a domain and E a point set lying in G. For each fixed K, 1~ K 00, we consider ~he following three classes of mappings:

<

1. The class Wl consisting of all quasiconformal mappings of G whose local maximal dilatation satisfies the condition

sup F(z) < K .

(3·1 )

zEG-E

2. The class W 2 consisting of all sense-preserving homeomorphisms of G satisfying (3.1).

3. The class Wa, which is only defined when G - E is a domain; it then consists of all K-quasiconformal mappings of G - E. The set E is called removable with respect to the classes Wl and W 2 , respectively, if all mappings of the class concerned are K-quasiconformal in the whole domain G. Correspondingly, we say that E is removable with respect to the class Wa if every mapping of this class can be extended to a K-quasiconformal mapping of G.46 If G - E is a domain, then in each of the above cases we have to continue a K-quasiconformal mapping of G - E to a mapping of G. The removability of E in the case of the class Wl then means that every quasiconformal extension is K-quasiconformal; for W 2 the set E is removable if every topological extension is K-quasiconformal and for Wa if a K-quasiconformal extension exists.

It should be noted that removability with respect to Wl or W 2 is a monotonic property of E, i.e. all subsets of a removable set are removable. In 3.5 we prove that the same is true of Wa ; this is not quite evident from the above definition. Of the results proved in 1.8, Theorem 1.8.1 states that a set consisting of a single point is removable with respect to Wa, and Theorem 1.8.3 46

For a fourth related problem see Renggli [1J.

200

v.

Quasiconformal Mappings with Prescribed Complex Dilatation

expresses the removability of an analytic arc with respect to W 2 . Our subsequent results will contain these theorems as special cases. Remark. To obtain an analogy between the cases 1, 2 and 3, the removability of E with respect to Wi' i = 1, 2, should be defined as follows: E is removable with respect to Wi if there corresponds to every mapping wE Wi a K-quasiconformal mapping of G whose restriction to G - E coincides with w. If E has no interior points, then the two definitions of removability coincide. In fact, in this case a homeomorphism of G is uniquely determined by its restriction to G - E. If E has interior points,then we can only deduce that a set which is removable in the sense of the first definition also satisfies the requirements of the second. The converse of this is false; for example, a set E containing the whole of G apart from a single point is removable according to the second definition but not to the first. It also follows from this example that removability in the sense of the second definition is not a monotonic property of the set. One can however avoid this disadvantage by excluding certain sets E. Since such restrictions carry with them additional difficulties we have preferred the first definition.

3.2. Independence of the removability on the maximal dilatation. \Ve have not discussed so far the dependence of the classes WI' W 2 , Wa on K. Actually, the value of K has no influence on our removability problems: if a set is removable with respect to Wi for some value of K, then the same is true for every other K. This assertion, which we shall return to later, is proved in part in the following lemma. L e m m a 3.1. Let E C G be a set which is removable with respect to one of the classes WI' W 2 , Wa for K = 1. Then E has no interior points and is removable relative to the class concerned for every K.

Proof: If E has interior points we can choose a disc D, DeE, and map the complement of D conformally onto a Jordan domain with a nonanalytic quasiconformal boundary curve (see II.8). By Theorem 11.8.3 this conformal mapping can be extended to a quasiconformal mapping w of the whole plane. However, w cannot be conformal on the boundary of D. Hence E is not removable with respect to WI and W 2 for K = 1. In the case of Wa, G - E is a domain by definition. It then follows by the uniqueness of analytic continuation that a conformal mapping of G cannot coincide with w in G - E, since both mappings would assume the same values on the boundary of D, and w (Fr D) is a non-analytic

§ 3. Removable Point Sets

201

Jordan curve. Hence for K = 1 the set E is not removable with respect to W 3 either, and the first part of the lemma is proved. In order to prove the second part we assume that E is removable with respect to one of the classes Wi' i = 1,2,3, for K = 1. Let w be a mapping which belongs to this class for some 1. We have to prove either that w is a K-quasiconformal mapping of G or, in the case of W 3' that it can be extended to such a mapping.

K>

In the case of W 3 the set E is closed in G by hypothesis. This assumption may also be made in the other two cases without loss of generality. In fact, it follows from the semicontinuity of the local maximal dilatation that the set E e = {z I F(z) ~ K s} is closed in G for every s o. As a subset of E, E e is also removable for K = 1. If the assertion holds with E e and K s instead of E and K, then F(z) ~ K s at every point of G, and so it is enough to prove this for every s o.

>

+

+

+

>

The set E may thus be assumed to be closed in G. The mapping w is then quasiconformal in every component of the open set G - E and consequently preserves null sets (see IV.1.4). If ~ denotes the complex dilatation of w, then we can define a measurable function ~* almost everywhere in the plane by writing ~*(w(z))

~*(')

= =

~(z) e2iargwz(z)

for for

0

z EG - E ,

'(f w(G -

E) .

By the Existence theorem, there is a K-quasiconformal mapping w* of the whole plane with complex dilatation ~* almost everywhere. The composite mapping / = w * 0 w is defined in the same domain as wand has complex dilatation zero almost everywhere in G - E by equation (5.6) of IV.5 .2. Since G - E is open it follows from Theorem 2.1 that the local maximal dilatation of / is equal to 1 everywhere in G - E. Since E is removable for K = 1 by hypothesis, there exists a conformal mapping lof G whose restriction to G - E coincides with /. The mapping (W*)-1 0/ is then K-quasiconformal in G and coincides with w in G - E. In the cases WE Wi' i = 1,2, we have w = (W*)-1 0/ everywhere in G, since E has no interior points. The lemma is thus proved.

3.3. Removability of a set with respect to the class W l' For the class WI the removability problem is solved in the following theorem. Theorem 3.1. Let G be a domain and E a subset sup F(z) ZEG

= sup F(z) zEG-E

0/ G.

Then

202

V. Quasiconformal Mappings with Prescribed Complex Dilatation

for the local maximal dilatation of every quasiconformal mapping of G if and only if every compact subset of E has zero area. Proof: The condition is necessary. For let Eo C E be a compact set with m(Eo} O. Then for every K ~ 1 the function F defined by F(z) = K + 1 for z E Eo, F(z} = K for z E G - Eo is upper semicontinuQus. Hence by Theorem 2.2 there exists a quasiconformal mapping of G whose local maximal dilatation has the value K 1 almost everywhere in Eo and does not exceed K anywhere in G - E.

>

+

To prove that the condition is sufficient suppose next that every compact subset of E is a null set. Then all subsets of E which are closed in G have zero area, since G can be exhausted by a sequence of compact sets. Let w be a quasiconformal mapping of G whose local maximal dilatation has the upper bound K in G - E. In order to prove that F(z} < K everywhere in G we again denote by E e , e 0, the set of points z where F(z} ~ K e. This set is closed in G and since E e C E we have m(E e } = O. Hence F(z} K e almost everywhere in G. By Theorem 2.1 we have therefore F(z} < K e everywhere in G, a,nd since e 0 was arbitrary, it follows that F(z) :::;; K in G as required.

+

>

< +

>

+

From Theorem 3.1 we see that the removability of a set with respect to the class WI is independent of K (d. 3.2).

3.4. Removability of a set with respect to W 2 • We now investigate the class W 2 . In this case it is not possible to give a necessary and sufficient condition for the removability of E in terms of Hausdorff measures (d. 3.7). Since the class WI is a subclass of W 2 , the condition of Theorem 3.1 is necessary for the removability of a set with respect to W 2 • The following theorem, which contains Theorem 1.8.3 as a special case, gives a sufficient condition (Strebel [1J; this work also contains results on the classes WI and W s') Theorem 3.2. If every compact subset of the set E C G has a-finite linear measure, then sup F(z} = sup F(z) ZEG

zEG-E

for every sense-preserving homeomorphism of the domain G. Proof: We may assume that w is a sense-preserving homeomorphism of G, whose local maximal dilatation has a finite upper bound K in G - E. To prove that w isK-quasiconformal in G we make use of the Analytic definition. We show first that w is absolutely continuous on lines in G.

§ 3. Removable Point Sets

203

Let us consider a rectangle R= {x+iYla<x
>°

Let N k be the net (see III.5.3) whose squares have sides of length 2- k • F or every k = 1, 2, ...• choose those squares of N k whose closures lie in R - E. and which are not contained in any square chosen earlier. The union of the squares Qh' h = 1, 2, ... , so chosen covers the rectangle R with the exception of a set Eo consisting of R n E. and the sides of Qh' Thus Eo also has a-finite linear measure and its area is zero. Next, we consider an arbitrary Qh and denote by Ih(y) the open horizontal segment with ordinate y which joins the vertical sides of Qh' Since W is quasiconformal in every component of the open set R - E. which contains Qh' we see by Lemma IV.1.1 that W is absolutely continuous on all segments Ih(y), except for a set A h of values of ywith zero linear measure.

+

< <

We next consider horizontal segments I(y) = {x i y Ia x b}, c y d. To prove the absolute continuity of W on almost every I (y) we need three preparatory results. First it follows from the above that for y E! U A h , thus for almost all y, c y d, W is absolutely continuous on all (non-empty) segments I(y) n Qh = Ih(y).

< <

< <

By Lemma III.1.3 a set of finite linear measure has at most finitely many points on almost all horizontal segments I(y). Since the linear measure of Eo = R - U Qh is a-finite, we conclude secondly that the set Eo n I(y) = I(y) - U Ih(y) is countable for almost all y, c y d. To obt~in the third preliminary result we consider the restriction of W to an arbitrary square Qh' Since it is (K + e)-quasiconformal, we have (Iwz(z)! + IW z(Z)1)2 ~ (K + e) J(z) for almost all z E Qh' Thus by Lemma III.3.3, If (lw.1 IW zl)2 da ~ (K e) m(w(Qh))

< <

+

+

Qk

for each h = 1,2, .... Summing over h and noting that m (R - U Qh) = m(Eo) = we deduce that

°

If (Iwzl + R

IW zl)2 da < (K

+ e) m(w(R)) .

Since w(R) is a bounded domain, Fubini's theorem (III.1.5) states that (lw.1 IW zI)2, and so also Iwxl = Iw. wzl, is integrable with respect to linear measure on almost all segments I(y), c y d.

+

+

< <

204

V. Quasiconformal Mappings with Prescribed Complex Dilatation

< <

Combining the above results we deduce that for almost all y, C y d, the segment I(y) = I has the following three properties: 1°. I consists of the countable set Eo n I = I o and a sequence of segments I h , It = 1,2, .... 2°. W is absolutely continuous on every I h , It = 1,2, .... 3°. The integral J IW"I dx is finite. I

From these conditions we shall deduce that W is absolutely continuous on I. By III.2.1-2 it is enough to show that l(w(I)) is finite and that w maps every set A C I of zero linear measure onto a set of zero linear measure. It follows from 2° that the length of the image of I h is

l(w(Ih ))

J IW"I Idxl

=

Ih

for each It = 1, 2, ... (see III.2.6-7). Since I o and w(Io) are countable and consequently have zero linear measure, it follows from 3° that

l(w(I)) = };

J IW"lldxl = J IW"lldxl < 00 .

h Ih

I

!fA C Iisasetwithl(A) = 0 thenitfollowsfrom2°thatl(w (I h n A)) = 0 for It = 1,2, .... Since w (Io n A) is countable, l(w(A)) vanishes, and the absolute continuity of w on I is proved. In the same way we can prove that w is absolutely continuous on almost all vertical segments lying in R. The first requirement of the Analytic definition is thus satisfied. It is now easy to verify that w satisfies the second requirement of the Analytic definition. At every point z of the open set G - E. we have F(z) < K c, i.e. w is (K c)-quasiconformal in every component of G - E •. By Theorem IV.1.1,

+

+

max lo",w(z) I :::;: (K

+ c) min Io",w(z) I '"

for almost all z E G - E •. Since every compact subset of E has a-finite linear measure, we have m(E.) = 0 and the inequality therefore holds almost everywhere in G. Hence w is (K + c)-quasiconformal in G by the Analytic definition. Since c 0 was arbitrary, the theorem is proved. As in the case of WI' we remark that the removability of a set E C G with respect to W 2 is independent of K. To see this we consider a set E which is removable with respect to W 2 for some K 1. Then every sense-preserving homeomorphism w of G with F(z) = 1 in G - E is a K-quasiconformal mapping of G. Since E is removable with respect to WI> and this is true of every K ~ 1, it follows that w is conformal

>

>

205

§ 3. Removable Point Sets

in G. Thus E is removable with respect to W 2 for K K by Lemma 3.1.

= 1 and so for every

3.5. Removability of a set with respect to W 3 • To deal with the removability problem for the class W 3 we need some results from the theory of analytic functions. Using such tools, we first show that every set removable with respect to W 3 has area zero. This follows from the following, somewhat sharper, result. Lemma 3.2. Let E be a subset of the domain G such that G - E is a domain. If the area of E is positive, then there exists a conformal mapping of G - E which cannot be quasiconformally extended to G. Proof: By a theorem of Koebe [1J every domain can be mapped onto a slit domain whose complement has zero area. Let f be such a conformal mapping of G - E. If f could be extended to a quasiconformal mapping of G then this would carry E onto a set of zero area. However, this contradicts the fact that a quasiconformal mapping preserves null sets (see IV.1.4), and the lemma is proved.

If m(E) = 0, then E cannot have interior points and a quasiconformal mapping of G - E can be extended to E in at most one way. By Lemma 3.2, this holds for every set E removable with respect to W 3 , and so such a set is removable also with respect to WI and W 2 . For the same reason every subset of a set removable with respect to W 3 is removable.

Every conformal mapping of G - E can thus be extended to a quasiconformal mapping of G only if m(E) = 0. If such an extension exists it is conformal by Theorem 3.1, and so E is removable with respect to W 3 for K = 1. By Lemma 3.1 the same holds for every K, and we have thus proved the following result (d. Pesin [1J): Theorem 3-3. Let G be a domain and E a subset of G such that G - E is also a domain. If every conformal mapping of G - E has a quasiconformal extension to G then every K-quasiconformal mapping of G - E can be extended to a K-quasiconformal mapping of G.

It follows from this theorem that the third type of removability is also independent of K. In fact, if E is removable for some K = K o' then in particular all conformal mappings of G - E have a Ko-quasiconformal (in fact conformal) extension to G. By Theorem 3-3, E is then removable for every K.

3.6. A function-theoretical removability problem. Since the removability problem for W 3 is independent of K, it is equivalent to the follow-

206

V. Quasiconformal Mappings with Prescribed Complex Dilatation

ing purely function-theoretical problem: Under what conditions can every conformal mapping of G - E be extended to a <:;onformal mapping of G? We limit ourselves to the case when E is compact. Since G - E is a domain, it lies in a component of - E. The boundary of each component of - E is contained in E, and so - E is connected and hence a domainY We say that E belongs to the null class 0AD ifthere exists no non-constant analytic function f defined in - E such that

fJ 11'1 2 da

-E

<

00 .

The significance of the class 0 AD for our removability problem can be seen from the following function-theoretical equivalence theorem (Sario [1], AhHors-Beurling [1J): The set E belongs to the null class 0 A D if and only if every conformal mapping of G - E can be extended to a conformal mapping of G.

By this theorem, the removability of E with respect to 'lfJ3 is equivalent to E E 0 A D, provided that E and G satisfy the above conditions. The null class 0 A D has been exhaustively investigated in function theory; see for example the work of Ahlfors and Beurling [1J mentioned above, where further references are given. Of the results proved we mention the following: Every set E of zero linear measure belongs to 0AD'

3.7. Examples of removable and non-removable sets. By Theorem 3.1, sets which are removable with respect to the class 'lfJI can be characterized in terms of Hausdorff measures. On the other hand, this is not possible for 'lfJ 2 and 'lfJ3 • By means of some examples we shall now test the sharpness of the necessary or sufficient conditions obtained above for these classes. By Theorem 3.2, a set is removable with respect to 'lfJ2 if all compact subsets of E have a-finite linear measure. This sufficient condition cannot be improved very much. In fact, in IV.2.2 we constructed a non-quasiconformal homeomorphism of a square with the property that F(z) = 1 outside a set of dimension 1. It follows from this example that there exist sets of dimension 1 which are not removable with respect to 'lfJ2 • By 3.6 a compact set of zero linear measure 48 is removable with respect to'lfJ3 • This sufficient condition cannot be sharpened very much either: 47 Conver3ely it follows directly from Lemma I, 1. 3 that if E is a compact subset of G and - E is a domain, then G - E is a domain. 48

Note that the complement of a compact set of zero linear measure is connected.

207

§ 4. Approximation of a Quasiconformal Mapping

There exist sets of finite linear measure which are not removable with respect to Wa. A simple example is a segment.

On the other hand, we saw in 3 A and 3.5 that the inner area of a set which is removable with respect to W 2 or Wa vanishes. As concerns the sharpness of this necessary condition we refer to a result of Sario [1]. He has proved that the Cartesian product E of two congruent Cantor sets Ex = EAP1' P2' ...) and E y = E y(P1' P2' ...), lying on the x- and y-axes, respectively, belongs to GAD (and so is removable for Wa by 3.6) provided that l(E x ) = 0. If we choose Pn = (n 1)-1, then l(E x ) = 0, while it follows from Lemma III.1.2 that dim E = 2. Since sets removable for Wa are also removable for W 2 we deduce that there exist sets of dimension 2 which are removable with respect to W 2 and Wa•

+

§ 4. Approximation of aQuasiconformal Mapping 4.1. Approximation by mappings with prescribed complex dilatation. The proof of the Existence theorem depends essentially on good approximation (see IV.5A) of the required mapping by mappings whose complex dilatations are step functions. With the help of the Existence theorem this approximation process can be generalized as we shall now show (d. Bers [1J). Theorem 4.1. Let w: G --+ G' be a quasiconformal mapping with complex dilatation u. Let un' n = 1, 2, ... , be a sequence of measurable functions in G with sUPz /un(z)/ ::;: k 1, which converges to U almost everywhere in G. Then given a domain D, D C G, there exists a sequence n. of positive integers and a sequence wn • of quasiconformal mappings of D, v = 1, 2, ... , with the following two properties:

<

1. The mappings wn • give a good approximation to w in D. 2. The complex dilatation of

W nv

coincides with u nv almost everywhere

inDo Proof: By applying Theorem II.8.1 we first construct a quasiconformal mapping w* of the plane which coincides with w in D. Let u* be the complex dilatation of w*.

By the Existence theorem, for every n = 1, 2, ... , there exists a quasiconformal mapping of the plane whose complex dilatation coincides almost everywhere in D with un and in - D with u*. Since the mappings are determined up to a linear transformation we may set W:(Zh) = W*(Zh) for three fixed points Zh' h = 1,2,3. It then follows is normal. Hence from Theorem II. 5.1 that the family of mappings

w:

w:

w:

208

V. Quasiconformal Mappings with Prescribed Complex Dilatation

we can find a subsequence w~v' 'IJ = 1, 2, ... , which converges uniformly in the plane (with respect to the spherical metric) to a limit function wt. By Theorem II.5.3, W6 is a quasiconformal mapping of the plane. The complex dilatations of the mappings w~v converge almost everywhere in the plane to the complex dilatation of w*. By Theorem IV.5.2 the complex dilatation of W6 therefore coincides with almost everywhere. It follows from the Uniqueness theorem that w* = wt, since w* and wt have the same values at the points zl> Z2' Z3' If we set wnv equal to the restriction of w~v to D, then we obtain a sequence which satisfies conditions 1 and 2 of our theorem.

,,*

,,*

Remark. If G is simply connected, we may replace D by G. In fact, by the Mapping theorem in 1.3 there is for every n a quasiconformal mapping w n : G -+ G' whose complex dilatation coincides with "n almost everywhere in G. If these mappings are suitably normalized, then the results of II. 5 show that we can choose a subsequence which converges to w uniformly in every compact subset of G. The sequence thus constructed is then a good approximation to w in G. If G is not simply connected, there may not exist a quasiconformal mapping of G onto G' with complex dilatation "n almost everywhere in G, and the above method of proof is not applicable if D = G. In any case, given a sequence of domains D1 C D 2 C ... , Di C G, lim D i = G, by Theorem 4.1 we can construct a sequence of quasiconformal mappings wni ' i = 1, 2, ... , of the plane which converges uniformly to w on compact subsets of G and has the property that wni has complex dilatation "n; almost everywhere in D i . However, this sequence is not a good approximation of w if the maximal dilatations of wni are not uniformly bounded in G.

4.2. Mappings whose complex dilatation is a polynomial. The significance of Theorem 4.1 lies in the possibility of choosing the functions "n so that the approximating mappings w n are not only regular but even real-analytic. This can be achieved if we approximate the complex dilatation" by polynomials (d. III.5.3). By definition a function is real-analytic in a domain G of the finite z-plane if it can be represented by an absolutely convergent power series in z and in a neighbourhood of every point of G. A real-analytic function in G belongs to the class Coo there.

z

Lemma 4.1. Let G and G' be finite domains and w: G -+ G' a quasiconformal mapping whose complex dilatation coincides almost everywhere in G with a polynomial Pin z and z. Then w is regular and real-analytic.

209

§ 4. Approximation of a Quasiconformal Mapping

Proof: We have to prove that w has a power series expansion in a neighbourhood of an arbitrary point Zo E G, and that the Jacobian J of w does not vanish at ZOo It follows from the Uniqueness theorem that instead of w we can consider an arbitrary quasiconformal mapping defined in a neighbourhood U of zo and having complex dilatation P almost everywhere in U.

w

Such a mapping series. Let

wcan be

constructed directly by means of a power N

P(z)

E

=

a mn

(z -

zo)m (z -

(4.1)

zot

m,n=O

and write ()()

w(z)

E

=

Chk

(z -

ZO)h

(z - zo)k .

(4.2)

h,k~o

The coefficients Chk are to be so chosen that differential equation wz(Z) = P(z) wz(z)

wwill satisfy the Beltrami (4·3)

at every point z with a neighbourhood where the series (4.2) converges absolutely. This will be done by a direct calculation. In what follows we shall set Chk = 0 if h or k is negative, and a mn = 0 if a mn does not appear in (4.1). Differentiating (4.2) termwise with respect to z and and substituting in (4.3) we obtain the equations

z

N

k

Chk

= E (h

+1 -

m)

(4.4)

a mn ch+l-m,k-I-n'

m,n=O

For k = 0 these are of the form 0 = O. Hence the coefficients ChO may be chosen arbitrarily. We set ChO = 0 for h =1= 1 and C10 = 1. Starting from these initial conditions we then determine all the other coefficients Chk recursively by (4.4). To investigate the convergence of (4.2) we show first that Chk =

0

for h

>kN +1.

(4.5)

For k = 0 this follows from the initial condition. For k = 1, ChI = ahO by (4.4), and so ChI = 0 for h N. The general case can be proved by induction: Suppose that (4.5) is true for k ko and set k = ko, h = ho ko N 1 in (4.4). It then follows from 0 ::;: m, n ~ N that

>

>

<

+

ho

+1 -

m

>k

o

N

+2-

m

> (k

o -

1 - n) N

+1,

and all the coefficients cho+l-m,ko-l-n vanish. Thus (4.5) holds for k too.

=

ko

210

V. Quasiconformal Mappings with Prescribed Complex Dilatation

In order to derive an upper bound for IChkl we write A = max lamnl. We first have IChOI ~ 1 because of the initial condition, and IChl1 :::;; A since Chi = ahO' For k 1 we obtain from (4.4)

>

h + 1 IChkl :::;; A -k~

N

};

/Ch+I-m,k-l-nl .

m,n=O

~

In view of (4.5) we may assume that h < N + 2/ k :::;; N + 1, and consequently

kN

+ 1.

Then (h

+ 1)/k

N

IChkl :::;; A (N

+

1) }; /Ch+I-m,k-l-nl .

(4.6)

m,n=o

From this we deduce that N

+ 1) }; (lch+l-m,ol + /Ch+l-m,ll) < A (A + 1) (N + 1)2 < (A (N + 1)2 + 1)2. If we write (X = A (N + 1)2 + 1, we have I

Ch2/ :::;; A (N

m~O

IChkl < (Xk

(4.7)

for k = 0, 1, 2. It is easily seen by induction that this inequality holds for every k. We have therefore IChk (z - zo)" (z - zo)kl < (Xk Iz - zo/h+ k ,

<

and the series (4.2) is absolutely convergent in the disc Iz - zol 1/(X. Furthermore it follows from (4.4) that satisfies the equation (4.3) there. For the Jacobian of at the point Zo we have

w

w

J(zo)

= Iwz (zo)1 2 - Iw z(ZO)!2 = Ic I012 - [c ol l2 =

1-

laoo l2 = 1 - IP(zo)1 2 ,

>

so that ](zo) 0, since IP(zo)/ < 1. Hence w is a regular quasiconformal mapping in a neighbourhood of ZOo 4.3. Quasiconformal mappings as limits of regular mappings. Let G and G' be finite domains and w : G -+ G' a K-quasiconformal mapping

+

with complex dilatation i!. Then Ii!I :::;; (K - 1)/(K 1) = k < 1. By III.5.3, we can construct in every domain D, D C G, a sequence of polynomials P n in z and Z, n = 1, 2, ... , with the properties that JPn(z) 1 :::;; k for zED and lim P n(z) = i!(z) n~oo

almost everywhere in D.

§ 5. Application of the Hilbert Transformation

211

By Theorem 4.1, there exists in D a good approximation w n ., = 1,2, ... , of W such that the complex dilatation of w n • coincides with the polynomial P n • almost everywhere in D. It follows from Lemma 4.1 that every w n • is a regular real-analytic quasiconformal mapping of D which thus has complex dilatation Pn.(z) for every zED. Since IPn .[ < k, wn • is K-quasiconformal. We thus conclude:

')I

Every finite K-quasiconformal mapping is locally the limit of a sequence of regular real-analytic K-quasiconformal mappings.

Conversely, it follows from Theorem 11.5.3 that a function defined in G which can be uniformly approximated by a sequence of K-quasiconformal mappings in every domain D, D C G, is either.a K-quasiconformal mapping of G or a constant. We have thus obtained the following new characterization of quasiconformality, emphasizing once more the connection between regular and general quasiconformal mappings. Theorem 4.2. A non-constant finite function W in a finite domain G is a K-quasiconformal mapping of G if and only if in every domain D, D C G, there exists a good approximation of W by regular K-quasiconformal mappings.

§ !. Application of the Hilbert Transformation to Quasiconformal Mappings 5.1. Transformation of the Beltrami equation into an integral equation. We shall now examine the local behaviour of the derivatives of a finite K-quasiconformal mapping W : G ---+ G' with the aid of the Hilbert transformation introduced in IIL7. In order to be able to apply Lemma IIL7.2 we first replace W by a K-quasiconformal mapping if; of the plane, which in a prescribed bounded domain D C G depends conformally on w. Using the Existence theorem we define the mapping if; as follows: The complex dilatation" of if; is defined to be that of w in D and zero elsewhere. By the Uniqueness theorem, if; is uniquely determined up to a linear transformation of the plane and is conformal outside D. Thus by a suitable normalization we can ensure that if; has the expansion co

if;(z) = z

+ 1: an z-n n=!

in a neighbourhood of infinity.

(5.1 )

v.

212

Quasiconformal Mappings with Prescribed Complex Dilatation

In D we have

w=rpow,

(5.2)

where rp is a conformal mapping of w(D). Thus from the local behaviour ~f the derivatives of if; in D we can draw conclusions about those of w.

l,

As a consequence of the normalization (5.1) the function defined by I(z) = w(z) - z, satisfies the hypothesis of Lemma III. 7.2 : 1is absolutely continuous on lines in the plane and has LZ-integrable derivatives I z and Iz. Since I z = Wz - 1, Iz = z' we deduce from Lemma III.7.2 that (5.3 ) Wz = 1 S Wz

w +

almost everywhere in the plane. On the other hand W, being a quasiconformal mapping with complex dilatation ", satisfies the Beltrami equation Wz = z almost everywhere. If we combine this with (5.3), we conclude that Wz satisfies the relation

"w

wz="+,,Sw z almost everywhere.

of

5.2. Solution the integral equation. We are thus led to the singular integral equation (5.4) W="+,,SW, where " is a measurable function in the finite plane Q with compact support and with the property that sup 1,,(z)1 Z

=

k

<1.

E!J

By a solution of this equation we shall mean a function

W

in the class

V(Q), which satisfies (5.4) almost everywhere in Q.

The solution 01 the equation (5.4) is uniquely determined up to its values in a set 01 area zero. In fact, if WI and W z are two solutions, then the difference "P = WI - W z satisfies the equation "P = " S "P almost everywhere, and so

11"Pllz

~ k

liS "Pllz .

By III.7.4 and 6, however, 11"Pllz = liS "Pllz for every function "P E V(Q). Since k 1, it follows that 11"Pllz = 0, and consequently WI = W z almost everywhere in Q.

<

The integral equation (5.4) can be solved in the following way by iteration. We set

So"=1,

Si"=S(,,Si-l")'

i=1,2, ... ,

§ 5. Application of the Hilbert Transformation

213

and n

wn

= E " 5i ;~o

"

n = 0, 1,2, ....

,

(5.5)

Then n

5

W n -l

= E 5; " , ;=1

and consequently n = 1,2, .... (5.6) = " + " 5 W n- 1 , Using the relation IIwI12 = 115 w11 2, we prove that the sequence w n Wn

converges. First we obtain

Since this inequality holds for every i = 2, 3, ... , we deduce that (5.7) where C2 equals the area of the support of ".

> m,

From (5.5) and (5.7) it follows that, for n n

Ilwn - wm l1 2;;:; k;=m+l EllS; "1/2

~

C

n

2; ki+ 1 <

;=m+l

k m +2

C-=k' 1

Since the space P(Q) is complete (see IIL7.5), this implies the existence of a function W E P(Q) such that lim

Ilw -

W n l/2 =

o.

Since 115 (w - w,,)112 = Ilw - wn l1 2 we also have lim 115 W - 5 w,,112 = O. Hence by IIL7.5 we can find a subsequence w n ; such that lim w,,;(z) i -+ 00

=

w(z) ,

lim 5 w,,;(z)

= 5 w(z)

i-co

for almost all z. It follows from (5.6) that W is a solution of the integral equation (5.4).

5.3. LP-integrability of the solution of the integral equation. In the above iteration method of solving equation (5.4) we made use of the fact that the Hilbert transformation leaves the L2-norm invariant. It is however only necessary that the transformation does not increase the norm too much. In fact, there is a more general result on the norms of the Hilbert transformation, which we state here without proof (Calder6n-Zygmund [1J; a simplified proof is due to Vekua [1J, pp. 55 -61, and also given by'Ahlfors [4J, pp. 106-112):

V. Quasiconformal Mappings with Prescribed Complex Dilatation

214

The Hilbert transformation is bounded in every space LP(Q), p other words, the number

> 1.

In

IISllp = sup 115 flip' Iltllp=1

the LP-norm of the Hilbert transformation, is finite for every p

> 1.

By Riesz's convexity theorem (see for example Dunford-Schwartz [1J, p. 525, Zygmund [1J, p. 95, or Ahlfors [4J, pp.113-115), log (1lSplll is a convex function of 11P which increases with p for p 2 and tends to infinity as p --+ 00. From the convexity it follows that IISpl1 is a continuous function of p. By IIL7.6 we have 115112 = 1. Furthermore Calderon and Zygmund [1J have shown that

>

as p ->-

IISllplP

<M <

(5.8)

00

00.

From 5.2 we obtain the following result (Bojarski [1J): Lemma 5.1. The solution of the integral equation (5.4) belongs to the class U(Q) for every p for which IISlip 11k.

<

Proof: Using the notation of 5.2 we have

115; "lip = 115 (" Si-l ")llp;;;; kllSll p 115;-1 "lip'

i = 2,

3, ... ,

and consequently

115; ~llp ;;;; (kIISllp)i C2!P I t follows that for n

.

>m n

IIwn - wmllp :s;: k C2 !P E (kIISllp)i . ;=m+l

<

For kllSll p 1, w n is thus a Cauchy sequence in LP. Since LP is complete (see IIL7.5), the solution of (5.4), being the U-limit of wn> belongs to LP. 5.4. Inte~rability of the derivatives of a quasiconformal mappin~. We now return to the given finite-valued K-quasiconformal mapping w of G.and the associated K-quasiconformal mapping of the plane. In 5.1 we showed that W, satisfies the integral equation (5.4) with sup 1,,(z)1 = k < (K - 1)/(K 1). Thus it follows from Lemma 5.1 that Wz E U for every p for which IISllp (K 1)/(K - 1). In view of (5.3) W. belongs locally to the same classes LP. From (5.2) we see finally that the derivatives of w also belong to these LP-classes locally in G.

w

+

<

+

§ S. Application of the Hilbert Transformation

215

Let P(K) be the supremum of all numbers p for which all K-quasiconformal mappings have LP-derivatives. By Theorem IV.1.2 we have P(K) ~ 2. From the above we obtain for P(K) the lower estimation

II 5 II P(K) Since 115112

>K+

1

K _ 1 .

=

= 1, we thus have P(K)

> 2 for every K.

On the other hand, the K-quasiconformal mapping w defined by w(z)

=

zlzll/K-l

(5.9)

shows that P(K) is at most 2 Kj(K - 1). This can be seen by considering the derivatives of w in the neighbourhood of z = O. Summarizing these results we obtain the following generalization of Theorem IV.1.2: Theorem 5.1. A K-quasiconformal mapping has LP-derivatives for every P P(K), where P(K) satisfies the inequalities

<

K+1

11 5 1I p(K) ~ K

_ 1 '

2

< P(K)

2K

~ K _

1 .

(5.10)

>

Although the value of 11511p is not known for P 2 and the lower bound for P(K) in (5.10) is therefore implicit, (5.10) does give rise to some further conclusions on P(K). The double inequality on the right in (5.10) shows that lim P(K) = 2. K-co

This result shows that in Theorem IV.1.2, which refers to an arbitrary quasiconformal mapping, the exponent P = 2 cannot be replaced by any larger number. From the first inequality (5.10) and the estimate (5.8) we deduce further that lim P(K) = 00 . K-l

In particular, P(K) ->- 00 in such a way that (K - 1) P(K) remains between two finite positive bounds. An interesting open problem is to determine the exact value of P(K). One might conjecture that the upper bound in (5.10) is the solution to this problem, i.e. that P(K) = 2 Kj(K - 1).

5.5. Change of area under quasiconformal mappings. By using Theorem 5.1 we can obtain a sharpened version of Theorem IV.1.3 on absolute continuity with respect to the area measure (Bojarski [2J).

216

V. Quasiconformal Mappings with Prescribed Complex Dilatation

Theorem 5.2. Let W be a K-quasiconformal mapping of a domain G and F a compact subset of G - {oo} - {w- 1 (oo)}. Then for every (j

<

1 -

2/P(K) there exists a finite number C such that m(w(E)) ~ C(m(E))O

(5.11)

for every measurable set E C F. Proof: By IV.1.4 we have m(w(E))

= JJJ da. E

From wz' Wz E LP it follows that J = IWzl2 - /w z/2 E LP/2. Applying Holder's inequality for an arbitrary p, 2 p P(K), we deduce that

< <

m(w(E))

where

(j =

<

(kJ JP/2 datP(m(EW '

1 - 2/P; (5.11) follows from this.

The least upper bound (j(K) of values (j for which Theorem 5.2 holds is unknown. Example (5.9) shows that (j(K) ::;: 1/K.49 5.6. LP-convergence of the derivatives. If a sequence of K-quasiconformal mappings w n of a domain G of the finite plane converges uniformly in every compact subset of G to a finite-valued function w, then the derivatives of w n converge weakly to the corresponding derivatives of w, by Lemma IV.5.1. We can now extend this result by showing that in the case of good approximation the convergence takes place in the following stronger sense:

Theorem 5.3. Let w n be a sequence of finite-valued K-quasiconformal mappings of a domain G of the finite plane. If W n is a good approximation

>

of the quasiconformal mapping w in G, then for some p 2 the derivatives (w n). and (wnh converge in the LP metric of any compact subset F of G to W z and w z' respectively. Proof: We may assume, without loss of generality, that F is a closed

disc. Let D be an open disc such that FeD, D c G. As in 5.1 we find mappings n and W of the plane which have the same complex dilatations as w n and w in D, and are conformal outside D. At infinity they have an expansion of the form (5.1).

w

By Lemma III.7.2, we have for the function

'ljJn

= W-

wn (5.12)

49

On the other hand, r5(K) ~ 1/K2/"', where

by 5.4 we have 0

< IX ~ 2.

IX

=

lim inf (K K-t-

See also Gehring-Reich [1].

1) P(K) (Lehto [2]);

217

§ 5. Application of the Hilbert Transformation

almost everywhere in D. Denote by x n and x the complex dilatations of n and W, respectively. Then

w

x n(1fn)z = x

ivz -

xn(wn)z

+ x n Wz -

X

Wz = (1fn)'

+ (x n -

x) Wz

almost everywhere. Thus up to a null set (1fnlz = x n S(1fn)z

+ (x -

x n) Wz

and consequently

II (1fnlzl Ip ~ k IISll p II (1fnlzl Ip + II(x - x n ) wzl[p with k

=

(K -

1)j(K

(5.13)

+ 1).

<

>

We now fix p so that simultaneously p 2 ~d k /ISlip 1; this is possible by 5.3. Since Ix - xnl vanishes in - D and is less than 2 in D, we conclude from (5.13) that

11(1fnlzllp:;;:

11(" -

"n)

1- k

wzll p < 1 - 2IISll (JJ Iw.l - p dtY )1 1P .

llSll p

k

pD

(5.14)

The right hand side is bounded, since Wz belongs locally to LP by Theorem 5.1. As n --+ 00, x n tends to x almost everywhere and by Lebesgue's convergence theorem (III.1.5) we have lim II(x - x n ) wzllp = o. It then follows from (5.14) that lim 11(1fn)'[[p = lim [Iw. - (wnlzl[p =

n-(X)

n-oo

By (5.12) we also have lim Ilwz

-

(wnLll p =

o.

(5.15)

o.

The derivative (1fn)z vanishes outside D, and owing to the normalization (5.1) we obtain from Lemma III.7.1 by proceeding to the limit

r

for every finite z. Hence, by Holder's inequality,

<: (IJlc ~(Jzlq

l1fn(z)/

q 11(1fnlzllp,

<

where q = pj(P - 1) 2. In view of (5.15) we deduce that 1f.. = W n tends uniformly to zero in the plane as n --+ 00. It follows that W;1 tends to w- 1 uniformly in w(D).

-w

In D we have w n = In wn' w = low, where I.. and I are conformal mappings of wn(D) and w(D). Since W.. tends to Wuniformly in D we see that if'(F) C w,,(D) from some n onwards. For these values of n the inverses W;1 map a neighbourhood U of w(F), U C w(D), into D. Since by hypothesis wn tends to w uniformly in D it follows that In --+ I uniformly in U. The derivatives I~ therefore also converge uniformly 0

218

V. Quasiconformal Mappings with Prescribed Complex Dilatation

in a neighbourhood V ( U of w(F), and since wn(F) lies in V from some n onwards, we have (5.16) uniformly in F. Next, from Wz(z) - (wn)z(z)

=

t~(wn(z)) (wz(z) - (wn)z(z))

+ (I'(w(z)) -

t~(wn(z))) wz(z)

it follows that P

P It~(wn(z))1 Ilwz - (wnlzll p ( FIf Jwz - (wnlzl da)I/ < max ZEF

+ (£1 I(I' (w(z)) -

t~(w n(z)))

wz(z)IP da

fP.

From (5.16) and (5.15) we see that the first term on the right tends to zero as n ~ 00. By Lebesgue's convergence theorem the same is true for the second term.

It can be shown in the same way that (wn)z tends to of F, and the theorem is proved.

W z in

the LP metric

As a consequence of the theorem we conclude that for a good approximation w n ~ win G there exists a subsequence w nv ' for' which the derivatives tend to the corresponding derivatives of the limit mapping W almost everywhere in G (see III.7.5).

5.7. Representation of a mappin~ with prescribed complex dilatation. Let x be a m~asurable function in the plane, having bounded support and satisfying sUPz Ix(z) I < 1. By the Existence theorem there is a quasiconformal mapping W of the plane with complex dilatation equal to x almost everywhere. Since W is conformal in a neighbourhood of infinity it can be normalized as in (5.1). Application of Lemma III.7.1 to the function 'IjJ, 'IjJ(z) gives'IjJ = T 'ljJz.50 In other words we have W(z)

= z + Tw(z)

= w(z) - z, now (5.17)

for every z, where w = W zcan be constructed as a solution of the integral equation (5.4) as we showed in 5.1 and 5.2. The formula (5.17) is then a representation of a quasiconformal mapping with complex dilatation x almost everywhere. On the other hand, starting with the integral equation (5.4), we can construct the right hand side of (5.17) without using the Existence 50

Here T

=

T D' where D contains the support of x (d. the notation in III. 7.2).

§ 6. Conformality at a Point

219

theorem. It is then possible to show directly that the function defined by (5.17) is a quasiconformal homeomorphism of the plane whose complex dilatation is equal to x{z) for almost all z (see Bojarski [1], or, for a detailed proof, Vekua [1J, pp. 68-85; see also Ahlfors-Bers [1J). In this way a new proof of the Existence theorem can be obtained (d. 1.4).

§ 6. Conformality at a Point 6.1. Presentation of the problem. In 4.2 we have already discussed the problem of deducing the regularity of a quasiconformal mapping from the properties of its complex dilatation. This section and the next are devoted to the investigation of this problem. To start with we consider the following situation:

<

Let x be a measurable function with sUPz Ix{z)1 1 in a domain G. By the Existence theorem there is a quasiconformal mapping w of G with complex dilatation equal to x almost everywhere. Let Zo be a fixed point of G. We seek sufficient conditions on the function x such that w will be regular at ZOo It follows from the Uniqueness theorem that the regularity of w at Zo depends on the function x and indeed only on its values in an arbitrarily small neighbourhood U of zo: two quasiconformal mappings whose complex dilatations coincide almost everywhere in U are simultaneously regular or not at zo0 Our problem is therefore meaningful.

Without loss of generality we may assume that Zo is the origin. Furthermore, by means of an affine mapping we can normalize our problem in such a way that w, if it is regular at the origin, has an expansion of the form

w{z)

=

wz{o) z

+ o{z) ,

wz{O)

=1= 0 .

(6.1)

If (6.1) holds, then w{z)jz tends to the finite, non-zero limit wz{O) as z -->- O. In this case the mapping w is said to be conformal at the origin.

In general, a homeomorphism w of a neighbourhood of a point Zo =1= 00 is called conformal at Zo if (w{z) - w{zo))j{z - zo) tends to a finite nonzero limit as z -->- zoo In this section we shall study conditions on x under which the mapping w is conformal at the origin. 51 The somewhat related problem on the effect of the dilatation quotient on the boundary behaviour of quasiconformal mappings has been studied by Carleson [1J.

51

220

V. Quasiconformal Mappings with Prescribed Complex Dilatation

6.2. Examples. By Lemma 4.1 w is conformal at z = 0 when " is a polynomial with "(0) = O. It is not difficult to see that this condition can be replaced by essentially weaker ones. For example, it would suffice that" be differentiable and "(0) = O. To obtain a still weaker condition we consider the following example. Let w,

w(z) = !(!zl)

eiargz

= t{J/z z) V~

,

(6.2)

be a mapping of the unit disc onto itself, where I is a non-negative function which increases strictly with /zl, vanishes for Izi = 0 and takes the value 1 for /zl = 1. If! has a positive continuous derivative in the open interval (0,1), then w is a regular Cl-mapping of the punctured unit disc 0 /z/ 1. For the complex dilatation x of w we obtain

<

<

.

['(/zl) Izi - 1(lzl) + 1(lzl) .

Z2

,,(z) = IZl2 f'{lzl) Izi

In order to obtain a simpler expression we consider the dilatation quotient D = (1 1"1)/(1 - Ixl). Assuming that !(lzl)/lzl does not decrease with /zl we have

+

D(z) = Izi

1'(lzl) 1(1zl) .

It follows that

-f

1

D(r~-~dr

w(z) 1(1zl) Izi --=--=e

Izi

z

(6·3)

The mapping w is therefore conformal at z = 0 if and only if the integral 1

fD

--,---(r-)r-_1 dr

(6.4)

o

has a finite value. The integral (6.4) can of course be infinite even if D(z) - 1, and hence also ,,(z), tends to zero as z ->- O. As an example we have the mapping w defined by

w(z) =

z -1-----::l-og---,I---,zl '

with dilatation quotient

D(z)

=

1

+1-

1

Iog II z

§ 6. Conformality at a Point

221

This mapping is not conformal at z

=

0 since

. w(z) 1I m - = O . z-o z

Conversely it is easy to see that there exist quasiconformal mappings which are conformal at the origin even though their maximal dilatations differ from 1 by an arbitrary amount in every neighbourhood of this point. In fact we can obviously find a function tJ>, non-negative and continuous for 0 < r < 1, such that lim sup tJ>(r) is arbitrarily large while the integral r- 0 1

Ic]);r)

dr

o

is finite. If we then set D(r) - 1

= tJ>(r) in (6.3), we obtain a mapping

w with the required properties.

For a general quasiconformal mapping w the dilatation quotient D is not a function of Izl alone, and we have to replace the integral (6.4) by

II

R 2n

o

i

D(r e ;) -

1

dr dcp =

II D(~~l;-

1

da,

(6.5)

U

0

U = {zl Izi < R}. We shall prove that the convergence of this integral is sufficient to ensure the conformality of w at the origin. Since D(z) - 1 = 2 Ix(z)I/(1 - Ix(z)l) the convergence of the integral (6.5) is equivalent to

6.3. Module estimations in terms of the means of D. In order to draw conclusions from the convergence of the integral (6.5), we first derive some estimations for the dilatations of quadrilaterals and ring domains using certain mean values of D. Let w : G --+ G' be a quasiconformal mapping with dilatation quotient D. \Ve consider a family t' of arcs or curves C e G and the family t" of their images w(C). By applying formula (3.8) in IV.3.3 to the inverse of w we obtain (6.6) M(t") ~ JJ D e2 da G

for every e admissible for t'. As a first application of (6.6) we consider a horizontal rectangle

Q = {x+iYla<x
222

V. Quasiconformal Mappings with Prescribed Complex Dilatation

for Z E Q, e(z) = 0 for z Et Q, then e is admissible with respect to the family of arcs C connecting the a-sides of Q within Q. In view of IV.3.4 we deduce therefore from (6.6) that

M(w(Q)) <

~

(d

ff

C)2

D da.

Q

=

Since M(Q)

(b - a)/(d - c) we can write this estimate in the form

If D da ,

M(w(Q)) 1 M(Q) :S: m(Q)

(6.7)

Q

where m(Q) denotes the area of Q. If we exchange the roles of the a- and b-sides, we obtain the same upper bound for M(Q)/M(w(Q)). In order to find a corresponding inequality for an annulus B =

< <

{z I r Izi R}, BeG, we set e(z) = 1/(2 nlzl) for ZE B, e(z) ='0 for z EI B. Then e is admissible with respect to the family of curves C C B which separate the boundary components of B (d. I.6.3)~ In view of IV.3.4 it follows therefore from (6.6) that

M(w(B)) <

1 2n

IfD(Z) W da. B

Since

M(B) =

1 2n

If W da

B

we have

M(w(B)) - M(B) ~

1 2n

IfD(Z)IZJ2-

1

(6.8)

da.

B

<

Another estimate will be obtained in the case M(B) 00 if we set e(z) = 1/(lzllog (R/r)) = 1/(lzl M(B)) for z E B. Then e is admissible for the family of arcs which join the boundary components of B. By (6.6) and IV.3.4 we have M(w(B))

:S:

1 2n (M(B))2

IfD(Z) W da , B

and so M(B) M(w(B))

< =

1 2 n M(B)

IfD(ZL da Izl2

=

1

+ 2n~(B)

B

z !z)

1-: 1 da.

B

From this and (6.8) we obtain

IM(w(B)) - M(B)I ~

If0.

1 2n

ffD(Z) ~- da. 1

B

(6.9)

§ 6. Conformality at a Point

223

6.4. Transition to the logarithmic plane. The proof that the convergence of the integral (6.5) implies the conformality of the quasiconformal mapping w at the origin will be divided into two parts: We first deduce that Iw(z}/z/ tends to a non-zero finite limit as z --+ 0, and secondly, that arg (w(z}/z) also converges. In order to simplify certain calculations it is advisable to consider log w as a function of log z. In performing this transformation of coordinates we shall simultaneously determine the branches of arg z and arg w appearing in the second limit problem. Let w be a K-quasiconformal mapping of the plane with w(O} = 0, w(oo} = 00. We map the slit plane Go = {z 10 < arg z < 2n} conformally onto the strip 5 = {C = ~ + i'YJ I < 'YJ < 2 n} l]sing a branch of the logarithm. Similarly the image w(Go} of Go is mapped by the logarithm onto a domain 5'. The composite mapping I: 5 --+ 5', I(C} = log w(e C}, is then K-quasiconformal and has a continuous extension to all finite boundary points of 5.

°

°

+

The exponential function maps a segment {C = ~o i'YJ I < 'YJ < 2 n} onto the circle Izl = eeo. Since w is sense-preserving, we see that I(C} = log Iw(eC}1 + i arg w(e C} increases by 2 n i as 'YJ goes from to 2 n. Hence we can extend 1 = u i v throughout the finite plane as a continuous function by the formula

°

+

I(C

+ 2 n n i} =

I(C)

+2nn i ,

n

=

0,

±

1, ... ,

(6.10)

so that v(C) coincides with a branch of arg w(e C). The function 1 is then a homeomorphism of the plane and since it is locally the composition of a K-quasiconformal mapping and two conformal mappings, 1 is K-quasiconformal. At the points log z 2 n n i, n = 0, ± 1, ... , the dilatation quotient D of 1has the same value as the dilatation quotient of w at the point z, provided that w is regular at z.

+

By this transition to the logarithmic plane we have determined a branch of arg z and arg w as a function of C. In what follows the notation 'YJ = arg z, v = arg w will always refer to these branches. We note that arg (w/z) is then a single-valued function of z for z =1= 0, 00. It is important to remark on the uniform Holder continuity of the family c'F of all K -quasiconformal mappings of the plane satisfying (6.10). i'YJ I This follows from the fact that the image of the strip T = {~ 'YJ 4 n} under every 1 E c'F contains no closed vertical segment of length 4 n. By Theorems 11.4.1 and 11.4.3, c'F is therefore equicontinuous in T and Holder continuous at every point Co of T. Since 1 and 1 - I(Co} are simultaneously members of c'F, the family c'F is Holder continuous with respect to the euclidean metric too. This holds not only

< <

+

°

224

V. Quasiconformal Mappings with Prescribed Complex Dilatation

in T but in the whole finite plane, since every point is equivalent modulo 2 n i to a point in T. Thus for every finite point Co there exist two positive numbers C and d such that (6.11) for f E c'F and Ie - Col ~ d. Since the family c'F is invariant under a translation of the C-plane, (6.11) holds for fixed C and d depending only on K and not on the choice of Co'

6.5. Convergence of the absolute value. After these preparations we shall now prove that w is conformal at z = 0 if the integral (6.5) is finite, by showing first the existence of the non-zero limit of Iw(z)/z/. This result is due to Teichmuller [1J and Wittich [1J with the slight restriction that the mapping is regular for z =1= O. Lemma 6.1. Let w be a K-quasiconformal mapping of the plane with w(O) = 0, w( (0) = 00, and

l(r) =

1 271:

ffD(Z) l2J

Z

1

da

<

for r

00

<

00 .

(6.12)

,

(6.13)

Izi < r

Then there exists a constant c, min Iw(z)1 e- 1 (1) Izi =

~

c < max Iw(z)[ e1 (1) Izi =

1

1

such that

I w;Z) 1- cl ~ ce(lzl) ,

(6.14)

where e(Jzl) - 0 as z _ 0 and the function e depends only on K and 1. Proof: We write min Iw(z)[

=

eh(o) ,

max [w(z)[ = eh(o) +'1'(0)

\ (6.15)

,

Izi ~ eO

Izi =eO

~ - - 00. S Consider an annulus B = {z [ e < [zl < et } and its image w(B). The module of B is t - s and for the module of w(B) formula (6.9) yields the estimations t - s - 1(et ) < M(w(B)) ~ t - s + l(e t ) . (6.16)

and show first that

1f!(~)

- 0 as

<

<

The ring domain w(B) lies in the annulus B' = {w I eh(s) [wi eh(t)+'I'(t)} and, if h(t) h(s) 1f!(s) , contains the annulus B" = {w I eh(s) + 'I'(S) [wi eh(t)}. From the monotonicity of the module it thus follows that h(t) - h(s) -1f!(s) ~ M(w(B)) < h(t) - h(s) 1f!(t) .

<

<

>

+

+

225

§ 6. Conformality at a Point

Consequently, by (6.16),

t - s - I(e t )

-

'lJ!(t) < h(t) - h(s) :::;; t - s

+ I(et ) + 'lJ!(s).

(6.17)

As in 6.4 we intrbduce the variables C = ; + i'YJ = log z, 1= u + i v = log w. The annulus B, slit along the positive real axis, corresponds to the horizontal rectangle Q = {; + i'YJ I s t, 0 'YJ 2 n} of the C-plane and to the Jordan domain I(Q) of the I-plane. Since w(B) is contained in the annulus B' the area of I(Q) satisfies the inequality

<; <

m(t(Q») :::;; 2 n (h(t) - h(s)

+ 'lJ!(t»)

< <

(6.18)

.

The integral in (6.12) transforms to ;

I(e~) = 2~

2'"

JJ

(D(C) -

-00

1) d; d'YJ,

0

where D is the dilatation quotient of

I.

The function I maps the vertical segment C; = {; + i'YJ I 0 :::;; 'YJ < 2 n} onto a Jordan arc C~ with the endpoints 1(;) and 1(; + 2 n i) = 1(;) + 2 n i (d. (6.10»). Since C~ connects the vertical lines u = h(;) and u = h(;) + 'lJ!(;), its length has the lower bound 2 Vn2 + ('lJ!(;))2 .

t

s

Fig. 12

At regular points max~ 12J(C)1 2 = D(C) !(C), where! is the Jacobian of I. If I is absolutely continuous on C~ and regular at almost all points of C;, we therefore have 2",

2Vn + ('lJ!(;»)2:::;; JVD (; + i'YJ)! (; + i'YJ) d'YJ. 2

(6.19)

o

By Lemma IV.1.1 and Theorem IV.1.4 this inequality holds for almost all;. Integrating with respect to ; and using Schwarz's inequality we thus obtain (

jVn + 2

2 s

('lJ!(;) )2 d;)2

~ II D da If ! Q

Q

da .

(6.20)

226

V. Quasiconformal Mappings with Prescribed Complex Dilatation

Here

JJ D dll =

m(Q)

+ JJ (D -

Q

1) dll

<

2 n (t -

S

+ I(e

l

)) ,

Q

and by (6.18) and (6.17)

JJ J dll =

m(f(Q)) < 2 n (t -

+

S

+ 1jJ(s) + 1jJ(t)) .

I(e t )

Q

In order to estimate the left-hand integral in (6.20) we refer to the inequality (9.1) in 11.9.2, which yields the upper bound n K for 1jJ(~). A simple calculation then shows that

Vn2 + (1jJ(~))2 ~ n + ~~~; . Combining the above estimations with (6.20) we obtain I

2n(t-s)

+2~KJ(1jJ(~))2d~ s

< =

(

)(

2n t -

S

< 2 n (t.Letting

1

+

t-s

s) +n [2 I(e

S --+ -

00

+ 'tjJ(s) + 'tjJ(t)

2 f(e l )

l

)

+

+1jJ(s) +1jJ(t)

f(e l ) (f(e l ) + 271: K))1/2 ( t-)s2

+ I(e

l

)

(I(e l )

+ 2nK)/(t -

s)].

we conclude that

I

J (1jJ(~))2 d~ ~ 2 n

2

K

(2 I(e l )

-00

+ 1jJ(t) + lim sup 1jJ(s)) s--oo

From this inequality we derive an upper bound for 1jJ, making use of the fact that by (6.11) the function 1jJ fulfills the Holder condition 11jJ(~

+ Ll~)

- 1jJ(~)1 < C1Ll~ll/K

for ILl~1 ~ d, where C and d depend only on K. If M = sup 1jJ(~) for ~ < t, it follows that 1jJ(~) ~ M /2 on an interval of length min (d, (M/(2 C))K). Hence t

J

(1jJ(~))2 d~ > ~2 min (d, G~ () =

-()()

~

M2+K -4-

min

(dK)K' (71:

M:+K min (:K'

1)

(2 C)K

(2

~)K)

.

In other words, I

J (1jJ(~))2 d~ ~

-()()

C' sup (1jJ(~))K+2, <~t

(6.22)

§ 6. Conformality at a Point

227

where C depends only on K. Since the left-hand integral converges by (6.21), we see that 1jJ(e) -+ 0 as -+ - 00.

e

Next we show that 1jJ(e) has a majorant which is independent of the mapping f and tends to zero as -+ - 00. Setting t = 0 in (6.21) we first see that for every x 0

e

<

min (1jJ(ej)2

X;;;;'~;;;;'X/2

e" < -II ' oX

+

where C" = 8 n K (1(1) n K). After this we choose t in (6.21) to be the point on the interval x < e.:s; xJ2 where 1jJ takes its minimum. Then (6.21) yields 2

t

J (1jJ (ej) 2 de <

2n 2K

(2 1(et ) + (C" Jlxl) 1/2) :s; 2n2K (2 1(e X/2)+ (C" Jlxl)!/2) ,

-00

and we obtain from (6.22) the desired estimation C(1jJ(x))K+2 < 2 n 2 K (2 1(exf2 ) (C" Ilxl)!/2) .

+

(6.23)

e

In order to complete the proof we first notice that h(e) tends to a limit as -+ - 00. This now follows from the double inequality (6.17), written in the form

e

- 1(e~) - 1jJ(t) :s; (h(t) - t) - (h(s) - s) ~ 1(et ) since 1jJ(t) and

1(e t )

tend to 0 as t

-+ -

lim (h(e) - e)

(6.24)

Denoting

00.

=

+ 1jJ(s) ,

log c ,

~~-oo

and using (6.23) we see from c has the property (6.14). Finally, (6.24) yields for t = 0, s

eh(~)-~:s;

-+ -

\w(z)lzl <

eh(~)-H'I'(~)

that

00,

h(O) - 1(1) :s; log c < h(O)

+ 1jJ(0) + 1(1) .

This agrees with (6.13), and the lemma is proved.

6.6. Convergence of the argument. The lemma which we have to prove is as follows (d. Belinskij [1J): Lemma 6.2. Under the hypothesis of Lemma 6.1, arg w(z) - arg z tends to a limit (X as z -+ o. The rate of convergence depends on K and 1 but not otherwise on the mapping w.

Proof: We go over to the logarithmic plane as in 6.4. Hypothesis (6.12) then becomes x 2"

1(e

X )

=

1 2n

JJ

-00

(D(C) -

0

1) de drJ

<

00

(6.25)

228

V. Quasiconformal Mappings with Prescribed Complex Dilatation

for x < 00, where D is the dilatation quotient of the mapping u + i v. We have to prove that v(~ + i 'fJ) - 'fJ tends to a limit ex as ~ --+ - 00 and in such a way that Iv(~ + i 'fJ) - 'fJ - exl is bounded by a number c(~) depending only on K, I and ~ and tending to zero as

!=

~ --+ -

00.

By Lemma 6.1 we know that

+

+

for every pair of points ~ i 'fJ, f i 'fJ' with ~, ~' < x, where C1 is an increasing function of x depending only on K and I and tending to zero as x --+ - 00. We shall consider only values of x so small that both C1(X) and 2:rt I(e X ) are less than min (1, d2), where d is the constant mentioned in connection with the distortion formula (6.11). We fix two values 'fJ1 and 'fJ2' write

and show that

b(~)

tends to fJ as

~ --+ -

00.

<

First we suppose that

'fJ2 'fJ1' Consider a horizontal rectangle Qx = {~ i'fJ I x - C2(X) ~ x, X 'fJ2 'fJ 'fJ1}' where C2(X) = max eVC1(X), 7t I(e ) ) . The module of Qx is C2(X)/fJ. to estimate the module of the image Q: = !(Qx)' we note that by (6.26) the b-sides of are separated by a vertical strip of width C2(X) - C1(X). If (! denotes the function having the constant value 1/h(x) - C1(X)) in and-vanishing elsewhere, then lQ(C) ~ 1 for every rectifiable arc CeQ: connecting the b-sides of Q:. Hence IV.3.4 gives

V2

< <

+

< <

5:

Q:

Q: n 5:

(6.27)

Q:

From (6.11) it follows that each a-side of lies in a horizontal strip of width C(C2(X) )l/K (Fig. 13). The maximal height of the set n is therefore not greater than Ib(x)1 + 2 C(C2(X))1/K, and its area is at most (C2(X) - C1(X)) (lb(x)1 + 2 C(C2(X))1/K). In view of (6.27) it follows that

Q: 5:

(6.28) On the other hand, for fJ < 4n we conclude from (6.7) and (6.10) that

M(Q:) :S M(Qx)

+ 4:~~~x)

I(eX)

= E2~)

X

+ 4n;(8 )

229

§ 6. Conformality at a Point

Consequently, Ib(x)1

+ 2 C(B (X))l/K > 2

-

~2(%) -~ ::2: (3 _ C2(%)

4:'1: l(e X )

p+-p2 -

-

C1(x)P _ 4:'1: l(eX) . C2(%) C2(x)

Since B2 (X) = max (VB1(X) , V2nI(e X )) and (3;;;'4n, we further obtain Ib(x)1 ::2:

fJ -

2 C(B2 (X) )l/K - 4 n VB 1(X) - 2 V2 n I(e X )

= fJ - B3(X) , (6.29)

~ _}",xn'/K 82 (X)- 81 (x)

Fig. 13

where B3(X) depends only on K and I and tends to zero monotonically as x ~ - 00. Henceforth we shall assume that x is so small that

B3(X) < 2 n. Inequality (6.29) holds for all pairs 1]1 and 1]2 such that 1]1 - 1]2 = (3 lies between 0 and 4 n. If x is fixed, then b(x) is a continuous function of 1]1 and 1]2 and therefore by (6.29) has the same sign for all pairs 1]1,1]2 such that B3(X) < 1]1 - 1]2 < 4 n. However, by (6.10), b(x) = 2 n when (3 = 2 n, and so (6.29) becomes b(x) > (3 - B3(X) .

(6·30)

Furthermore it follows from (6.10) that (6.30) holds not only for B3(X) (3 < 4 n, but for every (3.

<

To obtain an upper bound for b(x) we need only interchange 'f}1 and 'f}2' Then b(x) and (3 become - b(x) and - fJ and we obtain finally

230

V. Quasiconformal Mappings with Prescribed Complex Dilatation

In order to complete the proof we fix two numbers x and and consider the expression

Xl' Xl

< X,

as a function of'YJ and 'YJI' First it follows from (6.31) that (6·33) If fJ vanishes for some pair 'YJ, 'YJ1> then IfJ! < 283 (X). If not, then fJ assumes only positive or only negative values. We suppose first that fJ o.

>

(x,x+zn) I

I I I I 1 1

I 1

I

I I

I

,. X-Xt+£t

(x)

·1

Fig. 14

Consider the parallelogram G = g + i 'YJ I Xl ~ X, ~ 'YJ ~ + 2n} and its image G' = f(G) (Fig. 14). The area of G' can be estimated from (6.10) and (6.26) with the help of Fubini's theorem; this gives

< <

< <

(6·34) To obtain an estimate for fJ we cover the parallelogram G with parallel segments L" = {~ + i'YJ I Xl ~ X, 'YJ = ~ + y}, 0 y 2n. By Lemma IV.1.1 and Theorem IV.1.4, f is absolutely continuous on L y and regular at almost all points of L y , up to a null set of y-values. For y not belonging to this null set, let A(y) denote the length of the image arc L~ = f(L y ). Then (d. (6.19))

< <

TA(y) dy o

<

V2 T( JVD(C) ](C) d~) dy. 0

Ly

By Fubini's theorem, 2"

f A(y) dy < {i JJVD(C) ](C) da,

o

< <

G

231

§ 6. Conformality at a F'oint

and it follows from Schwarz's inequality and (6.10) that

Here

f f ](C) da =

ff da =

m(G') ,

G

m(G)

=

Xl) .

2n (x -

G

Hence, by (6.34), we have 2"

f A(y) dy S 2 n (2 (x -

o =

2

V2 n (x -

+ 8 1 (X) ) (x Xl) + 84(X) ,

Xl

Xl

+ 1 (eX)) )1/2 (6·35)

where 84(X) --+ 0 as X --+ - 00. To find a lower bound for A(y) we recall that A(y) is the length of the arc which joins the points l(x1 i (Xl y)) and I(x i (x y)). Hence, by (6.26) and (6.32), we have for X - Xl 81(X),

L;

+

+

>

A(y) ~ ((X - Xl - 81(X))2 + (X - Xl 1/info 2 V2 (X- Xl - 81(X)) + V2 .

For X - Xl

< 81(X) we have A(y)

+

+

+ inf 15)2)1/2

> inf 15 •

Inserting these estimates into (6.35) we see that •

mf!5 < 281(X)

+

£4(X)

1~

y2 n

in both cases. Thus in view of (6'}3) we have

15 < 281(X)

+ 28 (X) + 3

£4(X)

1~

y2 n

= 85(X).

(6.3 6)

>

O. If We have proved this relation under the assumption that 15 < 0 we replace the parallelogram G by {~+ i'YJ 1 Xl < ~ < X, - ~ < 'YJ < - ~ + 2 n} and use the above argument.

15

We then obtain the upper bound (6.}6) for -15. In the case when 15 assumes both positive and negative values, we have already stated that 115/ < 283(X). Hence in all cases we have the inequality 1151

for Xl

~

X.

= 1(v(x

+ i 'YJ)

- 'YJ) - (v(x 1

+ i 'YJ1)

- 'YJ1)1

< 85(X)

232

V. Quasiconformal Mappings with Prescribed Complex Dilatation

It follows that v(x such a way that

+ i 'fJ)

- 'fJ tends to a limit

(X

as x

->- -

(lO,

and in

for every finite 'fJ. Lemma 6.2 is now proved.

6.7. Summary of the results. The desired result on conformality at a point follows easily from Lemmas 6.1 and 6.2. In fact, for any two complex numbers Zl = r1 ei'Pl, Z2 = r2 ei'P, the inequality IZ1 - z21 < Ir1 - r 2 1 + r2 lif1 - if21 holds and it follows that w(z)jz - c eicr. tends to zero if Iw(z)ljlzl and arg (w(z)jz) tend to c =1= 0,00 and (x, respectively, as z ->- O. By definition w is then conformal at z = 0, and from the expansion w(z) = c eicr. z + o(z) we see that w.(O) = c eicr. and wz(O) = O. Theorem 6.1. Let w be a K-quasiconformal mapping of the finite plane onto itself with w(O) = 0 and

I(r)

=

~ 2n

f(1?(Z)[z1 - 1 dO" < 00

for

2

v

r

< 00 .

Izl < r

Then w is conformal at z W(Z)

II

- wz(O)

Z

=

0 and

I < IWz(O)1 e(lzl) ,

I

lim e(lzl) = 0 , Izl~O

where the function e depends only on K and I and not otherwise on the mapping w: The derivative wz(O) satisfies the inequalities min Iw(z)1 e-I(I) :s;; Iwz(O)1

~

max Iw(z)1 eI(I) . Izi =

Izi ~ 1

(6·37)

1

Remarks. From the proofs of Lemmas 1 and 2 we see that our result can be expressed in a slightly more general form: If I (r) :s;; ,1 (r), where ,1 (r) ->- 0 as r ->- 0, then the conclusions of the theorem hold for a function e depending only on K and ,1. This monotonic dependence of e on I will be needed in § 7. Secondly we remark that the inequality on the right hand side of (6.37) can be sharpened if max Iw(z)1 is much larger than min Iw(z)1 on Izi = 1. The restriction of w to the unit disc D can in fact be written in the form w = f W, where is a quasiconformal mapping of D onto itself with w(O) = 0, and f maps the unit disc conformally onto w(D). By Koebe's one-quarter theorem (d. 11.1.3) we then have 0

w

11'(0)1

~

4 min Iw(z)1 . Izl=1

§ 7. Regularity of a Mapping with Prescribed Complex Dilatation

2)';

By (6.37) IWz(O)1 ~ e1 (1), since W has the same dilatation quotient as w. For IWz(O)1 = 11'(0)[ Iwz(O) I we thus obtain the estimate Iwz(O)1 :::;;: 4 min Iw(z)1 e1(1) Izi

•

=1

For further generalizations of Koebe's one-quarter theorem for quasiconformal mappings we refer to Pfluger [1J and Juve [1].

§ 7. Regularity of a Mapping with Prescribed Complex Dilatation 7.1. Complex dilatation and regular points. Having dealt with the problem of finding a sufficient condition for local conformality, we can treat without difficulty the more general problem of the regularity of a quasiconformal mapping at a point. In what follows we consider only finite points; of course we can get rid of this restriction by- means of a linear mapping. Contrary to the local conformality, the regularity of a quasiconformal mapping at a point Zo cannot be deduced from the behaviour of its dilatation quotient for z '# zoo We must therefore consider the complex dilatation of the mapping. In 6.2 we remarked that the conformality condition in Theorem 6.1 is equivalent to the convergence of the integral

Il If ~d z2

(j .

[zl<'

This gives rise to the following generalization of Theorem 6.1 : Theorem 7.1. Let G. and G' be domains of the finite plane, Zo a point of G, and w : G ~ G' a quasiconformal mapping with complex dilatation x, where Ix(z)1 ~ k 1 almost everywhere in G. If there exists a number X o

<

such that

If

Iz-z,1 <,

for some r

lu(z) Iz -

Uol d 2

Zo 1

(j

<

(7.1)

00

> 0, then w is regular at Zo and x(zo) =

xo'

Proof: . By Theorem II.8.1 the restriction of w to a neighbourhood of Zo can be extended to a quasiconformal mapping of the plane. By means of a linear transformation we ensure that the point at infinity is mapped onto itself. Hence we may assume, without loss of generality, that G and G' are planes.

234

V. Quasiconformal Mappings with Prescribed Complex Dilatation

In order to be able to apply Theorem 6.1 we construct the auxiliary mappingj, (7.2) where Xo is the number appearing in (7.1). We show that! is conformal at C= 0. From (7.1) it follows that IXol < k 1 and so ! is quasiconformal. Therefore, by Theorem 6.1, it is enough to show that

<

1'1 If 1"/('1I 2

d

a

<

(7-3)

00

ICj
> 0, where x, denotes the complex dilatation of f.

holds for some r

To prove (7.3) we apply the transformation formula (5.6) of IV.5.2 to calculate Setting (7.4) we obtain

x,.

Ix (C)I = I

I 1,,(z) - "0 I < - "0 ,,(z) -

"01

I"(z) 1 -

k2

almost everywhere. Since the Jacobian of the affine mapping defined by (7.4) is 1 - IXoI2 , it follows that

ff I~f~:) I

da

I~I

<

(1


~ k2)2

ff 1"~i(z~2 "01

da ,

lC(z)J < r

where by (7.4) (7.5) Because

we obtain finally

If

I~j

1"1") 1 d < 1'12 a=


1

(1 _ k)2

If

I"(z) - "01 d Iz _ zol2

a.

(7.6)

Iz-zol < (1 +k) r

Thus (7.3) follows from (7.1). Hence by Theorem 6.1 we have

!(C) = !c(O) C + o(C) .

(7.7)

In view of (7.2), (7.4) and (7.5) this is equivalent to

w(z)

fda)

"ofc(O)

-

-

= w(zo) + 1 - "I012 (z - zo) + 1 - "I012 (Z - Zo) + O(Z - zo)·

(7.8)

From this we see that w has the required properties: it is regular at Zo and x(zo) = Xo'

§ 7.

Regularity of a Mapping with Prescribed Complex Dilatation

235

7.2. Continuous differentiability. We shall now investigate the question of when a quasiconformal mapping w with prescribed complex dilatation is regular in a domain G. By definition this means that w is to be regular at every point of G and have continuous partial derivatives in G. The latter property does not follow from the first and in fact we need a stronger condition than the validity of (7.1) for every Zo E G (d. 6.2). The continuous differentiability of w in G does, however, follow if we impose upon the integral of (7.1) the additional requirement that it converge uniformly in every compact subset F of G. This means that for every B 0 there exists a t5 0 such that

>

>

If

lu(z) - u(zo)! d [

z -

Zo 12

(J


Iz-zol
for every zo'= F, where by Theorem 7.1, u(zo) must be the complex dilatation of w at ZOo / As previously we restrict ourselves to domains of the finite plane. Theorem 7.2. Let u, lui G such that the integral

<

k

< 1, be a measurable function in a domain

converges uniformly in every compact subset of G. Then every quasiconformal mapping w: G ~ G' whose complex dilatation coincides with u almost everywhere in G is a regular quasiconformal mapping of G and has the complex dilatation u(z) for every Z E G. Prool: As in the proof of Theorem 7.1 we can extend w to a quasiconformal mapping of the plane and normalize this so that w( (0) = 00. We fix a domain U, U C G, and show that the derivatives wz and W z are continuous in U.

For an arbitrary point Zo E U the equation (7.8) holds with Uo = u(zo) .. Comparing this with the equivalent relation (7.7) and applying Theorem 6.1 we obtain for the remainder term in (7.8) the estimate

lo(z - zo)l s:; IIc(O) C! B(IW s:; 1'1 c(IW max 1/(')1 el (l) ICI=l

< ': ~

;1 (I: -=- Z~I) B

max

Iw(z) _ w(Zo) I el (l). (7.9)

Iz-zol =l+k

Here B and I are the functions appearing in Theorem 6.1, the function I being defined with respect to I. We can choose B to be decreasing with

IC!·

V. Quasiconformal Mappings with Prescribed Complex Dilatation

Since the maximal dilatation of the mapping (1 - k)2 we have

D8) - 1 = 2

IUf(C) I IUf(C) I

1

:s:; 2 (1

1 _

t

+ k2 _ k)2

is at most (1

+ k)2j

Ix/C)I .

In view of (7.6) it follows that I (r) < =

+

If

2

1 k :rr; (1 - k)4

Iu(z) - U(Zo) I

Jz -

da

=

,1 (r) .

ZOl2

Iz-z.l< (l+k)r

Here ,1 (r) depends on zo, but tends by hypothesis uniformly to zero as O. Hence ,1(1) and thus also eI(l) in (7.9) are uniformly bounded for Zo E U. By the first remark at the end of 6.7, the function c depends only on k and ,1. Since Iw(z) - W(Zo) I is uniformly bounded for Zo E U, Iz - zol :s:; 1 + k, we can thus write (7.9) in the form lo(z -.,- zo)1 ~ Iz - zol Cl (Jz - zol), where Cl (Jz - zol) --+ 0 as Iz - zol --+ O·and Cl is the same function of Iz - zol for every point Zo E U. From (7.8) we then obtain r --+

Iw(z) - w(zo) ~ wz(zo) (z - zo) - w:z(zo) (z - zo) I

< Iz For real

z -

Zo

=

zol Cl(JZ - zol) .

(7.10)

h it follows from (7.10) that

I w(zo + h~ -

w(zo)_ -

wx(zo)

I < cl(Jh!) .

Hence wx is continuous in U as a uniform limit of continuous functions. In the same way we can show that wy is continuous, and the theorem is proved. Besides the investigations of Teichmuller, Wittich and Belinskij mentioned in § 6, results related to Theorems 7.1 and 7.2 have been obtained by Sabat [1J; see also Volkovyskij [1J. In the above formulation these theorems can be found in Lehto [1J. By applying the representation formula (5.17) in 5.7, Bojarski[2J has demonstrated regularity at a point under the condition

JJ

I

U~)~z:o IP da <

00 ,

(7.11)

Iz-z.[
This result is contained in Theorem 7.1: By Holder's inequality (7.1) follows from (7.11), while the example Ix(z)1 = Izl l - 2 !P, Zo = X o = 0, shows that the converse is false. 7.3. Sharpness of the conditions. Simple examples suffice to show that the criteria given in Theorems 7.1 and 7.2 are not best possible.

§ 7. Regularity of a Mapping with Prescribed Complex Dilatation

237

In fact, the integral in (7.1) need not be finite even for a mapping w which is regular throughout G and not only at zoo To see this we choose as our domain G the square {x + i Y 1- 1/2 1/2, - 1/2 y 1/2} and define a mapping w by the formula

<

< <

w (x

+ i y)

!

+ i (Y -

= x

<x

lo;t1tl ) .

This mapping is regular in G and has complex dilatation " (x

1

+ i y)

= -1-1 2 og Y-1- 1.

An easy calculation shows that the integral 1/2

1/2

'I" (x + iy)1 dx d X2 +y2 Y

JJ

-1/2 -1/2

is infinite. It is thus to be expected that a non-trivial necessary regularity condition must depend not only on I"(z) - ,,(zo) I but also on the argument of ,,(z) - ,,(zo)' In fact, Reich and Walczak [1J have shown that for every measurable function g of the real variable r, r 1, such that g(r) ::;;; k 1, there exists a mapping of the unit disc onto itself which is conformal at the origin and whose complex dilatation has absolute value g(lzl) for almost all z, [zl 1. 52 An example can be constructed as follows:

°::; ;

°<

<

<

<

We go over to the C= ;

+ if) = h(;)

=

log z-plane and write

V1 -

2g(e~) (g(e<))2

(7.12)

First we define a function h* of; in the following way: If the integral o

J h d; -00

is finite, then we write h* = h. If not, then there exists a decreasing sequence of numbers ;0 = 0, ;1' ;2' ... , such that 11=1,2, ....

Reich and Walczak [1] have also given a refined version of the sufficient condition (7.1), in which the argument of " is considered.

52

238

V. Quasiconformal Mappings with Prescribed Complex Dilatation

Then we set h*(;) = h(;) for ;zn+l ;zn+z <; ::;: ;zn+l' n = 0, 1, ....

<; ::;: ;zn and

h*(;)

= - h(;)

for

The function 1p, ~

1p(;) =

J h* d; ,

o

tends to a finite limit iX as ; ~ - 00 in both cases. Being an integral, is absolutely continuous on every compact interval of the half-line and has the derivative

1p

; <°

for almost all ;. We now define a mapping 1 of the left half-plane onto itself by the formula This mapping is absolutely continuous on lines and its derivatives satisfy the equations

I,(C) =

i

1

+2

h*(;) ,

i

!c(C) = 2 h*(;)

+

for almost all C= ; i 1]. Since /h*(;)/ = h(;) is bounded, 1 is quasiconformal. By (7.12) the modulus of the complex dilatation of t satisfies

for almost all C. Finally we return to the z = e'-plane. The mapping the mapping w, w(z) = e!(logz) = z ei'l'(log\zl) ,

t corresponds to

of the unit disc onto itself. This has the required properties: its complex dilatation has absolute value g(lzl) for almost all z, and w(z)Jz tends to the limit eilX as z ~ 0.

VI. Quasiconformal Functions

Introduction to Chapter VI In previous chapters we considered quasiconformal homeomorphisms between plane domains. We now extend the concept of quasiconformality to mappings which are not necessarily one-to-one. The simplest generalization retaining the essential features of the theory of quasiconformal homeomorphisms is the class of functions which are composed of a quasiconformal mapping and an analytic function. Since we shall consider mappings into the whole plane we allow the analytic functions to have poles. Definition. A function f is called K-quasiconformal in a plane domain G if it permits a representation f = ({! w, 0

where w: G --+ G' is a K-quasiconformal homeomorphism and ({! is a nonconstant analytic function in G'. A function is quasiconformal if it is K-quasiconformal for some K. If f = ({! w is quasiconformal in a domain G, then f is locally homeomorphic in G up to points which w maps onto zeros of the derivative ({!' or onto multiple poles of ({!, i.e. f is locally homeomorphic with the possible exception of isolated points. In addition, f satisfies the module cond~ion M (t(Q)) S K M(Q) for every quadrilateral Q such that Q c G and Q is mapped topologically by f. These properties of f are called geometric in analogy with our previous terminology. 0

The following analytic properties of a K-quasiconformal function fare likewise immediate consequences of the definition: f has V-derivatives (or even V"-derivatives for every p < P(K); d. V.SA), and the dilatation condition max", Io",f(z) I :S K min", Io",f(z) I holds almost everywhere. In other words, f is a generalized LP-solution of a Beltrami equation fz = x fz' where Ixl ~ (K - 1)((K 1), P P(K).

+

<

In the present chapter we shall establish geometric and analytic properties which are sufficient to characterize a quasiconformal function.

240

VI. Quasiconformal Functions

Geometric conditions will be given m § 1 and analytic conditions in § 2. In view of the definition of a quasiconformal function it is obvious that many theorems of Chapters I, II, IV and V can be carried over to quasiconformal functions, either as such Of with suitable modifications. Here such generalizations will not be dealt with.

§

I.

Geometric Characterization of aQuasiconformal Function

1.1. Quasiconformality and module conditions. According to the definition given in the Introduction, a quasiconformal function is continuous and, with the possible exception of isolated points, locally homeomorphic. Conversely, let us consider a continuous function I which is locally homeomorphic except at isolated points in a domain G. If the restriction of I to a domain U eGis a sense-preserving homeomorphism, then a simple connectivity argument shows that I preserves the orientation in every subdomain of G where it is homeomorphic.

We now prove that the geometric properties of K-quasiconformal functions mentioned in the Introduction do in fact characterize these functions. Theorem 1.1. Let I be a continuous 53 mapping 01 a domain G which is locally homeomorphic and sense-preserving up to a set E C G 01 isolated points. II I satislies the module condition M(t(Q)) S K M(Q)

(1.1)

lor every quadrilateral Q such that Q C G and I maps Q topologically, then tis K-quasiconlormal in G. Prool: By hypothesis, every point Z E G - E possesses a neighbourhood U which is mapped topologically, and by (1.1) K-quasiconformally, onto I(U). The complex dilatation of I is therefore defined almost everywhere in U.

Let x be a function which coincides with the complex dilatation of I at every point where the latter is defined and vanishes elsewhere in G. Then x is measurable and satisfies K -

Ix(z)1

1

< K+ 1

everywhere in G. By the Existence theorem there is a K-quasiconformal homeomorphism w of G whose complex dilatation is equal to x(z) for almost all z E G. 53

Note that continuity here is in the sense of the spherical metric.

241

§ 1. Geometric Characterization

The composite mapping cp = 1 0 w- 1 is locally homeomorphic in w(G - E). It follows from the Uniqueness theorem (IV.5.2) that every point of w(G - E) has a neighbourhood which is mapped conformally by cpo Since cp is continuous throughout w(G) and w(E) consists of isolated points, cp is analytic in w(G). The function 1 = cp 0 w is hence K-quasiconformal in G. 1.2. Interior mappings. The topological condition in Theorem 1.1 that 1 be locally homeomorphic except at isolated points can be expressed in another non-trivially equivalent form. For this purpose we introduce the concept of an interior mapping. A mapping of a domain G into a domain G' is called open if the image of every open subset of G is open. If 1is continuous and open, then for every set A' we have (1.2) The mapping 1is caJled light if every component of the preimage 1- 1 (') of any point' E f(G) consists of a single point. If 1is continuous, open and light, then it is called an interior mapping of G. Non-constant analytic functions provide examples of interior mappings. On the other hand, by a theorem of Stollow [1J an interior mapping can always be represented in the form 1 = cp w, where w is a homeomorphism of G and cp is analytic in w(G). 0

From this it follows that an interior mapping of a domain G is locally homeomorphic in G with the possible exception of isolated points. Thus in Theorem 1.1 we can replace the words "continuous and locally homeomorphic up to a set E C G of isolated points" by the word "interior" . We shall establish this equivalent version of. Theorem 1.1 completely without reference to Stollow's theorem. For this it is necessary to prove directly the part of Stollow's theorem which states that an interior mapping is locally homeomorphic up to isolated points. For the proof we first give three lemmas on open, light and interior mappings, respectively. 1.3. A lemma on open mappings. The following property of an open mapping will be of frequent application in our proof. Lemma 1.1. Let 1 be a continuous open mapping of a domain G, D a subdomain 01 I(G), and A, A c G, a component 01 1-1(D). Then f(A) =D.

242

VI. Quasiconformal Functions

Prool: In view of the continuity of I, no boundary point of A is mapped onto a point of D. Consequently

I(A) = I(A)

n D.

Since I is an open mapping and A is an open set, I(A) is open. On the other hand, I(A) is closed, being the continuous image of the compact set A. The set I(A) is hence both open and closed in D. Since D is connected our assertion follows.

1.4. A lemma on light mappings. Of light mappings we need the following property: Lemma 1.2. Let I be a continuous light mapping 01 a domain G, Zo a point 01 G, and U a neighbourhood 01 Zo with U C G. Then there exists a neighbourhood Vol Wo = I(zo) such that the component 01 l-l(V) which contains Zo lies in U. Prool: We may assume that U is a disc. Let V n' n = 1, 2, ... , be discs with the same centre Wo and with radii r n' r 1 r2 lim r n = O. We show that the zo-component An of the set l-l(V n) is contained in U from some n onwards. Our assertion follows from this, since the zocomponent of l-l(Vn+ t ) is a closed subset of An'

> > ... ,

We suppose that on the contrary An does not lie in U for any value of n and derive a contradiction. First we have for the zo-component En of An n U the relation En n Fr U #- {). (1.3)

n

n

n

For otherwise En C U and so En An = En (An U). Since En' as a component of Ann U, is closed in this set, it follows that En = En n An' Consequently En is not only open but also closed in An' Since An is connected this is impossible and (1.3) follows. We now make use of the following well-known result on compact sets (Newman [1J, pp. 47 and 81): The intersection 01 a decreasing sequence 01 non-empty (connected) compact sets is non-empty (connected). From this we first deduce that E = n En is connected, since En is connected as the closure of a connected set. Further, in view of (1.3), it follows from E Fr U = (En Fr U) that E Fr U is nonempty. On the other hand, I(E) C V n for every n so that I(E) consists of Wo only. Hence E lies in the zo-component of l-l(WO) , which therefore contains points of Fr U. This contradicts the assumption that I is light, i.e. that the zo-component of l-l(WO) consists of the point Zo alone.

n

n

n

n

243

§ 1. Geometric Characterization

1.5. A lemma on interior mappings. Using the two Lemmas above we prove the following result on interior mappings: Lemma 1.}. Let j be an interior mapping oj a simply connected domain G such that f(G) is not the whole plane, and V, V C j(G), a disc. Let D l -and D 2 be discs, D l U D 2 C V, D l n D 2 = 0, and W E V a point outside 51 U~. If A, .if c G, is a component of f- l (V), then the sets j-l(W)' f- l (D l ), f-l(D 2 ) each have a finite positive number n, nl> n 2 oj components in A, and these numbers satisfy n < n 1 n 2 • Proof: We first remark that every component of the preimage of a set

E C V either lies completely in A or completely outside A. We also note that for a compact E C V the set f-l(E) n A is compact.

n

Let WI be an arbitrary point of D l . As a compact set A j-l(Wl ) is covered by a finite number of components of f-l(D l ). In fact, this finite covering consists of all components of f-l(D l ) lying in A, since every such component is mapped onto the whole of D 1 by Lemma 1.1. Consequently the number n l is finite and the same reasoning shows that n 2 is finite. From Lemma 1.1 it follows that n, n l and n 2 are positive. We suppose now that n n l n 2 , contrary to our assertion. In order to derive a contradiction we choose n l n 2 1 points Zi in the set f-l(W) n A and cover them with discs U i whose closures are disjoint and lie in A. By Lemma 1.2 there exists a disc D 3 , D 3 C V, with midpoint w, such that the zi-component of j-l(D3 ) is contained in U i for every i. The number n 3 of the components A 3k of f-l(D 3 ) n A is thus at least n l n 2 1. On the other hand we deduce as above that j-l(D 3 ) n A and by (1.2) also f-l(D 3 ) n A, has only finitely many components.

>

+

+

If necessary we extend the domains D l and D 2 so that both meet the disc D 3 , and carry out the extension in such a way that D l and D 2 remain disjoint subsets of V and the complement of D l U D 2 U D 3 is connected. We thus remove the restriction that D l and D 2 are discs; it was made only to ensure that the above extension would be possible. It follows from Lemma 1.1 that the number of components of f-l(D h ) n A, h = 1,2, is at most n h after the extension. By (1.2) the same is true for the components A hk , h = 1,2, of the sets f-l(D h ) n A.

n

n

Since D l D 3 and D 2 D 3 are non-empty, every A 3k has points in common with at least one A I i and one A Zi by Lemma 1.1. Since the number of pairs (Ali, A zi ) is at most n l n 2 there exists a pair AlP' A zq which meets two components A 3r and A 3s (Fig. 1-5). We can now apply

244

VI. Quasiconformal Functions

v

~

Fig. 15

Lemma 1.1.2' in the following way: We set F 1 = Alp U A 3 r> F 2 = A 2q U A 3s , and choose as the sets F., 11 = 3,4, ... , the remaining A ik • Then the conditions of Lemma 1.1.2' are satisfied and it follows that the complement of the set F = U A;k = 1-1 (U D;) disconnected.

nA

is

Since G is simply connected, - G is contained in a component of - F. Hence one component B of - F lies in G. If z* is a boundary point of the domain B, then z* E F, since z* would otherwise have a neighbourhood lying in the complement of F. Hence I(z*) E U D i , and therefore I(B) - U D; = I(B) - U D;. The set I(B) - U D; is consequently both open and closed in the complement of U D;. Since this complement is connected and I(G) is not the whole plane it follows that I(B) - U D; = e, and so I(B) CUD;. From I(A - F) n (U D;) = e we thus conclude that B n (A - F) = B n A must be empty. On the other hand B n A "# e, since the set F lies in A and the boundary of B lies in F. Hence we are led to a contradiction and the lemma is proved.

1.6. Local behaviour of interior mappings. The required result on interior mappings can now easily be proved.

•

Theorem 1.2. An interior mapping I 01 a domain G is locally homeomorphic in G with the possible exception 01 isolated points. Prool: Let Zo be an arbitrary point of G and W o = I(zo), It follows from Lemmas 1.2 and 1.3 that the points of l-l(WO) have no limit points in G. The point Zo lies therefore in a disc U, U C G, which contains no other

245

§ 1. Geometric Characterization

points of I-I(W O)' Let U be so small that I(U) is not the whole plane. By Lemma 1.2 there exists a disc V C I(U) with midpoint W o such that the closure of the zo-component A of I-I(V) lies in U. Let WI and W 2 be two other points in V and n; the number of the points of I-I(W;) n A, i = 1,2. We cover the points Wo and WI by discs Do and D I such that Do U D I C V, Do n D I = 0, and W2 El Do U D I· According to Lemma 1.1 I-I (Do) has precisely one component in A, and the number of components of I-I(D I ) in A is at most n l . Since U is simply connected, Lemma 1.3 can be applied to the restriction of I to U, and so n 2 S 1 . n l = n l • By interchanging the roles of WI and W 2 we also deduce that n l < n 2 . With the possible exception of the point wo, which has precisely one preimage point in A, all points of V have therefore the same finite number n of preimage points in A. If n

= 1, then the mapping I: A

--+

V is homeomorphic. In the case

n> 1 the mapping cannot be one-to-one in any neighbourhood of zoo

We show that every ZI oF Zo of A then has a neighbourhood in which I is homeomorphic. The point Zo is therefore the only exceptional point in A. We consider the set I-I (t(ZI) ) n A = {zv Z2' ,z,,} and cover the points z; with disjoint discs U; C A, i = 1,2, , n. By Lemma 1.2 there exists a disc VI with midpoint I(ZI) such that the z;-component of I-I(VI ) lies in U; for everyi = 1,2, ... ,n. Thus by Lemma 1.1, I-I(VI ) has exactly n components in A, which are all mapped onto VI' If now some point of VI had more than one preimage point in any of these n components, then it would have at least n 1 preimage points in A. This shows that the mapping of the zccomponent Of I-I(VI ) onto VI is homeomorphic, and the theorem is proved.

+

1.7. Other forms of the geometric characterization. Once in possession of Theorem 1.2 we can define when an interior mapping is sensepreserving (d. the remark at the beginning of 1.1). We then obtain from Theorem 1.2 the following modification of Theorem 1.1. Theorem 1.1'. Let I be a sense-preserving interior mapping 01 the domain G. II every quadrilateral Q, Q C G, which is mapped by I topologically onto a quadrilateral Q' satislies the module condition M(Q') < K M(Q), then I is K-quasiconlormal in G. By the hypothesis of Theorem 1.1 or Theorem 1.1' the domain G can be represented in the form G

=

EU (g\ U;) ,

246

VI. Quasiconformal Functions

where E consists of isolated points, the sets U i are domains, and the restriction of 1to Ui' i = 1, 2, ... , is a sense-preserving homeomorphism of U i . It follows from the proof of Theorem 1.1 that Theorems 1.1 and 1.1' remain true if the module condition (1.1) is replaced by any such condition which ensures the K-quasiconformality of 1 in every U i . For example we can replace (1.1) by a corresponding condition for analytic quadrilaterals, ring domains or rectangles, as follows from Theorems 1.5.3, 1.7.2 and IV').3. Furthermore it is possible to assume instead of (1.1) that sup F(z) < K zEG-E

(Theorem 1.9.1), or that H(z) is finite in G - E and < K almost everywhere in G - E (Theorem IV,4.2).

§

2.

AnalYtic Characterization of aQuasiconformal Function

2.1. V-solutions of the Cauchy-Riemann equation. We shall set out analytic conditions implying the quasiconformality of a function in two steps. In the simplest special case, that of a 1-quasiconformal, i.e. analytic function, we obtain the conditions from a previous result. The general case can then be reduced to this with the aid of the Existence theorem. Since the results of Chapter III which find application here were stated only for domains of the finite plane and finite-valued functions, we shall restrict ourselves to this case. In particular, this means that analytic functions will have no poles. As a special case of Lemma II1.7.1 we then have Theorem 2.1. Let 1 be a generalized V-solution 01 the Cauchy-Riemann equation in a domain G. Then 1 is analytic in G. This theorem is closely related to the Uniqueness theorem according to which a homeomorphic generalized solution of the Cauchy-Riemann equation is a conformal mapping. Since we have removed the assumption the the mapping be one-to-one here, we need the additional condition that the derivatives be locally integrable. 2.2. V-solutions of Beltrami equations. Let P(K) be the extremal exponent for K-quasiconformal mappings as defined in V.5A, and 1 - 1/P(K) = 1jq(K). We remind the reader that P(1) = 00 and P(K) decreases monotonically to 2, as K ->- 00. Hence q(K) increases monotonically from 1 to 2 as K increases from 1 to 00.

247

§ 2. Analytic Characterization

For K-quasiconformal functions we can generalize Theorem 2.1 as follows: Theorem 2.2. In a domain G let f be a non-constant generalized Lqsolution of a Beltrami equation h = uf z (2.1)

where u is a measurable function in G with lu(z)1 < (K - 1)j(K and q q(K). Then f is a K-quasiconformal function in G.

>

+ 1)

Proof: By the Existence theorem there is a K-quasiconformal homeomorphism w of G with complex dilatation u(z) for almost all z E G. We shall show that the function cp = f w- 1 is analytic in the domain w(G), which will prove the theorem. 0

> +

By hypothesis f has Lq-derivatives for a q q(K). On the other hand wand w- 1 , being K-quasiconformal mappings, have LP-derivatives for every p P(K). We fix p such that 1jp 1jq = 1 and apply Lemma III.6.4 to the composite function cp = f 0 w- 1 . It follows that cp has Uderivatives in w(G).

<

In view of Theorem 2.1 the analyticity of cp thus follows if we show that cpc(C) = 0 for almost all 1; E w(G). For this we need only repeat the calculations inlV.5.2 without the assumption that f be a homeomorphism. Formula (5.9) in IV.5.2 then yields the desired result. Hence rp is analytic in w(G). The function f formal as was asserted.

=

rp

0

w is then K-quasicon-

<

Remark. As was remarked above, q(K) 2 for every K. Hence Theorem 2.2 holds under the assumption that f is a generalized V-solution of the Beltrami equation (2.1). The proof now becomes simpler: Without drawing on the results of V.5, we deduce the existence of Ll_ derivatives of f w- 1 from Lemma III.6.4 using the fact that a quasiconformal mapping has V-derivatives. 0

2.3. Other forms of the analytic characterization. In view of the connection between the Beltrami equation and the dilatation condition (1.6) in IV.1-3 we see immediately that Theorem 2.2 can also be expressed in the following form:

Let f be a non-constant function with U-derivatives for a q domain G, and suppose that the inequality

> q(K) in a

max lo"f(z)[2 ~ K J(z)

" holds almost everywhere in G. Then f is a K-quasiconformal function in G.

248

VI. Quasiconformal Functions

A K-quasiconformal function f = q; 0 w naturally has LP-derivatives P(K). Hence we conclude from Theorem 2.2: for every P

<

If the function f is a generalized LP-solution of the Beltrami equation "f., with ',,(z)1 :S (K - 1){(K + 1) for some P q(K), then f is a generalized LP-solution for every p < P(K).

Iz =

>

This theorem naturally yields the more information the closer K lies to 1, since q(1) = 1, P(1) = 00, q(K) -')- 2, P(K) -')- 2 as K -')- 00. The analytic characterizations of a K-quasiconformal function given here are implicit in the sense that the precise value of P(K) is unknown. Apart from this we do not know how high a degree of integrability of the derivatives must be assumed in Theorems 2.1 and 2.2.

Bibliography

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Index

Absolutely continuous - additive set function 117 - ' function on an arc 126 -: function on an interval 125 -, homeomorphism 118, 162 - in the sense of Tonelli 143 -, locally 117 - on lines 127, 162 Agard 181 Ahlfors 2, 16,21,28,44,66, 79, 82, 85, 9~ 10~ 13~ 15~ 166, 195, 20~ 213, 214, 219 Analytic -, arc 20 . -, characterization of a quasiconformal function 246, 247 -, curve 20 -, definition of quasiconformality 168 -, quadrilateral 27 -, ring domain 36 Angular distortion 181 Annulus 31 Apostol 149 Approximation -, good 185 in the LPmetric 141 of a function with LP-derivatives 144, 147 of a measurable function 136, 138 of a quadrilateral from inside 26 of a quasiconformal mapping 207 of a quasisymmetric function 91 of a ring domain from inside 36 theorem of Weierstrass 138 Arc -, analytic 20 -, Jordan 6 - length 122 -, locally rectifiable 122

-, of bounded turning 102 -, quasiconformal 97 -, rectifiable 122 Area measure 110 Areal derivative 120, 130 Behnke 13, 56 Belinskij 227 Beltrami differential equation 184 -, generalized solution 184 -, generalized LP -solution 184 Bers 2,165,170,187,195,207,219 Beuding 21, 79, 82, 85, 132, 159, 166, 206 Bojarski 195,214,215,219,236 Borel function 112 Borel set 111 Boundary - behaviour - - of conformal mappings 13 - of quasiconformal mappings 42, 79 - of topological mappings 11 correspondence theorem (for conformal mappings) 14 -, positively oriented 8 - . sets, corresponding to each other 11 value problem - - for a Jordan domain 79 - - fora multiply connected domain 85 - - for the half-plane 80, 82 Bounded turning 100, 102 Bounded variation 123, 126 -, on lines 127 Calderon 159,213,214 Canonical annulus 31 Canonical mapping - of a quadrilateral 15 - of a ring domain 31 Canonical rectangle 15

254 Cantor set 125 Carleson 219 Characteristic function 112 Characterization of a (homeomorphic) quasiconformal mapping 16, 30,41, 48,168,171,173,174,176,177,185, 211 Characterization of a quasiconformal . function 239,240,245,247 Circular dilatation 105, 177 Class cn 139 Coo 139 Cgo 139 LP 135 0AD 206 Closed polygon 6 Completely additive class 111 Completeness of LP 158 Complex derivatives 49 Complex dilatation 182 Conformal equivalence classes of simply connected domains 13 Conformal module, see Module Conformality at a point 219 Conjugate quadrilaterals 99 Connectivity 10 Continuation, see Extension Continuity -, absolute 117,118,125,126, 127, 143, 162 -, Holder 70 - of the module 26,27, 36 Convergence in the sense of Frechet 27 - of quadrilaterals 26, 27 - of ring domains 35 -, weak 186 Convexity theorem of Riesz 214 Curve -, analytic 20 -, Jordan 6 - of bounded turning 100 - ,oriented 8 -, quasiconformal 97 -, rectifiable 122 Degenerate quadrilateral 25 Degenerating ring domains 34 Density, see Point of density Derivative -, areal 120, 130

Index -, complex 49 - in the direction IX 1 7 -,LP- 143,162 - of a set function 119 Differentiability 9, 49 of a homeomorphism 128 - of a quasiconformal mapping 164, 233, 235 Dilatation -, circular 105,177 -, complex 182 - condition 50, 164 -, maximal 16 -, -, at a point 47 -, -, of quadrilaterals 16 -, -, of ring domains 38 - quotient 17, 49 Dimension of a point set 116 Distance - between a-sides (b-sides) 22 -, hyperbolic 66 - ,spherical 5 Distortion function - A 81, 82 - CPK 63,64 Distortion theorem of Koebe 56 Dunford 214 Egoroff's theorem 113 Equicontinuity 68 Existence theorem 194 - for the local maximal dilatation 197 Extension of quasisymmetric functions 89 over a free boundary arc 47, 98 to an isolated point 41 to free boundary arcs 42 to the exterior of a compact set 96 Extremal domain of Grotzsch 53 - of Mory 58 - of Teichmiiller 55 Extremal length 22, 133 Finite plane 5 Free boundary arc 11 Fubini's theorem 113 Fuglede 132, 134, 135 Function -, absolutely continuous

255

Index -, -, --, -, -, -, -, -, -,

- on an arc 126 - on an interval 125 - on lines 127, 162 -, set- 117 admissible for a curve family 132 Borel 112 Holder continuous 70 integrable 113 measurable 112 of bounded variation - on an arc 126 ~ on an interval 124 - on lines 127 -, quasiconformal 239 -, quasisymmetric 88 -, singular 125 - with LP-derivatives 143 Gauss 195 Gehring 39,105,128, 170,174,177,178, 181,198,216 General module condition 171 Generalized Cauchy integral formula 154 Generalized solution of a Beltrami equation 184 Geometric characterization of a quasiconformal function 240, 245 Geometric definition of quasiconformality 16 Golusin 27 Good approximation 185 Green's formula 148 Gross 116 Grotzsch 1, 17, 18, 53 -, extremal domain 53 -, inequality 18 -, module theorem 54

Hausdorff dimension 116 Hausdorff measure 116 Hersch 59, 63 Hilbert transform 1 57 -, differentiation 157 -, integral representation 156, 159 -, L2-norm 157 Hilbert transformation 157 -, extension to L2 159 -, LP-norm 214 Holder condition 70 Holder continuous 70

Holder's inequality 141 Homeomorphism 6 -, absolutely continuous 118 -, differentiability 128 -, locally absolutely continuous 118 -, sense-preserving 9 Horizontal rectangle 164 Hyperbolic metric 66 Inequality of Gr6tzsch 18 - of Holder 141 - of Rengel 22 Inner measure 112 Integrable 113 -, locally 139 Integral 113 -, Lebesgue 113 -, Legendre normal 60 - , over an oriented arc 126 -, Stieltjes 126 Interior mapping 241 Interval 6 Invariance - of domains 6 - of open sets 6 Isolated boundary point 41 Jacobian 9, 49, 130 Jordan arc 6 -, oriented 8 Jordan curve 6 -, oriented 8 Jordan curve theorem 7 Jordan domain 8 Juve 233 Kelingos 88 Kernel of a sequence of sets 76 Koebe 205 -, distortion theorem 56 -, one-quarter theorem 56, 233 K-quasiconformal function 239 K-quasiconformal mapping 16 k-quasisymmetric function 88 Kiinzi 3 Kuusalo 195 Landau's symbols 0 and Lavrentieff 2, 66, 194

0

49

256 Lebesgue - area measure 110 - convergence theorem 114 -, extended measure 132 integral 113 linear measure 110 theorem 120 volume measure 111 Legendre normal integral 60 Lehto 92, 106, 108, 128, 170, 216, 236 Length of an arc 122, 123 Length preserving parametrization 123 Light mapping 241 Limit exponent P(K) 215 Limit function of quasiconformal mappings 29, 73, 74, 210 Line 6 -, extended 6 -, finite 6 - segment 6 Linear measure 116 -, Hausdorff 116 -, Lebesgue 110 Local maximal dilatation 48 Locally absolutely continuous 117 Locally of bounded variation 124 Locally rectifiable 122 LP-derivatives 143, 162 LP metric 135 LP-norm 135 - of Hilbert transformation 214 Lusin's theorem 137 Mapping 5 -, affine 19 -, antiquasiconformal 16 -, canonical -, -, of a quadrilateral 15 -, -, of a ring domain 31 -, conformal 13 -, homeomorphic 6 -, interior 241 -,light 241 -, open 241 -, quasiconformal 16 -, -, K- 16 -,-,1- 28 -, -, regular 17 Mapping theorem 194 -, Riemann 13 Marstrand 116 Martio 195

Index Maximal dilatation at a point 47 of a homeomorphism 16 of quadrilaterals 16 of ring domains 38 Measurable function 112 Measurable set 111 Measure -, area 110 -, Hausdorff 116 -, inner 112 -, Lebesgue 110 -, linear 110, 116 -, outer 110 Metric -, hyperbolic 66 -, LP 135 -, spherical 5, 34 Module -, conformal invariance 173 - of a family of arcs or curves 133 -, -, monotonicity 133 -, -, subadditivity 133 - of a quadrilateral 15, 22, 173 - -, characterization without conformal mapping 21 - -, continuity 26, 27 - -, monotonicity 25 - -, superadditivity 25 - of a ring domain 31,32,173 - -, characterization without conformal mapping 32 - -, continuity 35 - -, monotonicity 35 - -, superadditivity 35 Module condition 171 Module theorem - of Gr6tzsch 54 - of Mori 59 - of Teichmuller 56 Mori 2, 58, 66, 106, 162, 165, 169 - extremal domain 58 - module theorem 59 Morrey 2, 165, 195 Munroe 109, iii, 112, 113, 114, 115, 117,118,158 Naatanen 182 Net 138 Newman 4,6,7,8,9,10, ii, 12,242 Nikodym 118 Norm

257

Index

-, LP 135 -, -, of Hilbert transformation 214 Normal family 72 -, closed 72 - of quasiconformal mappings 72 n-tuply connected 10 Null set 111 One-quarter theorem of Koebe 56, 233 Open mapping 241 Orientation 8 Orientation theorem 9 Oriented arc 8 Oriented curve 8 Outer measure 110 -, Hausdorff 116 -, -, a-dimensional 116 -, -, linear 116 - in the sense of Caratheodory 110 -, Lebesgue 110 - , -, area 110 -, -, linear 110 -, -, volume 111 -, regular 110 Pesin 205 Pfluger 2,16,63,92,99,104,162,170, 233 Piranian 105 Plane 5 -, finite 5 Point of density 115 -, linear 11 5 -, xy-

115

Polygonal arc 6 Polynomial approximation 138, 211tJ Positively oriented boundary 8 Quadrilateral 14 -, analytic 27 -, approximation from inside 26 -, a-side (b-side) 14 -, canonical mapping 15 -, canonical rectangle 15 -, conformal mapping 15 -, conjugate pair 99 -, convergence 26,27 -, convergence from inside 26 -, degenerate 25 -, dilatation under a homeomorphism 16 - ,homeomorphism 15 -, module 15,22,173

-, quasiconformal mapping 30 -,side 14 -, vertex 14 Quasieonformal arc 97 Quasiconformal curve 97 Quasiconformal extension 96, see also Extension Quasiconformal function -, analytic characterization 247 -, definition 239 -, geometric characterization 240,245 Quasiconformal mapping (quasiconformal homeomorphism) 16 -, absolute continuity on arcs 170 -, absolute continuity on lines 162 -, absolute continuity with respect to area measure 165 -, analytic definition 168 -, approximation 185,207 -, boundary behaviour 42, 79 -, characterization 16, 30, 41, 48, 168, 171,173,174,176,177,185,211 -, convergence of derivatives 186, 216 -, degree of continuity 71 -, differentiability 164,233,235 -, dilatation condition 50, 164 -, geometric definition 16 -, good approximation 185,207 -, K- 16 -, LP-derivatives 165,215 -, limit mapping 29, 73, 74,210 -, of a quadrilateral 30 -, of a ring domain 38 -,1- 28 -, preservation of null sets 165, 216 -,regular 17,210,235 -, regular points 166,199,233 -, representation 218 Quasiconformal reflection 98 Quasisymmetric function 88 Rademacher-Stepanoff theorem 165 Rad6 27 Radon-Nikodym theorem 118 Real-analytic 208 Rectangle 15 -, horizontal 164 Rectifiable 122 Reed 85 Reflection - in a circular arc 46 -, quasiconformal 98

258

Index

Reflection principle 47 Regular point 10 Regular quasiconformal mapping 210,235 Reich 27, 39, 216, 237 Removability - of an analytic arc 44 - of an isolated point 41 Removability problems 199 Removable 41, 199 Rengel's inequality 22 Renggli 177, 199 Rickman 103 Riemann mapping theorem 13 Riemann sphere 5 Riesz convexity theorem 214 Ring domain 30 -, analytic 36 -, canonical mapping 31 -, convergence from inside 35 -, degenerating 34 -, module 31,32,173 -, quasiconformal mapping 38

17,

v

Sabat 236 Saks 109,115,120,123, 127, 130, 165 Sario 206, 207 Schiffer 55 Schwartz 214 Sense-preserving homeomorphism 9 Separation·theorems 6 Set function 110, 132 Sewing theorem 92 a-finite 112 Simply connected 10 Singular function 125 Sommer 13, 56 Spherical metric 5, 34

721110173

Springer 98 Step function 138 Stepanoff 130, 165 Stollow 241 Strebel 162, 202 Support 139 Taari 181 Teichmiiller 2, 54, 56, 59, 224 -, extremal domain 55 -, module theorem 56 Tienari 99, 104 Turning 100, 102 Uniform Holder continuity 70 Uniformization theorem 195 Uniqueness theorem 183 Viiisiilii 39, 86, 104, 105, 106, 108, 153, 171,174,177 Variation -, bounded 123,126 -, -, on lines 127 -, on an arc 126 -, on an interval 124 Vekua 213,219 Vertex of a quadrilateral 14 Virtanen 92, 106, 108 Volkovyskij 2, 181,236 Walczak 237 Weak convergence 186 Weierstrass's approximation theorem 138 Wittich 224 Yl1j6b6 2, 170, 178

Zygmund 159, 213, 214

V112/6