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Area Distortion of Quasiconformal Mappings
CHAPTER
5
Area Distortion of Quasiconformal Mappings
D.H. Hamilton Department of Mathematics, University of Maryland, College Park, MD 20742, USA E-mail: dhh @math. umd. edu
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Gr6tzsch to Bojarski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphy ................................................... The class Z'(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. A strange Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Theorem 1, part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. T h e o r e m 1, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Bounds on the B e u r l i n g - A h l f o r s transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Further applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H A N D B O O K O F C O M P L E X A N A L Y S I S : G E O M E T R I C F U N C T I O N THEORY, VOLUME 1 Edited by R. Kfihnau 9 2002 Elsevier Science B.V. All rights reserved 147
149 150 152 153 154 155 156 158 159 159
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1. Introduction
A homeomorphism f : C --+ C is called K-quasiconformal if its directional derivatives satisfy max~ IO~f(x)l <<.K mind IO~f(x)l a.e. (with respect to area measure). It is also required that f be area preserving, i.e., if we use IEI to denote the area of any (measurable) planar set E:
IEI--0
~
If(E)[ -- O. 1,2
For quasiconformal mappings this means that f is in the Sobolev class Wloc , i.e., functions whose first derivatives are square integrable. There is the equivalent geometric definition that f have "bounded distortion". This is measured by two measures of stretching: t f(~,r)-
maxl f(x + Ihl=r
h)
-
f(x) l,
If(x,r)-
min IhI-r
If(x
§ h)-
f(x)l.
The infinitesimal distortion of f at x is H i ( x ) = limsuPr~oLf(x,r)/lf(x,r). If f is conformal, then Hf(x) = 1 (and the converse is true). This reflects the fact that infinitesimally conformal mappings preserve circles. Unfortunately this elegant geometric definition is tricky to work with so the analytic definition is more common. The equivalence between the geometric definition and the analytic definition was shown by Pesin [21] in 1956. If quasiconformal mappings were merely generalizations of conformal maps their theory would be a curiosity. In fact they are important for some of the most significant mathematical theory of the century. It has been known since the work of Ahlfors [1] and Mori [18] in 1955 that K-quasiconformal mappings are locally H61der continuous with H61der exponent 1/K. The mapping z --+ zlzl 1/K-1 shows that this is best possible. Bojarski [7], as a consequence of his fundamental existence theorem, showed that K-quasiconformal mappings actually belong Wllo'cp for some p -- p ( K ) > 1. This is equivalent to the fact that f distorts area by a power depending only on K. The above example shows one might guess that the optimal exponent in area distortion is also 1/K. The following theorem was conjectured and formulated by Gehring and Reich [9]. Let U be the unit disk {Izl < 1}. In 1992 Astala [4] proved. THEOREM 1. Suppose f : U - - + U is a K-quasiconformal mapping with f (O)= O. Then we have ]f(E)[ ~< MIE] 1/K for all Borel measurable sets E C U. Moreover, the constant M = M ( K ) depends only on K with M ( K ) = 1 + O(K - 1). Although this theorem is only for self maps of the disk a simple trick allows a factorization which proves similar bounds depending only on the normalization. As a straightforward consequence of Astala's theorem:
sup/p'! f ~ Wll~ } -
2K K- 1
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D.H. Hamilton
Also there is the optimal result for the distortion of Hausdorff dimension: 1t
1 Dim(E)
1
~)
~< D i m ( f (E))
1 2 ~< K
(
1 Dim(E)
~) "
The original proof of Astala depended on deep ideas from Dynamics: relating Hausdorff dimension of holomorphically varying Cantor type sets. Shortly afterwards a short direct proof was obtained by A.E. Eremenko and D.H. Hamilton [8], as a direct consequence of the holomorphy of the class of all quasiconformal mappings. In this case the normalization is f is conformal off U with f (z) = z + o(1) near o~ (i.e., the classical class 27). THEOREM 2. Suppose f E r is a K-quasiconformal mapping. Then we have [f(E)l ~< K 1/K Jr 1-1 / K IEI 1/ x for all Borel measurable sets E C U. We shall see that this bound is a combination of Theorem 5 (the case where f is conformal on E and Theorem 6 (where f is conformal off E), both of which are sharp. Theorem 2 corrects a typo in [8] where the 1 / K power of K was omitted. (Also it was stated that Theorem 1 in [8] had best possible constants, whereas it was meant that inequalities of Theorems 5 and 6 (only) were sharp.) As well as having applications to analysis (we shall later give sharp bounds for the Beurling Ahlfors Transform) these inequalities, in particular, the constant of Theorem 5, have proved important in applied mathematics: namely to the theory of determining optimal bounds of physical properties (heat/electrical conductivity, magnetic permeability, elastic stiffness) of compound solids consisting of two or more different materials combined together with some microstructure.
2. From Griitzsch to Bojarski Diffeomorphisms with uniform bounded distortion were first studied around 1928 by H. Gr6tzsch [11]. In 1939 O. Teichmtiller [25] found a fundamental connection between quasiconformal mappings and quadratic differentials in his studies of extremal mappings between Riemann surfaces. However, the class of quasiconformal diffeomorphisms is not closed under uniform limits. Thus the generalization to Sobolev spaces is necessary if one is to solve extremal problems. We then find the limit of a bounded sequence of quasiconformal mappings is either quasiconformal or constant. Shortly after Gr6tzsch, Lavrentieff [14] showed the importance of quasiconformal mapping for problems in partial differential equations. Here we study elliptic equations of the form div(AVu) = 0
a.e. in s
where A -- A(z) is uniformly elliptic matrix field measurable in z. If one can find a homeomorphism f with Jacobian matrix D f satisfying the equations ( D f ) t ( D f ) -- d e t ( D f ) 2 A -1
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then it turns out that u -- v ( f ) where v is harmonic. These problems were solved by Lavrentieff for continuous A and finally by C.B. Morrey [ 19], in general. It took another 20 years before Bers recognized that homeomorphic solutions are quasiconformal mappings. Nowadays it is fundamental that the correct approach is to use complex notation. From the analytic definition of a quasiconformal mapping we see that there is a measurable function # defined in S2 such that O f ( z ) - lZ(z)Of (z).
Indeed [I/z l]ec - (K - 1) / (K + 1) < 1. This is what we call the complex Beltrami equation. Notice that when / z - 0, or equivalently K -- 1, we obtain the usual Cauchy-Riemann equations. The function # f -- Of/O f is called the Beltrami coefficient of f or the complex dilatation of f . The Beltrami equation has a long history. Gauss first studied the equation, with (real) analytic #, in the 1820s while investigating the problem of existence of isothermal coordinates on a given surface. In studying solutions to the Beltrami equation an operator now known as the Beurling Ahlfors transform has proved to be important. It is defined by a singular integral of Calder6n-Zygmund type: d_x dZ) y .2 g(z) - Sco(z) - ~- 1 f f c -(~-co(r
Beurling observed that this was a unitary transformation of L 2 (C). It was one of the early successes of Calder6n-Zygmund theory that S is a bounded operator of L p (C). Now, in the sense of distributions, S is the 0 derivative of the Cauchy Transform
h(z) - Too(z) -- -i---~ 1 ffc
~(ff~o(~) d x- d z) y.
However T is not singular but for p > 2 transforms functions in L p (C) to the H61der continuous functions h"
]h(z) -
h(w)] ~< Clz - wl '~,
with oe - o r ( p ) by Sobolev's Theorem. These two operators allow a surprisingly explicit solution to the complex Beltrami equation. Beginning with # with compact support we see that O f = p is of compact support and lies in LP(C) if f has integrable derivatives in L p (C). As the mapping f is conformal at ec we can assume it is asymptotic to z at ec. Thus by the generalized form of Cauchy Theorem f ( z ) -- z + T ( p ) and differentiating Of -- 1 + S(p). So if f is to satisfy the Beltrami equation we obtain m
p - of - ~ + ~s(p). Now suppose that II~ll~llSIIp < 1. Then the equation may be solved for p to give p - (I - # S ) - l # and hence f--z
+ T(I
-/s
+ T(Iz)
+ T(tzSlz) + . . . .
D.H. Hamilton
152
Since ]lSll2 : 1 and IlSllp is logarithmically convex in p then for every # there is a p > 2 so that II~ll~llSIIp < 1. Thus the representing series is convergent. It is not too difficult to go from the representation formula to the existence theorem. The existence theorem for quasiconformal mappings, more recently called the "measurable Riemann mapping theorem", is one of the most fundamental results in the theory and has come to play a central role in modern complex analysis, Teichmtiller theory and complex dynamics. THEOREM 3. Let lz be a measurable function in a domain F2 C C andsuppose II~ll~ < 1. Then there is a quasiconformal mapping f " I-2 --+ C such that lZ is the complex dilatation off. The proof sketched above was done by Ahlfors [2] in 1955 in the case that/z is H61der, the all important measurable case is due to Bojarski (1957). However it was Ahlfors and Bers [3] in 1960 who recognized the crucial fact that f depends holomorphically on/z and used this in Teichmtiller Theory. Let f, g:I-2 --+ C be quasiconformal mappings. The transformation formula for the Beltrami coefficient of a composition of quasiconformal mappings is
lZf(z)-I~g(z) ( O g ( z ) ) 2 ~ f o g - ' (~) --- 1 - lZf(z)~g(z) [Og(z)l
= g(z).
Thus i f / z f --/Zg we conclude that f o g - I 9g(S2) --+ f(12) is a conformal mapping since it is analytic and injective. So we get uniqueness up to a conformal mapping. This formula is a crucial step in the proof of Theorem 1.
3. Holomorphy Astala's original proof made use of notions that appeared in a paper [16] of R. Marl6, P. Sad and D. Sullivan in 1983 that has been dubbed "holomorphic motions". Basically the idea is that a holomorphic family of injections A --+ C of a set A C C is necessarily a quasiconformal mapping. Here is the precise definition. m
DEFINITION 1. A holomorphic motion of a set A C C is a map f ' U x A ~ C such that (i) for each fixed z ~ A, the map )~ --+ f ()~, z) is holomorphic in U, (ii) for each fixed )~ ~ U, the map z ~ f ()~, z) - fz (z) is an injection, and (iii) the mapping f0 is the identity on A. Note that there is no assumption regarding the continuity of f as a function of z or the pair ()~, z). That such continuity occurs is a consequence of the )~-lemma of Marl6, Sad and Sullivan, given here as extended by Slodkowski [24]
Area distortion of quasiconformal mappings
153
m
THEOREM 4. I f f ' U
• A --~ C is a holomorphic motion o f A C C, then f has an extension to F ' U • C -+ C such that (i) F is a holomorphic motion o f C, (ii) F is continuous in U • C -+ C, and (iii) F z ' C --~ C is K - q u a s i c o n f o r m a l w i t h K <~ (1 + I&l)/(1 - I ) ~ l ) f o r e a c h )~ ~ U.
Astala constructs a holomorphic motion of a Cantor type set then obtains his result with the Ruelle-Bowen thermodynamic formalism. In fact holomorphy was important right from the beginning of quasiconformal theory when Teichmtiller used quasiconformal mapping to consider the space of Riemann surfaces of some fixed compact topological type. He showed that so called Teichmtiller space has a natural holomorphic structure. In fact the modern form of this theory was used by Bers and Royden in their proof of one form of the )~-lemma. Actually for the planar quasiconformal mappings the appearance of holomorphy is completely obvious. We have only to think of the space of admissible # as the open unit ball B in L ~ and realize that the mapping 9 "# ----> f -- z + T ( # ) -+- T ( # S ( # ) )
+...
is a holomorphic mapping from B to Wllo'c2(C). This is a central fact in much of the Teichmtiller theory as well as applications to Complex dynamics.
4. The class Z'(k) We define the class Z;(k) functions f which are quasiconformal on C with dilatation # supported on unit disk U = {Izl < 1} satisfying I/zl ~< k, normalized by f ( z ) = z + o(1) at cx~. Variational problems for classes of quasiconformal mappings were considered, from 1960 (see [23] for references). It was proved that S ( k ) is exactly the class of Kquasiconformal mappings represented by f -- z + T(lz) + T ( # S ( # ) ) + . . . with dilatation supported on U with k -- (K - 1) / (K + 1). As k --+ 1 in the limit we obtain the class Z' the classical family of maps conformal on {Izl > 1} with the above normalization. This class is famous for providing the classical distortion theorems for conformal mappings from the famous area theorem: i.e., for f (z) = z + b l z -1 + . . . (X)
~--~nlb,zl 2 ~ 1. n--1
The other side of the area inequality also shows IC - f(Izl > 1)1 ~ Jr with equality only for f = z. Somewhat surprisingly we shall find that this first theorem of geometric function theory plays a role in proving the area theorem for quasiconformal mappings. There is a straightforward way of obtaining estimates on Z' (k) from bounds on Z'. This was first done by Ktihnau [15], see [23] for complete references. From the area theorem we have [bj[ ~< 1 in Z'. Now we apply Schwarz lemma to the holomorphic functional/z --+ bl and obtain the bound Ibjl ~< k in r ( k ) . This is the germ of the idea which enabled a
D.H.Hamilton
154
proof of the area distortion theorem by holomorphy. This discussion is for quasiconformal mappings with dilatation supported on U, however it is valid for the analogous class of quasiconformal mappings with dilatation supported on any compact set A of span cr = 1, i.e., for functions f ( z ) = z + blz -1 . . . . conformal off A the bound I C - f ( C - A)I ~< re is best possible. The case where A is connected (and so has transfinite diameter 1) is essential to the proof of Theorem 1.
5. A strange Harnack inequality The area distortion theorem requires more than Schwarz lemma: LEMMA 1. Let al . . . . . an be positive functions on the unit disk, such that log aj is harmonic and
•
aj ()~) <~ 1,
I)~1 < 1.
j=l
Then for I~l < 1 1
aj(1.)
log
<~ i)-------1+ ~
j=l
j=l
The proof is based on the following well known "entropy" inequality occurring in statistical mechanics. This also occurs in the work of Astala and provides a common ground. Let p j, qj > 0 be probability distributions on {1, 2 . . . . . n }. Then n
pj
--
log qj + E
j=l
PJ log pj >/O.
j=l
The proof is trivial: using the convex function r = x log(x), the left of the inequality becomes Y~ q j ~ ( p j ~q j). So the inequality follows from
Z qjr
/qj ) >/dp(Zqjpj/qj ) =0.
To prove the lemma, for
aj O0 P J = ~aj(~.)
I~1 < 1 and Izl < 1 define and
aj (z) p j = ~_.aj(z)
Then for fixed ~. the function
H (z) = - E
PJ log aj (z) + E
PJ log pj
Area distortion of quasiconformal mappings
155
is harmonic in z. By the "entropy" inequality H (z) ~> - log E
aj (z) >~O.
So that the classical Harnack inequality gives H (z) ~>
1-1zl
1 + Izl
H (0).
Finally putting z -- )~ and using the "entropy" inequality again H()~)---l~
1-
1+
~ 1 +l)~ ( - Z p j l ~ 1 7 6 ( -l~176
(0)),
which proves Lemma 1.
6. Theorem 1, part I Actually Theorem 1 has two distinct cases. The first part is the heart of the problem. Let A be a compact set of span 1. Define S* to be the normalized conformal mappings and ZT*(k) the corresponding quasiconformal mappings f conformal off A (if there are any). THEOREM 5. Suppose f ~ Z* is a K-quasiconformal mapping. Then for all Borel measurable sets E C A such that f is conformal on E (i.e., the dilatation lZ = 0 a.e. on E): If(E)] ~< 7Cl-l/glEll/g. Without loss of generality f is smooth, since smooth quasiconformal mappings are Wllo'2(C) dense in the space of all K-quasiconformal mappings. (This is not true for dimension n ~> 4.) In particular, we may assume the dilatation p is smooth and supported on A. So proving the theorem for the smooth case gives a uniform bound for the general case and proves the theorem. Now for Ikl < 1 define Kz = (1 + I~1)/(1 - I~1) and K+I K-1 so with )~ = ( K - 1)/(K + 1) we have #x = # . N o w let f z ( z ) E S * have dilatation/zz by using the standard solution of the Beltrami equation: fz = z + T(lzz) + T(lzxSlzz) + . . . .
156
D.H. Hamilton
The function fz has Jacobian
Jz- IOzf~12(1-1~zl 2) which is everywhere nonzero as # is smooth. However by Holomorphy morphic in )~. Therefore the function
Ozfz
is holo-
1 12 a(z,~.)----IOzfz has the property that l o g a ( z , ) 0 is harmonic in )~. Furthermore if f is conformal on E we have lzz = 0 on E and hence Jz/rr = a(z, )0. Also by the Area theorem for 27* (i.e., definition of span)
f f E a (~., z ) d x d y -- f f F --J~d x d y <~f f --J~d x d y
If~(A)l
~<1.
Therefore a (z,)~) satisfies the continuous version of L e m m a 1 (i.e., we integrate over E instead of sum over j ) and so log
Jx dx dy :r
<~ ~ ~ log 1 +lXl
.
Setting )~ = (K - 1) / (K + 1) then proves the theorem.
7. Theorem 1, part 2 The complementary result we need to prove is
Suppose f E r* is a K-quasiconformal mapping. Then for all Borel measurable sets E C A such that f is conformal on C - E:
THEOREM 6.
I f ( E ) [ ~< KIEI. The argument here is due to Gehring and Reich. It begins with the observation that as the Beurling transform S is unitary then for any set G, by Cauchy-Schwarz,
f f lS(xG)] dx dy <~IG[. Also S is (almost) self-adjoint so for any function p supported on G we have
f f [s(p)l dx dy <~Ilpllo~lGI, which can be regarded as the basic lemma.
Area distortion of quasiconformal mappings
157
The main idea is to set up a deformation family of quasiconformal mappings and integrate this inequality. As before the uniform bound is proved for sufficiently smooth mappings and this is enough. Here the deformation parameter is a real variable t 6 [0, 1). For fixed function /z, supported on E define /zt = t/z with corresponding normalized mappings ft 6 r * ( t ) . In particular f0 = z and f is obtained for t = (K - 1 ) / ( K + 1). Taking the infinitesimal form of the composition formula for dilatations we may write
a f, Ot
- - g t o ft,
where gt is the infinitesimal deformation given by
gt (z) = z + T (Pt ),
Pt =
i~ o ft -1
e2i arg(0r, j~-l )
1 - t2l/z o f t - l l 2
Now as 0 T -- S we see that dlft(E)l = 2~(ff
dt
t S(pt)dxdy). .)(E)
The above bound on S easily implies
Ift(E)l 1 -t 2 '
dlfi(E)l
dt which integrates to give l+t
~' ~tJ"E'~ ~< 1 - t
IEI,
proving the theorem. It remains to complete the proof of Theorem 1. Thus f is quasiconformal with dilatation # supported on compact set A of span 1. We decompose f into g o h where h is conformal on C - E and g is conformal on h(E) (and C - h(A)). Thus by the composition formula h has dilatation # on E and zero elsewhere. While g has dilatation v(z) on C - h(E) and zero elsewhere. Now ]#(h -1 (z))] = Iv(z)l. This both g and h are K-quasiconformal. We may take g and h to be normalized at o,z, in particular h (A) has span 1. Therefore applying Theorem 5 to h yields ]h(E)] < KIEI while Theorem 4 gives
[g o h(E)[ <~rc 1-1/g K1/glE]l/K, which is what we wanted.
D.H. Hamilton
158
8. Bounds on the Beurling-Ahlfors transform
It is well known that Astala's theorem and the consequent regularity theory for quasiconformal mappings would follow, in the sharpest possible form, from the conjectured values of the p-norms of the Beurling-Ahlfors transform. The following is the natural conjecture (see T. Iwaniec and G.J. Martin [13]). The p-norms of the Beurling-Ahlfors transform S:L p ~ L p satisfy IISIIp = P - 1 if p ~> 2 and (p - 1) -1 if p ~< 2. The calculation of the p-norms of the Beurling-Ahlfors transform remains one of the outstanding problems in the area. For the case p = oc we have S is an operator of BMO. Thus the maximal function of a function co 6 L ~ satisfies
m(t) -
I{z 6 U" 9](So)) > t }l ~< e x p ( - C t ) .
Sharp bounds come from considering any measurable E C U and proving
f l u -E
]S(XE) [ dx
dy <~IEI log
Jr
IEI
This sharp bound due to Eremenko and Hamilton refines an earlier result of Astala. The proof is also by holomorphic deformation which like the other proofs in this article is simple enough to include: For any function # supported on U - E with II/zlloc ~ 1 we define/zz -- )~/z and the corresponding family of normalized mappings fz. This time we let positive )~ --+ 0 to find that Ifzl = IEI
+29]()~ffs(#)dxdy)+
o()~)~ jr2)~+~
1-2)~+~
7[
= IEI + 2l)~llfl log ~
+ o()0,
by Theorem 4. Hence we obtain Jr
f f s(lz) dx dy
Igl log IE--/'
for all # supported on U - E and bounded by 1. This gives our result as S is unitary.
REMARK. In the twentieth century the operator here referred to as the Beurling Ahlfors transform has been also called complex (or the two dimensional) Hilbert transform, in harmonic analysis it is usually just the Beurling transform.
Area distortion of quasiconformal mappings
159
9. Further applications Astala's theorem has applications to the L 1-theory of analytic functions, quadratic differentials and critical values of harmonic functions, see Iwaniec [13]. Also, by results of Lavrentieff, Bers and others the solutions to the elliptic differential equations d i v ( A ( x ) V u ) = 0 can also be considered. Therefore Astala's theorem yields sharp exponents of integrability on the gradient Vu; note that the dilatation of f and so necessarily the optimal integrability exponent depends in a complicated manner on all the entries of the matrix A rather than just on its ellipticity coefficient. We conclude by sketching some results of applied mathematics where the sharp constants of Theorem 5 are needed to produce optimal bounds. Composite materials consist of two (or more) phases of materials with different physical properties (conductivity, stiffness etc.) with some fine scale structure (e.g., laminate). The problem is to determine the large scale physical properties. The material can be realized as a measurable matrix field cr = XEO'I + XEc~r2
where E, E C are complementary subsets of fundamental square [0, 1] 2 and o-j are measurable positive definite matrices subject only to the restriction that crj has prescribed eigenvalues )~j,1, )~j,2 and also that E has prescribed volume fraction IEI = p. Then cr is extended to C by periodicity. For smooth f we solve the pde div(o ( Z ) v u ~ ) - f and letting e --+ 0 seek the weak limit ue --+ u0 which satisfies div(cr0Vu0) = f for some constant matrix o0. The problem is to determine the region of possible values for the eigenvalues )~1, ~,2 of or0 in terms of )~j, 1, ~,j,2 and p. Using the sharp bounds of the Eremenko-Hamilton inequality, the first results were given by Nesi [20], extended by Astala and Mietten [6], before the optimal bounds were completed by Milton and Nesi [ 17].
References [1] L.V. Ahlfors, On quasiconformal mappings, J. Analyse Math. 3 (1954), 1-58. [2] L.V. Ahlfors, Conformality with respect to Riemannian metrics, Ann. Acad. Sci. Fenn. Ser. AI Math. 206 (1955). [3] L.V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. Math. 72 (1960), 385404. [4] K. Astala, Area distortion ofquasiconformal mappings, Acta Math. 173 (1994), 37-60. [5] K. Astala and M. Mietten, On quasiconformal mappings and 2-dimensional G-closure problems, Arch. Rational Mech. Anal. 143 (1998), 207-240.
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[6] J. Becker and C. Pommerenke, On the Hausdorff measure of quasicircles, Ann. Acad. Sci. Fenn. Ser. AI Math. 12 (1987), 329-333. [7] B.V. Boyarskii, Generalised solutions of differential equations of first order and elliptic type with discontinuous coefficients, Mat. Sb. 43 (85) (1957), 451-503 (in Russian). [8] A.E. Eremenko and D.H. Hamilton, On the Area distortion by quasiconformal mappings, Proc. Amer. Math. Soc. 123 (9) (1995), 2793-2797. [9] EW. Gehring and E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. AI Math. 388 (1966), 1-14. [ 10] EW. Gehring and J. Vais~il~i,Hausdorffdimension and quasiconformal mappings, J. London Math. Soc. (2) 6 (1975), 504-512. [11] H. Gr6tzsch, Ober die Verzerrung bei schlichten nichtkonformen Abbildungen und iiber eine damit zusammenhiingende Erweiterung des Picardschen Satzes, Ber. Verh. S~ichs. Akad. Wiss. Leipzig 80 (1928), 503-507. [12] T. Iwaniec, LP-theory of quasiregular mappings, Quasiconformal Space Mappings, A Collection of Surveys, 1960-1990, Lecture Notes in Math., Vol. 1508, Springer, Berlin (1992), 39--40. [13] T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1992), 29-81. [14] M.A. Lavrentieff, Sur une classe de representations continues, Rec. Math. 42 (1935), 407--423. [15] R. Ktihnau, Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsabhiingiger Dilatationsbeschriinkung, Math. Nachr. 40 (1969), 1-11. [16] R. Marl6, E Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. t~cole Norm. Sup. (4) 16 (2) (1983), 193-217. [17] G.W. Milton and V. Nesi, Optimal G-closure bounds via stability under lamination, Preprint (1998). [18] A. Moil, On an absolute constant in the theory of quasiconformal mapping, J. Math. Soc. Japan 8 (1956), 156-166. [19] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166. [20] V. Nesi, Quasiconformal mappings as a tool to study certain two dimensional G-closure problems, Arch. Rational Mech. Anal. 143 (1996), 17-51. [21] I.N. Pesin, On the theory of generalized K-quasiconformal mappings, Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 223-224 (in Russian). [22] E. Reich, Some estimates for the two dimensional Hilbert transform, J. Analyse 18 (1967), 279-293. [23] G. Schober, Univalent Functions - Selected Topics, Lecture Notes in Math., Vol. 478, Springer, Berlin (1975). [24] Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (2) (1991), 347355. [25] O. Teichmiiller, Gesammelte Abhandlungen, Springer, Berlin (1982).
5
Area Distortion of Quasiconformal Mappings
D.H. Hamilton Department of Mathematics, University of Maryland, College Park, MD 20742, USA E-mail: dhh @math. umd. edu
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Gr6tzsch to Bojarski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphy ................................................... The class Z'(k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. A strange Harnack inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Theorem 1, part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. T h e o r e m 1, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Bounds on the B e u r l i n g - A h l f o r s transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Further applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H A N D B O O K O F C O M P L E X A N A L Y S I S : G E O M E T R I C F U N C T I O N THEORY, VOLUME 1 Edited by R. Kfihnau 9 2002 Elsevier Science B.V. All rights reserved 147
149 150 152 153 154 155 156 158 159 159
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1. Introduction
A homeomorphism f : C --+ C is called K-quasiconformal if its directional derivatives satisfy max~ IO~f(x)l <<.K mind IO~f(x)l a.e. (with respect to area measure). It is also required that f be area preserving, i.e., if we use IEI to denote the area of any (measurable) planar set E:
IEI--0
~
If(E)[ -- O. 1,2
For quasiconformal mappings this means that f is in the Sobolev class Wloc , i.e., functions whose first derivatives are square integrable. There is the equivalent geometric definition that f have "bounded distortion". This is measured by two measures of stretching: t f(~,r)-
maxl f(x + Ihl=r
h)
-
f(x) l,
If(x,r)-
min IhI-r
If(x
§ h)-
f(x)l.
The infinitesimal distortion of f at x is H i ( x ) = limsuPr~oLf(x,r)/lf(x,r). If f is conformal, then Hf(x) = 1 (and the converse is true). This reflects the fact that infinitesimally conformal mappings preserve circles. Unfortunately this elegant geometric definition is tricky to work with so the analytic definition is more common. The equivalence between the geometric definition and the analytic definition was shown by Pesin [21] in 1956. If quasiconformal mappings were merely generalizations of conformal maps their theory would be a curiosity. In fact they are important for some of the most significant mathematical theory of the century. It has been known since the work of Ahlfors [1] and Mori [18] in 1955 that K-quasiconformal mappings are locally H61der continuous with H61der exponent 1/K. The mapping z --+ zlzl 1/K-1 shows that this is best possible. Bojarski [7], as a consequence of his fundamental existence theorem, showed that K-quasiconformal mappings actually belong Wllo'cp for some p -- p ( K ) > 1. This is equivalent to the fact that f distorts area by a power depending only on K. The above example shows one might guess that the optimal exponent in area distortion is also 1/K. The following theorem was conjectured and formulated by Gehring and Reich [9]. Let U be the unit disk {Izl < 1}. In 1992 Astala [4] proved. THEOREM 1. Suppose f : U - - + U is a K-quasiconformal mapping with f (O)= O. Then we have ]f(E)[ ~< MIE] 1/K for all Borel measurable sets E C U. Moreover, the constant M = M ( K ) depends only on K with M ( K ) = 1 + O(K - 1). Although this theorem is only for self maps of the disk a simple trick allows a factorization which proves similar bounds depending only on the normalization. As a straightforward consequence of Astala's theorem:
sup/p'! f ~ Wll~ } -
2K K- 1
150
D.H. Hamilton
Also there is the optimal result for the distortion of Hausdorff dimension: 1t
1 Dim(E)
1
~)
~< D i m ( f (E))
1 2 ~< K
(
1 Dim(E)
~) "
The original proof of Astala depended on deep ideas from Dynamics: relating Hausdorff dimension of holomorphically varying Cantor type sets. Shortly afterwards a short direct proof was obtained by A.E. Eremenko and D.H. Hamilton [8], as a direct consequence of the holomorphy of the class of all quasiconformal mappings. In this case the normalization is f is conformal off U with f (z) = z + o(1) near o~ (i.e., the classical class 27). THEOREM 2. Suppose f E r is a K-quasiconformal mapping. Then we have [f(E)l ~< K 1/K Jr 1-1 / K IEI 1/ x for all Borel measurable sets E C U. We shall see that this bound is a combination of Theorem 5 (the case where f is conformal on E and Theorem 6 (where f is conformal off E), both of which are sharp. Theorem 2 corrects a typo in [8] where the 1 / K power of K was omitted. (Also it was stated that Theorem 1 in [8] had best possible constants, whereas it was meant that inequalities of Theorems 5 and 6 (only) were sharp.) As well as having applications to analysis (we shall later give sharp bounds for the Beurling Ahlfors Transform) these inequalities, in particular, the constant of Theorem 5, have proved important in applied mathematics: namely to the theory of determining optimal bounds of physical properties (heat/electrical conductivity, magnetic permeability, elastic stiffness) of compound solids consisting of two or more different materials combined together with some microstructure.
2. From Griitzsch to Bojarski Diffeomorphisms with uniform bounded distortion were first studied around 1928 by H. Gr6tzsch [11]. In 1939 O. Teichmtiller [25] found a fundamental connection between quasiconformal mappings and quadratic differentials in his studies of extremal mappings between Riemann surfaces. However, the class of quasiconformal diffeomorphisms is not closed under uniform limits. Thus the generalization to Sobolev spaces is necessary if one is to solve extremal problems. We then find the limit of a bounded sequence of quasiconformal mappings is either quasiconformal or constant. Shortly after Gr6tzsch, Lavrentieff [14] showed the importance of quasiconformal mapping for problems in partial differential equations. Here we study elliptic equations of the form div(AVu) = 0
a.e. in s
where A -- A(z) is uniformly elliptic matrix field measurable in z. If one can find a homeomorphism f with Jacobian matrix D f satisfying the equations ( D f ) t ( D f ) -- d e t ( D f ) 2 A -1
Area distortion of quasiconformal mappings
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then it turns out that u -- v ( f ) where v is harmonic. These problems were solved by Lavrentieff for continuous A and finally by C.B. Morrey [ 19], in general. It took another 20 years before Bers recognized that homeomorphic solutions are quasiconformal mappings. Nowadays it is fundamental that the correct approach is to use complex notation. From the analytic definition of a quasiconformal mapping we see that there is a measurable function # defined in S2 such that O f ( z ) - lZ(z)Of (z).
Indeed [I/z l]ec - (K - 1) / (K + 1) < 1. This is what we call the complex Beltrami equation. Notice that when / z - 0, or equivalently K -- 1, we obtain the usual Cauchy-Riemann equations. The function # f -- Of/O f is called the Beltrami coefficient of f or the complex dilatation of f . The Beltrami equation has a long history. Gauss first studied the equation, with (real) analytic #, in the 1820s while investigating the problem of existence of isothermal coordinates on a given surface. In studying solutions to the Beltrami equation an operator now known as the Beurling Ahlfors transform has proved to be important. It is defined by a singular integral of Calder6n-Zygmund type: d_x dZ) y .2 g(z) - Sco(z) - ~- 1 f f c -(~-co(r
Beurling observed that this was a unitary transformation of L 2 (C). It was one of the early successes of Calder6n-Zygmund theory that S is a bounded operator of L p (C). Now, in the sense of distributions, S is the 0 derivative of the Cauchy Transform
h(z) - Too(z) -- -i---~ 1 ffc
~(ff~o(~) d x- d z) y.
However T is not singular but for p > 2 transforms functions in L p (C) to the H61der continuous functions h"
]h(z) -
h(w)] ~< Clz - wl '~,
with oe - o r ( p ) by Sobolev's Theorem. These two operators allow a surprisingly explicit solution to the complex Beltrami equation. Beginning with # with compact support we see that O f = p is of compact support and lies in LP(C) if f has integrable derivatives in L p (C). As the mapping f is conformal at ec we can assume it is asymptotic to z at ec. Thus by the generalized form of Cauchy Theorem f ( z ) -- z + T ( p ) and differentiating Of -- 1 + S(p). So if f is to satisfy the Beltrami equation we obtain m
p - of - ~ + ~s(p). Now suppose that II~ll~llSIIp < 1. Then the equation may be solved for p to give p - (I - # S ) - l # and hence f--z
+ T(I
-/s
+ T(Iz)
+ T(tzSlz) + . . . .
D.H. Hamilton
152
Since ]lSll2 : 1 and IlSllp is logarithmically convex in p then for every # there is a p > 2 so that II~ll~llSIIp < 1. Thus the representing series is convergent. It is not too difficult to go from the representation formula to the existence theorem. The existence theorem for quasiconformal mappings, more recently called the "measurable Riemann mapping theorem", is one of the most fundamental results in the theory and has come to play a central role in modern complex analysis, Teichmtiller theory and complex dynamics. THEOREM 3. Let lz be a measurable function in a domain F2 C C andsuppose II~ll~ < 1. Then there is a quasiconformal mapping f " I-2 --+ C such that lZ is the complex dilatation off. The proof sketched above was done by Ahlfors [2] in 1955 in the case that/z is H61der, the all important measurable case is due to Bojarski (1957). However it was Ahlfors and Bers [3] in 1960 who recognized the crucial fact that f depends holomorphically on/z and used this in Teichmtiller Theory. Let f, g:I-2 --+ C be quasiconformal mappings. The transformation formula for the Beltrami coefficient of a composition of quasiconformal mappings is
lZf(z)-I~g(z) ( O g ( z ) ) 2 ~ f o g - ' (~) --- 1 - lZf(z)~g(z) [Og(z)l
= g(z).
Thus i f / z f --/Zg we conclude that f o g - I 9g(S2) --+ f(12) is a conformal mapping since it is analytic and injective. So we get uniqueness up to a conformal mapping. This formula is a crucial step in the proof of Theorem 1.
3. Holomorphy Astala's original proof made use of notions that appeared in a paper [16] of R. Marl6, P. Sad and D. Sullivan in 1983 that has been dubbed "holomorphic motions". Basically the idea is that a holomorphic family of injections A --+ C of a set A C C is necessarily a quasiconformal mapping. Here is the precise definition. m
DEFINITION 1. A holomorphic motion of a set A C C is a map f ' U x A ~ C such that (i) for each fixed z ~ A, the map )~ --+ f ()~, z) is holomorphic in U, (ii) for each fixed )~ ~ U, the map z ~ f ()~, z) - fz (z) is an injection, and (iii) the mapping f0 is the identity on A. Note that there is no assumption regarding the continuity of f as a function of z or the pair ()~, z). That such continuity occurs is a consequence of the )~-lemma of Marl6, Sad and Sullivan, given here as extended by Slodkowski [24]
Area distortion of quasiconformal mappings
153
m
THEOREM 4. I f f ' U
• A --~ C is a holomorphic motion o f A C C, then f has an extension to F ' U • C -+ C such that (i) F is a holomorphic motion o f C, (ii) F is continuous in U • C -+ C, and (iii) F z ' C --~ C is K - q u a s i c o n f o r m a l w i t h K <~ (1 + I&l)/(1 - I ) ~ l ) f o r e a c h )~ ~ U.
Astala constructs a holomorphic motion of a Cantor type set then obtains his result with the Ruelle-Bowen thermodynamic formalism. In fact holomorphy was important right from the beginning of quasiconformal theory when Teichmtiller used quasiconformal mapping to consider the space of Riemann surfaces of some fixed compact topological type. He showed that so called Teichmtiller space has a natural holomorphic structure. In fact the modern form of this theory was used by Bers and Royden in their proof of one form of the )~-lemma. Actually for the planar quasiconformal mappings the appearance of holomorphy is completely obvious. We have only to think of the space of admissible # as the open unit ball B in L ~ and realize that the mapping 9 "# ----> f -- z + T ( # ) -+- T ( # S ( # ) )
+...
is a holomorphic mapping from B to Wllo'c2(C). This is a central fact in much of the Teichmtiller theory as well as applications to Complex dynamics.
4. The class Z'(k) We define the class Z;(k) functions f which are quasiconformal on C with dilatation # supported on unit disk U = {Izl < 1} satisfying I/zl ~< k, normalized by f ( z ) = z + o(1) at cx~. Variational problems for classes of quasiconformal mappings were considered, from 1960 (see [23] for references). It was proved that S ( k ) is exactly the class of Kquasiconformal mappings represented by f -- z + T(lz) + T ( # S ( # ) ) + . . . with dilatation supported on U with k -- (K - 1) / (K + 1). As k --+ 1 in the limit we obtain the class Z' the classical family of maps conformal on {Izl > 1} with the above normalization. This class is famous for providing the classical distortion theorems for conformal mappings from the famous area theorem: i.e., for f (z) = z + b l z -1 + . . . (X)
~--~nlb,zl 2 ~ 1. n--1
The other side of the area inequality also shows IC - f(Izl > 1)1 ~ Jr with equality only for f = z. Somewhat surprisingly we shall find that this first theorem of geometric function theory plays a role in proving the area theorem for quasiconformal mappings. There is a straightforward way of obtaining estimates on Z' (k) from bounds on Z'. This was first done by Ktihnau [15], see [23] for complete references. From the area theorem we have [bj[ ~< 1 in Z'. Now we apply Schwarz lemma to the holomorphic functional/z --+ bl and obtain the bound Ibjl ~< k in r ( k ) . This is the germ of the idea which enabled a
D.H.Hamilton
154
proof of the area distortion theorem by holomorphy. This discussion is for quasiconformal mappings with dilatation supported on U, however it is valid for the analogous class of quasiconformal mappings with dilatation supported on any compact set A of span cr = 1, i.e., for functions f ( z ) = z + blz -1 . . . . conformal off A the bound I C - f ( C - A)I ~< re is best possible. The case where A is connected (and so has transfinite diameter 1) is essential to the proof of Theorem 1.
5. A strange Harnack inequality The area distortion theorem requires more than Schwarz lemma: LEMMA 1. Let al . . . . . an be positive functions on the unit disk, such that log aj is harmonic and
•
aj ()~) <~ 1,
I)~1 < 1.
j=l
Then for I~l < 1 1
aj(1.)
log
<~ i)-------1+ ~
j=l
j=l
The proof is based on the following well known "entropy" inequality occurring in statistical mechanics. This also occurs in the work of Astala and provides a common ground. Let p j, qj > 0 be probability distributions on {1, 2 . . . . . n }. Then n
pj
--
log qj + E
j=l
PJ log pj >/O.
j=l
The proof is trivial: using the convex function r = x log(x), the left of the inequality becomes Y~ q j ~ ( p j ~q j). So the inequality follows from
Z qjr
/qj ) >/dp(Zqjpj/qj ) =0.
To prove the lemma, for
aj O0 P J = ~aj(~.)
I~1 < 1 and Izl < 1 define and
aj (z) p j = ~_.aj(z)
Then for fixed ~. the function
H (z) = - E
PJ log aj (z) + E
PJ log pj
Area distortion of quasiconformal mappings
155
is harmonic in z. By the "entropy" inequality H (z) ~> - log E
aj (z) >~O.
So that the classical Harnack inequality gives H (z) ~>
1-1zl
1 + Izl
H (0).
Finally putting z -- )~ and using the "entropy" inequality again H()~)---l~
1-
1+
~ 1 +l)~ ( - Z p j l ~ 1 7 6 ( -l~176
(0)),
which proves Lemma 1.
6. Theorem 1, part I Actually Theorem 1 has two distinct cases. The first part is the heart of the problem. Let A be a compact set of span 1. Define S* to be the normalized conformal mappings and ZT*(k) the corresponding quasiconformal mappings f conformal off A (if there are any). THEOREM 5. Suppose f ~ Z* is a K-quasiconformal mapping. Then for all Borel measurable sets E C A such that f is conformal on E (i.e., the dilatation lZ = 0 a.e. on E): If(E)] ~< 7Cl-l/glEll/g. Without loss of generality f is smooth, since smooth quasiconformal mappings are Wllo'2(C) dense in the space of all K-quasiconformal mappings. (This is not true for dimension n ~> 4.) In particular, we may assume the dilatation p is smooth and supported on A. So proving the theorem for the smooth case gives a uniform bound for the general case and proves the theorem. Now for Ikl < 1 define Kz = (1 + I~1)/(1 - I~1) and K+I K-1 so with )~ = ( K - 1)/(K + 1) we have #x = # . N o w let f z ( z ) E S * have dilatation/zz by using the standard solution of the Beltrami equation: fz = z + T(lzz) + T(lzxSlzz) + . . . .
156
D.H. Hamilton
The function fz has Jacobian
Jz- IOzf~12(1-1~zl 2) which is everywhere nonzero as # is smooth. However by Holomorphy morphic in )~. Therefore the function
Ozfz
is holo-
1 12 a(z,~.)----IOzfz has the property that l o g a ( z , ) 0 is harmonic in )~. Furthermore if f is conformal on E we have lzz = 0 on E and hence Jz/rr = a(z, )0. Also by the Area theorem for 27* (i.e., definition of span)
f f E a (~., z ) d x d y -- f f F --J~d x d y <~f f --J~d x d y
If~(A)l
~<1.
Therefore a (z,)~) satisfies the continuous version of L e m m a 1 (i.e., we integrate over E instead of sum over j ) and so log
Jx dx dy :r
<~ ~ ~ log 1 +lXl
.
Setting )~ = (K - 1) / (K + 1) then proves the theorem.
7. Theorem 1, part 2 The complementary result we need to prove is
Suppose f E r* is a K-quasiconformal mapping. Then for all Borel measurable sets E C A such that f is conformal on C - E:
THEOREM 6.
I f ( E ) [ ~< KIEI. The argument here is due to Gehring and Reich. It begins with the observation that as the Beurling transform S is unitary then for any set G, by Cauchy-Schwarz,
f f lS(xG)] dx dy <~IG[. Also S is (almost) self-adjoint so for any function p supported on G we have
f f [s(p)l dx dy <~Ilpllo~lGI, which can be regarded as the basic lemma.
Area distortion of quasiconformal mappings
157
The main idea is to set up a deformation family of quasiconformal mappings and integrate this inequality. As before the uniform bound is proved for sufficiently smooth mappings and this is enough. Here the deformation parameter is a real variable t 6 [0, 1). For fixed function /z, supported on E define /zt = t/z with corresponding normalized mappings ft 6 r * ( t ) . In particular f0 = z and f is obtained for t = (K - 1 ) / ( K + 1). Taking the infinitesimal form of the composition formula for dilatations we may write
a f, Ot
- - g t o ft,
where gt is the infinitesimal deformation given by
gt (z) = z + T (Pt ),
Pt =
i~ o ft -1
e2i arg(0r, j~-l )
1 - t2l/z o f t - l l 2
Now as 0 T -- S we see that dlft(E)l = 2~(ff
dt
t S(pt)dxdy). .)(E)
The above bound on S easily implies
Ift(E)l 1 -t 2 '
dlfi(E)l
dt which integrates to give l+t
~' ~tJ"E'~ ~< 1 - t
IEI,
proving the theorem. It remains to complete the proof of Theorem 1. Thus f is quasiconformal with dilatation # supported on compact set A of span 1. We decompose f into g o h where h is conformal on C - E and g is conformal on h(E) (and C - h(A)). Thus by the composition formula h has dilatation # on E and zero elsewhere. While g has dilatation v(z) on C - h(E) and zero elsewhere. Now ]#(h -1 (z))] = Iv(z)l. This both g and h are K-quasiconformal. We may take g and h to be normalized at o,z, in particular h (A) has span 1. Therefore applying Theorem 5 to h yields ]h(E)] < KIEI while Theorem 4 gives
[g o h(E)[ <~rc 1-1/g K1/glE]l/K, which is what we wanted.
D.H. Hamilton
158
8. Bounds on the Beurling-Ahlfors transform
It is well known that Astala's theorem and the consequent regularity theory for quasiconformal mappings would follow, in the sharpest possible form, from the conjectured values of the p-norms of the Beurling-Ahlfors transform. The following is the natural conjecture (see T. Iwaniec and G.J. Martin [13]). The p-norms of the Beurling-Ahlfors transform S:L p ~ L p satisfy IISIIp = P - 1 if p ~> 2 and (p - 1) -1 if p ~< 2. The calculation of the p-norms of the Beurling-Ahlfors transform remains one of the outstanding problems in the area. For the case p = oc we have S is an operator of BMO. Thus the maximal function of a function co 6 L ~ satisfies
m(t) -
I{z 6 U" 9](So)) > t }l ~< e x p ( - C t ) .
Sharp bounds come from considering any measurable E C U and proving
f l u -E
]S(XE) [ dx
dy <~IEI log
Jr
IEI
This sharp bound due to Eremenko and Hamilton refines an earlier result of Astala. The proof is also by holomorphic deformation which like the other proofs in this article is simple enough to include: For any function # supported on U - E with II/zlloc ~ 1 we define/zz -- )~/z and the corresponding family of normalized mappings fz. This time we let positive )~ --+ 0 to find that Ifzl = IEI
+29]()~ffs(#)dxdy)+
o()~)~ jr2)~+~
1-2)~+~
7[
= IEI + 2l)~llfl log ~
+ o()0,
by Theorem 4. Hence we obtain Jr
f f s(lz) dx dy
Igl log IE--/'
for all # supported on U - E and bounded by 1. This gives our result as S is unitary.
REMARK. In the twentieth century the operator here referred to as the Beurling Ahlfors transform has been also called complex (or the two dimensional) Hilbert transform, in harmonic analysis it is usually just the Beurling transform.
Area distortion of quasiconformal mappings
159
9. Further applications Astala's theorem has applications to the L 1-theory of analytic functions, quadratic differentials and critical values of harmonic functions, see Iwaniec [13]. Also, by results of Lavrentieff, Bers and others the solutions to the elliptic differential equations d i v ( A ( x ) V u ) = 0 can also be considered. Therefore Astala's theorem yields sharp exponents of integrability on the gradient Vu; note that the dilatation of f and so necessarily the optimal integrability exponent depends in a complicated manner on all the entries of the matrix A rather than just on its ellipticity coefficient. We conclude by sketching some results of applied mathematics where the sharp constants of Theorem 5 are needed to produce optimal bounds. Composite materials consist of two (or more) phases of materials with different physical properties (conductivity, stiffness etc.) with some fine scale structure (e.g., laminate). The problem is to determine the large scale physical properties. The material can be realized as a measurable matrix field cr = XEO'I + XEc~r2
where E, E C are complementary subsets of fundamental square [0, 1] 2 and o-j are measurable positive definite matrices subject only to the restriction that crj has prescribed eigenvalues )~j,1, )~j,2 and also that E has prescribed volume fraction IEI = p. Then cr is extended to C by periodicity. For smooth f we solve the pde div(o ( Z ) v u ~ ) - f and letting e --+ 0 seek the weak limit ue --+ u0 which satisfies div(cr0Vu0) = f for some constant matrix o0. The problem is to determine the region of possible values for the eigenvalues )~1, ~,2 of or0 in terms of )~j, 1, ~,j,2 and p. Using the sharp bounds of the Eremenko-Hamilton inequality, the first results were given by Nesi [20], extended by Astala and Mietten [6], before the optimal bounds were completed by Milton and Nesi [ 17].
References [1] L.V. Ahlfors, On quasiconformal mappings, J. Analyse Math. 3 (1954), 1-58. [2] L.V. Ahlfors, Conformality with respect to Riemannian metrics, Ann. Acad. Sci. Fenn. Ser. AI Math. 206 (1955). [3] L.V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. Math. 72 (1960), 385404. [4] K. Astala, Area distortion ofquasiconformal mappings, Acta Math. 173 (1994), 37-60. [5] K. Astala and M. Mietten, On quasiconformal mappings and 2-dimensional G-closure problems, Arch. Rational Mech. Anal. 143 (1998), 207-240.
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