CHAPTER 1. Preliminaries

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1 CHAPTER 1 Preliminaries We collect here some definitions and properties of plane quasiconformal mappings. Two basic references for this material are the books by Ahlfors [7] andlehto and Virtanen [117], to which we refer the reader for further details. A more recent title is the book by Astala, Iwaniec, and Martin [16]. In what follows R denotes the Euclidean plane with its usual identification with the complex plane C. The one-point compactification R = R { } is equipped with the chordal metric ch(z, w) = z w z +1 w +1, where we employ the usual conventions regarding. Let D and D be subdomains of R. We will assume, unless stated otherwise, that card(r \ D). The exterior of D is denoted by D = R \ D. LetB(z, r) be the open Euclidean disk with center z R and radius r, and let B be the unit disk B(0, 1). Finally, H will denote the upper or right half-planes {z = x + iy : y>0} or {z = x + iy : x>0} Quasiconformal mappings There are several different ways to view a quasiconformal mapping. Perhaps the most geometrically intuitive is in terms of the linear dilatation of a homeomorphism. Suppose that f : D D is a homeomorphism. For z D \{,f 1 ( )} and 0 <r<dist(z, D) welet l f (z, r) = min f(z) f(w), z w =r (1.1.1) L f (z, r) = max f(z) f(w) z w =r and call L f (z, r) H f (z) = lim sup r 0 l f (z, r) the linear dilatation of f at z. See Figure 1.1. Recall that a homeomorphism in R is either sense-preserving or sense-reversing [117]. Menchoff showed in 1937 [19] thatifd, D R, a sense-preserving homeomorphism f : D D is analytic, and hence conformal, whenever (1.1.) H f (z) =1 for all but a countable set of z D. The following definition for quasiconformality is a natural counterpart of Menchoff s theorem (Gehring [47]). 3

2 4 1. PRELIMINARIES Figure 1.1 Definition A homeomorphism f : D D is K-quasiconformal where 1 K< if H f (z) < for every z D \{,f 1 ( )} and H f (z) K almost everywhere in D. The inequality in the above definition can be weakened significantly to yield the same class of mappings. Letting h f (z) = lim inf r 0 L f (z, r) l f (z, r), where l f and L f are as in (1.1.1), Heinonen and Koskela [81] and Kallunki and Koskela [96] obtained the following surprising result. Theorem A homeomorphism f : D D is K-quasiconformal, where 1 K<, ifh f (z) < for every z D \{,f 1 ( )} and h f (z) K almost everywhere in D. The next results, which can be found in Lehto-Virtanen [117], identify mappings which are 1-quasiconformal or which are the composition and inverses of quasiconformal mappings. Theorem A homeomorphism f : D D is 1-quasiconformal if and only if f or its complex conjugate f is a conformal mapping, i.e., analytic in D \ {,f 1 ( )}. Theorem If f : D D is K 1 -quasiconformal and g : D D is K -quasiconformal, then g f : D D is K 1 K -quasiconformal. The inverse of a K-quasiconformal mapping is K-quasiconformal. Menchoff s theorem asserts that a sense-preserving homeomorphism f of D is a conformal mapping if, except at a countable set of points z D, f maps infinitesimal circles about z onto infinitesimal circles about f(z). Theorems and extend this result by first replacing the countable exceptional set where

3 1.1. QUASICONFORMAL MAPPINGS 5 (1.1.) was not required to hold by a set of measure zero and then by requiring that f preserves only a sequence of infinitesimal circles about the remaining points z D. Definition A real-valued function u is absolutely continuous on lines, or ACL, inadomaind if for each rectangle [a, b] [c, d] D, 1 u(x + iy) is absolutely continuous in x for almost all y [c, d], u(x + iy) is absolutely continuous in y for almost all x [a, b]. A complex-valued function f is ACL in D if its real and imaginary parts are ACL in D. If a homeomorphism f is ACL in D, then a measure theoretic argument shows that f has finite partial derivatives a.e. in D and hence, in fact, a differential a.e. in D by Gehring-Lehto [63]. A quasiconformal mapping can then be described in terms of its analytic properties as follows. See e.g. Lehto-Virtanen [117]. Theorem A homeomorphism f : D D is K-quasiconformal if and only if f is ACL in D and (1.1.9) max f(z) K J f (z) almost everywhere in D. Here f(z) denotes the derivative of f at z in the direction and J f (z) denotes the Jacobian of f at z. Moreover, if f is quasiconformal, we have that J f (z) 0a.e. in D and that it satisfies Lusin s property ( N), i.e. m(f(e)) = 0 whenever m(e) =0for the planar Lebesgue measure m. If f : D D is K-quasiconformal, then inequality (1.1.9) can also be written max f(z) K min f(z). If we assume also that f is sense-preserving, then max f = f z + f z, min f = f z f z, where f z and f z are the complex derivatives f z = 1 (f x if y ) and f z = 1 (f x + if y ). In this case (1.1.9) takes the form (1.1.10) f z K 1 K +1 f z. Then since f z f z = J f > 0 a.e. in D, we may also consider the quotient μ f = f z. f z The function μ f (z) isthecomplex dilatation of f at z. It satisfies the relations 1+ μ f (z) 1 μ f (z) = H f (z) and μ f (z) K 1 K +1 a.e. in D. Hence μ f =0a.e. ind if and only if f is conformal.

4 6 1. PRELIMINARIES If f : D D and g : D D are both sense-preserving and quasiconformal, then μ g f = μ f a.e. in D if and only if g is conformal. It is possible to prescribe the complex dilatation μ f (z), and hence the linear dilatation H f (z), at almost every point z of a domain D. This result, known as the measurable Riemann mapping theorem, has turned out to be a powerful tool in complex analysis. See Ahlfors-Bers [9], Lehto-Virtanen [117], Morrey [133], and Bojarski [5]. Theorem If μ is measurable with μ L = ess sup μ(z) < 1, D then there exists a sense-preserving quasiconformal mapping f of D with μ f = μ a.e. in D. Moreover f is unique up to post composition with a conformal map. 1.. Modulus of a curve family The conditions for quasiconformality in Definition and Theorem involve the local behavior of a homeomorphism. We need a way to integrate the inequality in Theorem in order to derive global properties of the mapping. When K =1,f or its complex conjugate f is conformal and the Cauchy integral formula is available. The tool most often used to replace the Cauchy formula when K>1isthe method of extremal length, first formulated by Ahlfors and Beurling in [3]. Suppose that Γ is a family of curves in R. We say that ρ is an admissible density for Γ, or is in adm(γ), if ρ is nonnegative and Borel measurable in R and if ρ(z) dz 1 γ for each locally rectifiable γ Γ. The modulus and extremal length of the family Γ are then given, respectively, by mod(γ) = inf ρ(z) 1 dm and λ(γ) = ρ R mod(γ), where the infimum is taken over ρ adm(γ). Theorem If f : D D is conformal and if Γ is a family of curves in D, then mod(f(γ)) = mod(γ). Proof. We consider the case where D, D R.Foreachρ adm(f(γ)) let { ρ (f(z)) f (z) if z D, ρ(z) = 0 if z R \ D. Then ρ is nonnegative and Borel measurable in R.Ifγis locally rectifiable, then f(γ) f(γ) is locally rectifiable and ρ(z) dz = ρ (f(z)) f (z) dz = ρ (w) dw 1. γ γ f(γ)

5 1.3. MODULUS ESTIMATES 7 Thus ρ adm(γ), mod(γ) ρ(z) dm = ρ (f(z)) f (z) dm R D = ρ (w) dm ρ (w) dm, D R whence mod(γ) inf ρ (w) dm =mod(f(γ)). ρ R Now take the infimum over all such ρ. Finally we obtain mod(γ) = mod(f(γ)) by applying the above argument to f 1. If the curves γ Γ are disjoint arcs, we may think of them as homogeneous electric wires. Then the modulus mod(γ) is a conformally invariant electrical transconductance for the family of wires γ and the extremal length λ(γ) is the total electrical resistance of the system. In particular, mod(γ) is big if the curves γ Γareshort and plentiful and small if the curves γ are long or scarce. The following properties show that mod(γ) is also an outer measure on the curve families Γ in R : 1 mod( ) =0. mod(γ 1 ) mod(γ )ifγ 1 Γ. 3 mod( j Γ j) j mod(γ j). Finally the conformal invariant mod(γ) yields a third characterization for quasiconformal mappings. Theorem 1.. (Ahlfors [7]). A homeomorphism f : D D is K-quasiconformal if and only if 1 mod(γ) mod(f(γ)) K mod(γ) K for each family Γ of curves in D Modulus estimates Estimates for the moduli of various curve families are useful tools for studying geometric properties of conformal and quasiconformal mappings. We derive here three simple modulus estimates and a distortion theorem for quasiconformal mappings of the plane which we will need later. Lemma Suppose that R = R(0,a,a+ i, i) is the rectangle with vertices at 0,a,a+ i, i where a>0 and suppose that Γ is the family of curves which join the horizontal sides of R in R. Then mod(γ) = a. Proof. The segment γ = {z : x + iy :0<y<1} is in Γ for 0 <x<a. Hence if ρ adm(γ), then by the Cauchy-Schwarz inequality, ( ρ(x + iy) dy) ρ(x + iy) dy 0 0

6 8 1. PRELIMINARIES Figure 1. for 0 <x<a.thus and Next the function is in adm(γ) and R ρ(z) dm a 0 ( 1 0 ) ρ(x + iy) dy dx a mod(γ) = inf ρ ρ(z) dm a. R { 1 if z R, ρ(z) = 0 otherwise R ρ(z) dm = a, completing the proof for Lemma Lemma If Γ is a family of curves and if for each t with a<t<bthe circle {z : z = t} contains a curve γ Γ, then mod(γ) 1 π log b a. Proof. See Figure 1.3. If ρ adm(γ), then ( ( π 1 ρ(z) dz ) ρ(te iθ ) tdθ) πt γ whence 1 π log b b a = 1 b a πt dt a Now take the infimum over all such ρ. 0 ( π 0 π 0 ρ(te iθ ) tdθ, ) ρ(te iθ ) tdθ dt ρ(z) dm. R Lemma If Γ is a family of curves which join continua C 1 and C where dist(c 1,C ) a>0, diam(c 1 ) b, then b ). mod(γ) π( a +1

7 1.3. MODULUS ESTIMATES 9 Figure 1.3 Proof. Choose z 1 C 1 and z C so that z 1 z =dist(c 1,C )andset { 1/a if z B(z1,a+ b), ρ(z) = 0 otherwise. Then each γ Γ either joins C 1 to C in B(z 1,a+b) orjoins B(z 1,b)to B(z 1,a+ b). In either case γ contains a subarc of length at least a which lies in B(z 1,a+ b). Thus ρ adm(γ) and ( b ). mod(γ) ρ(z) dm = π R a +1 We now apply the modulus estimate established above in Lemma 1.3. to prove an elementary distortion theorem for quasiconformal mappings which we will need in what follows. Figure 1.4

8 10 1. PRELIMINARIES Figure 1.5 Theorem If f : R R is K-quasiconformal and if z z 0 z 1 z 0, then (1.3.5) f(z ) f(z 0 ) c f(z 1 ) f(z 0 ) where c = e 8K. Proof. By means of preliminary similarity transformations, we may assume that z 0 = f(z 0 )=0andthat z 1 = 1, whence z 1. We may also assume that f(z 1 ) < f(z ) since otherwise there is nothing to prove. Let Γ be the family of circles {w : w = t} where f(z 1 ) <t< f(z ). Then 1 π log f(z ) f(z 1 ) mod(γ ) by Lemma To estimate the modulus of Γ = f 1 (Γ ), let φ denote the stereographic projection of R onto the Riemann sphere S = {x R 3 : x =1}. Ifγ Γ=f 1 (Γ ), then γ separates the points 0 and z 1 from and z ; hence φ(γ) is a closed curve on S which separates the points φ(0) and φ(z 1 )fromφ( ) andφ(z ). Since each arc on S which joins φ(0) to φ(z 1 )orφ( ) toφ(z ) has length at least π/, and hence the density γ dz =length(φ(γ)) π 1+ z 1+ z ρ(z) = 1 π is admissible for Γ. Thus mod(γ) ρ(z) dm = 1 R π and we obtain 4 R (1 + z ) dm = 4 π 1 π log f(z ) f(z 1 ) mod(γ ) K mod(γ) 4K π from which (1.3.5) follows.

9 1.3. MODULUS ESTIMATES 11 A more detailed reasoning yields the following sharp estimate for the constant c in (1.3.5), namely c = λ(k) where ( ) 1 (1.3.6) λ(k) = 4 eπk/ e πk/ + δ(k), 0 <δ(k) <e πk. See Anderson-Vamanamurthy-Vuorinen [11] and Lehto-Virtanen-Väisälä [118]. Corollary If f : R R is K-quasiconformal and if (1.3.8) z z 0 k z 1 z 0 where k is an integer, k 0, then (1.3.9) f(z ) f(z 0 ) c(c +1) k f(z 1 ) f(z 0 ) where c = e 8K. Proof. By Theorem 1.3.4, (1.3.8) implies (1.3.9) when k = 0. Suppose this implication is true for some k 0andsetz = 1 (z + z 0 ). Then z z = z z 0 k z 1 z 0 and f(z ) f(z) c f(z) f(z 0 ) again by Theorem Since f(z) f(z 0 ) c(c +1) k f(z 1 ) f(z 0 ) by hypothesis, we obtain f(z ) f(z 0 ) f(z ) f(z) + f(z) f(z 0 ) (c +1) f(z) f(z 0 ) c(c +1) k+1 f(z 1 ) f(z 0 ). Thus (1.3.8) implies (1.3.9) for k + 1 and hence for all k by induction. Theorem and its corollary are also consequences of the following general result (Gehring-Hag [57]), the proof of which is less elementary and depends on theorems due to Teichmüller [157] and Agard [1]. Theorem If f : R R is K-quasiconformal, then ( ) K f(z ) f(z 0 ) z z K 1 f(z 1 ) f(z 0 ) z 1 z 0 +1 for z 0,z 1,z R. The coefficient 16 K 1 cannot be replaced by any smaller constant. The property in Theorem is called quasisymmetry (Heinonen [80], Astala- Iwaniec-Martin [16]). We conclude by listing two properties of quasiconformal mappings that we will need in what follows. See, for example, Lehto-Virtanen [117]. Theorem If f : D D is quasiconformal and if D and D are Jordan domains, then f has a homeomorphic extension which maps D onto D. Theorem Suppose that E D is closed and contained in a countable union of rectifiable curves. If f : D D is a homeomorphism which is K- quasiconformal in each component of D \ E, thenf is K-quasiconformal in D.

10 1 1. PRELIMINARIES 1.4. Quasidisks We come now to the principal object of study in this book. Definition A domain D is a K-quasidisk if it is the image of a Euclidean disk or half-plane under a K-quasiconformal self-mapping of R. D is a quasidisk if it is a K-quasidisk for some K. We present next three Jordan domains that we will use in what follows to illustrate various properties of quasidisks. The first of these is an angular sector. Example For 0 <<π let S() denote the angular sector S() ={z = re iθ :0<r<, θ < }. Then S() isak-quasidisk where ( ) π (1.4.3) K =max,. π The bound in (1.4.3) is sharp. To prove this, let f(re iθ )=r p e iφ(θ) for 0 <r< and θ π where π p = ( π ) and φ(θ) = πθ π if 0 θ, π (π θ) if π θ π, φ( θ) if π θ 0. An elementary calculation shows that f is K-quasiconformal, where K is as in (1.4.3), and that f maps S() onto the right half-plane S(π). To show that the bound in (1.4.3) is best possible, suppose that f is a K- quasiconformal mapping of R which maps S() onto the right half-plane S(π) and let h = f 1 g f where g denotes reflection in the imaginary axis. Then h is a K -quasiconformal mapping of R which maps S() onto its exterior S (). Next fix 0 <a<b< and let Γ denote the family of arcs which join the circles {z : z = a} and {z : z = b} in {z : a z b, arg(z) </}. Then it is not difficult to check that mod(γ) = log(b/a). Similarly, mod(γ π )= log(b/a)+log(c) where Γ is the family of arcs which join {z : z = a/c} and {z : z = bc} in {z : a/c z bc, / < arg(z) π}.

11 1.4. QUASIDISKS 13 Figure 1.6 Hence if c =8e K, then Theorem implies that for each arc γ Γ there exists an arc γ Γ such that h(γ) γ.thusadm(h(γ)) adm(γ ), whence mod(h(γ)) mod(γ ) and K mod(h(γ)) mod(γ) We conclude that π K π log(b/a) log(b/a)+log(c). by letting b/a. Finally reversing the roles of S() ands () in the above argument yields K π and hence (1.4.3). Definition A domain D is a sector of angle if it is the image of S() under a similarity mapping. Our second example is a simple Jordan domain that is not a quasidisk. Example The half-strip D = {z = x + iy :0<x<, y < 1} is not a quasidisk. We shall show that there exists no quasiconformal self-mapping f of R which maps H onto D. By performing a preliminary Möbius transformation, we need only consider the case where f( ) =. Suppose that f is a K-quasiconformal self-mapping of R with f(h) =D, set w 1 = x + i, w =0,w 3 = x i, and let z i = f 1 (w i )fori =1,, 3. Then z 1,z,z 3 is an ordered triple of points on H with z 1 z < z 1 z 3

12 14 1. PRELIMINARIES Figure 1.7 for each choice of x in (0, ). On the other hand, x< f(z 1 ) f(z ) c f(z 1 ) f(z 3 ) =c by Theorem where c = c(k) and we have a contradiction. If D is a K-quasidisk, then D is the image of a circle under a self-homeomorphism f of R which is differentiable a.e. Thus D is a Jordan curve which is a circle or line when K = 1. Hence it is natural to ask if D has any nice analytic properties when 1 <K<. For example, is D locally rectifiable? Our third example shows that the answer is no and that, from the standpoint of Euclidean geometry, the boundary of a quasidisk can be quite wild. See Gehring- Väisälä [70]. Figure 1.8 Example For each 1 <a< there exists a quasidisk D such that dim( D) a where dim denotes Hausdorff dimension. We will sketch a proof of this. We say that a square is oriented if its sides are parallel to the coordinate axes and we let Q and Q denote the open squares Q = Q = {z = x + iy : x < 1, y < 1}.

13 1.4. QUASIDISKS 15 Next set z 1 = 3 4, z = 1 4, z 3 = 1 4, z 4 = 3 4 and w 1 = 1+i, w = 1+i, w 3 = 1 i, w 4 = 1 i, and fix 0 <r<1/. Then choose 0 <s<1sothat log 4 log(/s) = a. Finally for j =1,, 3, 4letQ j denote the open oriented square with center z j and side length r and let Q j be the open oriented square with center w j and side length s. Then we can choose a piecewise linear homeomorphism f 0 : Q \ 4 Q j Q \ j=1 such that f 0 is the identity on Q and is of the form a j z + b j,a j > 0, on Q j with f 0 ( Q j )= Q j.thenf 0 is K-quasiconformal in Q \ j Q j where K = K(r, s). Next for each j choose oriented squares Q j,k in Q j and Q j,k in Q j in the same way as the squares Q j and Q j were chosen in Q and Q, respectively. By scaling we can extend f 0 to obtain a piecewise linear homeomorphism f 1 : Q \ 4 j,k=1 Q j,k Q \ 4 j=1 4 j,k=1 Q j Q j,k which is K-quasiconformal in Q \ j,k Q j,k. Continuing in this way, we obtain a homeomorphism f : Q \ E Q \ E where E and E are Cantor sets. Then f can be extended by continuity to give a K-quasiconformal mapping which maps Q onto Q and is the identity on Q. Set f(z) =z in R \ Q. Then f is a K-quasiconformal self-mapping of R which maps the upper half-plane H onto a quasidisk D with Hausdorff dimension dim( D) log 4 log(/s) = a. See Beardon [17] or page 67 in Mattila [17]. Although the Hausdorff dimension of the boundary D of a quasidisk D can be arbitrarily close to, it always satisfies m( D) = 0 where m is planar Lebesgue measure. This follows from Lusin s property (N) of quasiconformal mappings in Theorem On the other hand, a result due to Astala [13] gives the estimate dim( D) K K +1 for the Hausdorff dimension of the boundary D of a K-quasidisk D. Our final example, or rather class of examples, in this section illustrates how quasidisks arise naturally in complex dynamics.

14 16 1. PRELIMINARIES Figure 1.9 Example For a nonconstant meromorphic function f : R R,the iterates f n (z) =f f n 1 (z), n, f 1 (z) =f(z) are all defined and meromorphic. The Fatou set F f of f is the largest open set where the sequence (f n ) is a normal family, while its complement, J f = R \ F f, is called the Julia set. If p is a polynomial function of degree two, we may assume without loss of generality that it has the form p c (z) =z + c. If c = 0, the Julia set is the unit circle, and if c < 1/4, it can be shown that the Fatou set has exactly two components F 0 and F,with0 F 0 and F.See Beardon [19] or Carleson-Gamelin [31]. Arguments using Theorem in an ingenious way reveal that in fact F 0 is a quasidisk. See e.g. Carleson-Gamelin [31] What is ahead Though quasidisks can be quite pathological domains, they occur very naturally in surprisingly many branches of analysis and geometry. We will describe in what follows some thirty different properties of quasidisks which generalize corresponding properties of Euclidean disks and which characterize this class of domains. See also Gehring [51] and[54]. The properties of a quasidisk D that we will discuss fall into the following categories: 1 geometric properties of D, conformal invariants defined in D, 3 injectivity criteria for functions defined in D, 4 criteria for extension of functions defined in D, 5 two-sided criteria for D and D, 6 miscellaneous properties.

15 1.5. WHAT IS AHEAD 17 In the remainder of Part 1 (Chapters to 7) we will consider properties in each of these categories. A number of them can be used to characterize Euclidean disks or half-planes. We will indicate when this is the case. In Part (Chapters 8 to 11) we will present proofs for some of the characterizations mentioned above. Many of the arguments follow a series of implications. There are four main series of implications, as well as some additional equivalences proved. Some proofs not in the main series of implications belong naturally to the discussion of the results and are presented in Part 1.