Area dstorton of quasconformal mappngs K. Astala 1 Introducton A homeomorphsm f : Ω Ω between planar domans Ω and Ω s called K-quasconformal f t s contaned n the Sobolev class W2,loc 1 (Ω) and ts drectonal dervatves satsfy max α α f(x) Kmn α α f(x) a.e. x Ω. In recent years quasconformal mappngs have been an effcent tool n the study of dynamcal systems of the complex plane. We show here that, n turn, methods or deas from dynamcal systems can be used to solve a number of open questons n the theory of planar quasconformal mappngs. It has been known snce the work of Ahlfors [A] and Mor [Mo] that K-quasconformal mappngs are locally Hölder contnuous wth exponent 1/K. The functon f 0 (z) = z z 1 K 1 (1) shows that ths exponent s the best possble. In addton to dstance, quasconformal mappngs dstort also the area by a power dependng only on K, as shown frst by ojarsk [j]. Snce f 0 (r) = π 1 1 K (r) 1 K, where (r) = {z C : z < r}, t s natural to expect that the optmal exponent n area dstorton s smlarly 1/K. In ths paper we gve a postve answer to ths problem and prove the followng result whch was conjectured and formulated n ths precse form by Gehrng and Rech [GR]. We shall denote by the open unt dsk and by E the area of the planar set E. Theorem 1.1 Suppose f : s a K-quasconformal mappng wth f(0) = 0. Then we have fe M E 1/K (2) for all orel measurable sets E. Moreover, the constant M = M(K) depends only on K wth M(K) = 1 + O(K 1). For the proof of (2) we consder famles { } n 1 of dsjont dsks = (λ) whch depend holomorphcally on the parameter λ (n a sense to be defned later). After an approxmaton (2) now becomes equvalent to ( n ) 1 λ (λ) C (0) 1+ λ, (3) 1
where C depends only on λ. Furthermore, teratng the confguraton one s led to measures on Cantor sets and there we shall apply the Ruelle-owen thermodynamc formalsm [w]; f we wrte (3) n terms of the topologcal pressure, then the proof comes out n a transparent way. The functon f 0 s extremal n the dstorton of area as well as dstance, and therefore t s natural to ask [I, 9.2] f for quasconformal mappngs the Hölder contnuty alone, rather than the dlataton, mples the nequalty (2). However, ths turns out to be false, as shown recently by P.Koskela [K]. As s well known the optmal control of area dstorton answers several questons n ths feld. For example, n general domans Ω one can nterpret (2) n terms of the local ntegrablty of the Jacoban J f of the quasconformal mappng f. Ths leads to a soluton of the well known problem [LV], [Ge] on the value of the constant p(k) = sup{p : J f L p loc (Ω) for each K quasconformal f on Ω}. Corollary 1.2 In every planar doman Ω, p(k) = K K 1. In other words, for each K-quasconformal f : Ω Ω, f W 1 p,loc(ω), p < 2K K 1. The example (1) shows that ths s false for p 2K K 1. Theorem 1.1 governs also the dstorton of the Hausdorff dmenson dm(e) of a subset E. Corollary 1.3 Let f : Ω Ω be K-quasconformal and suppose E Ω s compact. Then 2 K dm(e) dm(fe) 2 + (K 1) dm(e). (4) Ths nequalty, as well, s the best possble. Theorem 1.4 For each 0 < t < 2 and K 1 there s a set E C wth dm(e) = t and a K-quasconformal mappng f of C such that dm(fe) = 2 K dm(e) 2 + (K 1) dm(e). The estmate (4) was suggested by Gehrng and Väsälä [GV]. It can also be formulated [IM2] n the symmetrc form 1 ( 1 K dm(e) 1 ) 1 2 dm(fe) 1 ( 2 K 1 dm(e) 1 ). (5) 2 The results 1.3 and 1.4 are closely related to the removablty propertes of quasregular mappngs, snce n plane domans these can be represented as compostons of analytc functons and quasconformal mappngs. The strongest removablty conjecture, due to Iwanec and Martn [IM1], suggest that sets of Hausdorff d measure zero, d = n K+1, are removable for bounded quasregular mappngs n Rn. Here we obtan the followng. 2
Corollary 1.5 In planar domans sets E of Hausdorff dmenson dm(e) < 2 K + 1 are removable for bounded quasregular mappngs. Conversely, for each K 1 and t > 2/(K + 1) there s a t dmensonal set E C whch s not removable for some bounded K quasregular mappngs. In addton to [IM1] removablty questons have recently been studed for nstance n [JV], [KM] and [R]. Fnally, we menton the applcatons to the regularty results of quasregular mappngs. Recall that a mappng f W 1 q,loc(ω), 1 < q < 2, s sad to be weakly quasregular, f J f 0 almost everywhere and max Df(x)h K mn Df(x)h a.e. x Ω. h =1 h =1 Then f s K quasregular n the usual sense f f W2,loc 1 (Ω),.e. f J f s locally ntegrable. We can now consder the number q(k), the nfmum of the q s such that every weakly K quasregular mappng f Wq,loc 1 (Ω) s actually K quasregular. Corollary 1.6 q(k) = K 2K + 1. Indeed, Lehto and Vrtanen [LV] have proven that the precse estmate on the L p ntegrablty, Corollary 1.2, mples that q(k) 2K 2K K+1. The opposte nequalty q(k) K+1 was shown by Iwanec and Martn n [IM1]. Quasconformal mappngs are also the homeomorphc solutons of the ellptc dfferental equatons f(z) = µ(z) f(z); here µ s the complex dlataton or the eltram coeffcent of f wth µ = K 1 K+1 < 1. Hence there are close connectons to the sngular ntegral operators and especally to the eurlng-ahlfors operator Sω(z) = 1 π C ω(ζ) dm(ζ) (ζ z) 2, see [I],[IM1],[IK] for example. In fact, ths operator was the man tool n the work of ojarsk [j] and Gehrng-Rech [GR], c.f. also [IM2]. elow we shall use mostly dfferent approaches and the role of the S operator remans mplct. Stll the area dstorton nequalty has a number of mplcatons on the propertes of S. In partcular, we have Corollary 1.7 There s a constant α 1 such that for any measurable set E, Sχ E dm E log α E. (6) It s for ths consequence that we must show the asymptotc estmate M(K) = 1 + O(K 1) and then, actually, Theorem 1.1 s equvalent to 1.7, cf. [GR]. 1 1 Davd Hamlton and Tadeusz Iwanec have ponted out that now (6) holds wth α = eπ. 3
If we consder general functons ω L ( ) then the nequalty (6) mples the correct exponental decay for {z : R Sω > t} when t. As a consequence, for each δ > 1 there s a constant M δ < such that Sv dm δ v log(1 + M δ v v ) dm, v L log L( ). (7) Here v = 1 π v dm s the ntegral mean of v. It s a natural queston whether (7) holds at δ = 1 as n (6) wth characterstc functons. Ths would also mply the Iwanec-Martn removablty conjecture n the planar case. However, n the last secton we show, agan by consderng the nequaltes arsng from the thermodynamc formalsm, that n fact (7) fals when δ = 1. Further results equvalent to the Gehrng-Rech conjecture have been gven by Iwanec and Koseck [IK]. These nclude applcatons to the L 1 theory of analytc functons, quadratc dfferentals and crtcal values of harmonc functons. Moreover, by results of Lavrentev, ers and others the solutons to the ellptc dfferental equatons A(x) u = 0 can be nterpreted n terms of quasregular mappngs f. Therefore Corollary 1.2 yelds sharp exponents of ntegrablty on the gradent u; note that the dlataton of f and so necessarly the optmal ntegrablty exponent depends n a complcated manner on all the entres of the matrx A rather than just on ts ellptcty coeffcent. Acknowledgements Durng the preparaton of ths manuscrpt a number of people found smplfcatons n the frst prelmnary notes. Especally I would lke to thank Alexander Eremenko and Jose Fernandez who both ponted out Corollary 2.4. Also, I would thank Tadeusz Iwanec and Mchel Znsmester for mportant dscussons and comments on the topcs n ths paper and many people, ncludng Alexander Eremenko, Fred Gehrng, Seppo Rckman, Juha Henonen and Pekka Koskela for readng and makng correctons to the frst draft. 2 Holomorphc deformatons of Cantor sets Let us frst consder a famly { } n of nonntersectng subdsks of. We shall study the quasconformal deformaton of such famles and, n partcular, estmate sums r( ) t, t R, (8) where r( ) denotes the radus of. Lookng for the extremal phenomena we can terate the confguraton { } n and are thus led to Cantor sets. There one needs measures µ whch reflect n a natural manner the propertes of the sums (8). It turns out that such measures can, ndeed, be found by usng the thermodynamc formalsm ntroduced by Ruelle and owen, c.f. [w], [W]. To descrbe ths n more detal suppose hence that we are gven smlartes γ, 1 n, for whch = γ. Snce the γ are contractons, there s a unque compact 4
subset J of the unt dsk for whch n γ (J) = J. Thus J s self-smlar n the termnology of Hutchnson [H]. We can also reverse ths pcture and defne the mappng g : by g = γ 1. Now J = g k, k=0 and g s a n-to-1 expandng mappng wth J completely nvarant, J = gj = g 1 J. Furthermore, g represents the shft on J; n a natural manner we can dentfy the pont x J wth the sequence (j k ) k=0 {1,..., n}n by defnng j k = f g k (x). Then, n ths dentfcaton, g : (j k ) k=0 (j k+1) k=0. In the sequel we use the notaton J = J(g) for our Cantor set and say also that t s generated by the smlartes γ. Next, let s = dm(j(g)), the Hausdorff dmenson of J(g). Then the Hausdorff s measure s nonzero and fnte on J(g) and after a normalzaton t defnes a probablty measure µ s whch s nvarant under the shft g,.e. µ s (g 1 E) = µ s (E) for all borelan E J(g). A general and systematc way to produce further nvarant measures s provded by the Ruelle-owen formalsm: Gven a Hölder contnuous and real valued functon ψ on J(g) there s a unque shft-nvarant probablty measure µ = µ ψ, called the Gbbs measure of ψ, for whch the supremum P (ψ) = sup { h µ (g) + J(g) ψ dµ : µ s g nvarant} (9) s attaned, see [w] or [W]. Here h µ (g) denotes the entropy of µ and the quantty P (ψ) s called the topologcal pressure of ψ. Let us then look for the Gbbs measures that are related to the sums (8). Recall that s = dm(j(g)) s the unque soluton of P ( s log g ) = 0, and ths suggest the choces ψ t = t log g. It then readly follows from [w, Lemma I.1.20], that ( P ( t log g n ) = log γ t) ( n = log r( ) t). (10) 5
In fact, the functons ψ = ψ t are n our stuaton locally constant and therefore t can be shown that the system g : (J(g), µ ψ ) (J(g), µ ψ ) s ernoull. In other words, the numbers p = µ ψ (J ) satsfy n p = 1 and on J(g) = {1,..., n} N µ ψ s the product measure determned by the probablty dstrbuton {p } n of {1,..., n}. Ths enables one to make the dynamcal approach more elementary, as ponted to us by Alexander Eremenko. We are grateful to hm for lettng us to nclude ths smplfcaton here. For the readers convenence let us recall the proof of the varatonal prncple, the counterpart of (9), n the elementary settng of product measures. Then also the entropy of µ = µ ψ attans the smple form h µ (g) = p log p. 1 Lemma 2.1 Let ν be a product measure on J(g) determned by the probablty dstrbuton {q } n. Then for each t R, wth equalty f and only f h ν (g) t log g dν log( r( ) t ) J(g) q = r( ) t n r( ) t, 1 n. Proof: Snce the logarthm s concave on R +, h ν (g) t J(g) log g dν = q log γ t q = q log r( ) t q log( r( ) t ) where the equalty holds f and only f q r( ) t 1 n. has the same value for each Remark. In the elementary settng, one can use (10) as the defnton of the pressure P ( t log g ). Note also that f s = dm(j(g)), then n r( ) s = 1, or P ( s log g ) = 0, and the extremal measure n Lemma 2.1 s agan the normalzed Hausdorff s measure. We shall next consder holomorphc famles of Cantor sets or pars (g λ, J(g λ )), λ. y ths we mean that each set J(g λ ) s generated as above by smlartes γ,λ (z) = a (λ)z + b (λ), 1 n, where the coeffcents a (λ) 0, b (λ) now depend holomorphcally on the parameter λ. On the other hand, we can also consder the (λ) = γ,λ and say that { (λ)} n 1 s a holomorphc famly of dsjont dsks n. oth of these confguratons can be descrbed as holomorphc motons; recall that a functon Φ : A C s called a holomorphc moton of a set A C f 6
() for any fxed a A, the map λ Φ(λ, a) s holomorphc n () for any fxed λ, the map a Φ λ (a) = Φ(λ, a) s an njecton, and () the mappng Φ 0 s the dentty on A. In fact, (global) quasconformal mappngs and holomorphc motons are just dfferent expressons of the same geometrc quantty. For nstance, accordng to Slodkowsk s generalzed λ lemma ([Sl], see also [AM], 3.3) the correspondence γ,0 (z) γ,λ (z) for z and 1 n, extends to a quasconformal mappng Φ λ : C C wth K(Φ λ ) 1+ λ 1 λ. Therefore the estmate ( n ) 1 λ 1+ λ (λ) C (0) (11) s a specal case of the Gehrng-Rech conjecture. ut after smplfyng arguments, gven later n Lemmas 3.1 and 3.3, we wll see that the conjecture s n fact equvalent to (11). Expressng ths nequalty now n terms of the topologcal pressure (10) we end up wth the followng formulaton. Theorem 2.2 Suppose that (g λ, J(g λ )) depends holomorphcally on the parameter λ. Then 1 + λ 1 λ P ( 2 log g 0 ) P ( 2 log g λ ) 1 λ 1 + λ P ( 2 log g 0 ). Proof: y the varatonal nequalty 2.1 for each λ there s a unque (product) measure µ λ such that P ( 2 log g λ ) = h µλ (g λ ) 2 log g λ (z) dµ λ (z) (12) and clearly log g λ (z) s harmonc n λ. To use Harnack s nequalty we freeze the measure µ λ. In other words, gven a probablty dstrbuton {p } n 1 on {1,..., n}, defne for each λ a product measure µ λ on J(g λ ) by the condton µ λ (J(g λ ) (λ)) = p ; ths s possble snce the dsks (λ) reman dsjont. y the constructon, h µλ (g λ ) s also constant n λ. Moreover, we have that P ( 2 log g λ ) < 0, snce P ( s log g λ ) s strctly decreasng n s and t vanshes for s = dm(j(g)) < 2. Alternatvely, we may also use here the dentty (10) to P ( 2 log g λ ) = log( n r( (λ)) 2 ) < 0. If now the numbers {p } are so chosen that µ 0 = µ 0 (the maxmzng measure n (12) when the parameter λ = 0), then Harnack s nequalty wth 2.1 mples that J(g λ ) 1 + λ 1 λ P ( 2 log g 0 ) = 1 + λ ( h µ0 (g 0 ) 2 1 λ h µλ (g λ ) 2 J(g λ ) J(g 0 ) log g λ dµ λ P ( 2 log g λ ) 7 log g 0 dµ 0 )
whch proves the frst of the requred nequaltes. symmetry n λ and 0. The second follows smlarly by When t > 2 the same nequaltes hold for P ( t log g λ ) as well. However, smaller exponents must change wth λ and we shall later see how ths reflects n the precse dstorton of Hausdorff dmenson under quasconformal mappngs. Corollary 2.3 If (g λ, J(g λ )) s as above and 0 < t 2, set t(λ) = Then 1 t(λ) P ( t(λ) log g λ ) 1 λ 1 1 + λ t P ( t log g 0 ). t(1 + λ ) (1 λ + t λ ). Proof: If {µ λ } λ s a famly of product measures on J(g λ ), all defned by a fxed probablty dstrbuton {p } n 1 lke n the prevous theorem, then by 2.1 1 t(λ) h µ λ (g λ ) log g λ dµ λ = h µλ (g λ )( 1 t(λ) 1 2 ) + 1 2 h µ λ (g λ ) log g λ dµ λ J(g λ ) 1 λ ( h µ0 (g 0 )( 1 1 + λ t 1 2 ) + 1 2 h µ 0 (g 0 ) 1 λ 1 1 + λ t P ( t log g 0 ) J(g 0 ) J(g λ ) log g 0 dµ 0 ) and takng the supremum over the product measures on J(g λ ) proves the clam. The above estmates for the topologcal pressure hold actually n a much greater generalty. We can consder, for nstance, polynomal-lke mappngs of Douady and Hubbard [DH]. More precsely, suppose we have a famly of holomorphc functons f λ defned on the open sets U λ, λ, such that U λ f λ U λ. We need to assume that J(f λ ) = f n λ U λ n=0 s a mxng repeller for f λ. That s, f λ 0 for z J(f λ) and J(f λ ) s compact n C wth no proper f λ nvarant relatvely open subsets. Then the f λ are expandng on J(f λ ) and the thermodynamc formalsm extends to f λ : J(f λ ) J(f λ ), see [w] or [Ru]. To consder the dependence on the parameter, let U λ depend contnuously on λ and let (λ, a) f λ (a) be holomorphc whenever defned. ecause the functons are expandng, we have a holomorphc moton of the perodc ponts ([MSS], p.198). Snce these are dense n the repeller J(f λ ), by the λ lemma of Mañé, Sad and Sullvan we obtan a holomorphc moton Φ of J(f 0 ) such that J(f λ ) = Φ λ J(f 0 ) and f λ Φ λ = Φ λ f 0. Combnng these facts we conclude that 1 t(λ) P ( t(λ) log f λ ) 1 λ 1 1 + λ t P ( t log f 0 ). (13) 8
Namely, snce the varatonal prncple generalzes to ths settng, the proof of (13) s as n Lemma 2.3. In ths case to show that P ( t(λ) log f λ ) < 0 we may use Mannng s formula [M] h µ (f λ ) dm(µ) = log f λ dµ J(f λ ) and the fact [Su] that dm(µ) nf{dm(e) : µ(e) = 1} dm(j(f λ )) < 2. These hold for any ergodc f λ nvarant measure on J(f λ ). Especally, startng from a measure µ on J(f 0 ) we can take the mages µ λ = Φ λ µ under the holomorphc moton, and snce the entropy s an somorphsm nvarant, (13) follows. On the other hand, f one looks for the mnmal approach to the quasconformal area dstorton, then the above leads also to a proof for (11) that avods the thermodynamc formalsm. In fact, ths was shown to us by A. Eremenko and J.Fernandez, who ndependently ponted out the followng result on the (nonharmonc!) functon log f(z). Corollary 2.4 Let n = {z C n : z < 1}. If f : n s a holomorphc mappng such that all of ts coordnate functons f are everywhere nonzero, then 1 + z 1 z log f(0) log f(z) log f(0). 1 z 1 + z Proof: If f = (f 1,..., f n ), consder numbers p > 0 wth n 1 p = 1 and set p u(z) = p log f (z) 2. Then u(z) s harmonc and by Jensen s nequalty, e u(z) n f 1 p (z) 2 p < 1, u s also postve. Hence usng the concavty of the logarthm and the Harnack s nequalty we may deduce log f(z) 2 p log f (z) 2 p 1 + z 1 z p log f (0) 2 p. proves the frst nequalty. The second follows by symme- Choosng fnally p = f (0) 2 f(0) 2 try. 3 Dstorton of area We shall reduce the proof of the area dstorton estmate fe M E 1/K nto two dstnct specal cases. In the frst, where we use the nequaltes of the prevous secton, let us assume that E s a fnte unon of nonntersectng dsks = (z, r ), 1 n. 9
Lemma 3.1 Suppose that f : s K quasconformal wth f(0) = 0. If f s conformal n E = n 1, then ( n ) 1 f C(K) K, where the constant C(K) depends only on K. Moreover, C(K) = 1 + O(K 1). Proof: Extend f frst to C by a reflecton across S 1 and assume wthout loss of generalty that f(1) = 1. Then we can embed f to a holomorphc famly of quasconformal mappngs of C. However, n order to control the dstorton as K we need to modfy f near. Thus, f µ s the eltram coeffcent of (the extended) f, defne new dlatatons by { λ K+1 µ λ (z) = K 1 µ(z), z 2, 0, z > 2. y the measurable Remann mappng theorem there are unque µ λ quasconformal mappngs f λ : C C normalzed by the condton f λ (z) z = O( 1 ) as z. z Then f λ s conformal n E, f λ (z) and ts dervatves (when z E) depend holomorphcally on λ [A, Theorem 3], f 0 (z) z and f λ 0 = K 1 K+1, then f λ0 = Φ f, (14) where Φ s conformal n f(0, 2). To apply Theorem 2.2 note that by Koebe s 1/4 theorem ( D (λ) f λ (z ), r ) 4 f λ(z ) f λ (z, r ) and smlarly f λ (0, 2) (0, 8). Also D (λ) = ψ λ, D (0), where ψ,λ (z) = f λ(z ) (z z ) + f λ (z ), and thus {D (λ)} n 1 s a holomorphc famly of dsjont dsks contaned n (0, 8). Therefore we need only choose extra smlartes φ : (0, 8) D (0), 1 n, set γ,λ = ψ,λ φ and note that these generate a holomorphc famly of Cantor sets J(g λ ) (0, 8). y Theorem 2.2 P ( 2 log g λ 1 λ ) 1+ λ P ( 2 log g 0 ) or, n other words, by (11) 4 λ ( f λ(z ) 2 r 2 1+ λ n ) 1 λ 1+ λ 32. r 2 The Lemma wll then be completed by smple approxmaton arguments. Snce the mages of crcles under global quasconformal mappngs have bounded dstorton, f λ π max z f λ (z) f λ (z ) 2 πc 0 ( λ ) mn z f λ (z) f λ (z ) 2 10
πc 0 ( λ ) f λ(z ) 2 r 2, where the last estmate follows from the Schwarz lemma. Moreover, the correct expresson for the constant C 0 ( λ ), see [L, p.16], shows that C 0 ( λ ) = 1+O( λ ). If we choose λ 0 = K 1 K+1, t then follows that f λ 0 E C 1 (K) E 1 K wth C 1 (K) = 1 + O(K 1). It remans to show that the functon Φ n (14) satsfes Φ (z) C 2 (K) = (1 + O(K 1)) 1 for all z. Frst, snce the dameter of f λ0 (0, 2) s at least four [P, 11.1], the basc bounds on the crcular dstorton, see [L, I.2.5], mply that (f λ0 (0), ρ(k)) f λ0 = Φ for a ρ(k) > 0 dependng only on K. As above, Φ (0) ρ(k) by the Schwarz lemma. Yet another applcaton of the Schwarz lemma, ths tme to the functon λ f λ (z) z, gves f λ (z) z 10 λ, z (0, 2). Ths shows that we may choose ρ(k) = (1 + O(K 1)) 1. Furthermore, as fs 1 = S 1 and f 1 s unformly Hölder contnuous wth constants dependng only on K, f (0, 2) (0, R) for an R = R(K) > 1. Then Koebe s dstorton theorem combned wth Lehto s majorant prncple [L, II.3.5] proves that and the requred estmates follow. ( Φ (z) Φ (1 z R (0) )3 ) K 1 K+1 1 + z R, z, Remark 3.2 The above proof gves us the followng varatonal prncple for planar quasconformal mappngs: Suppose we are gven numbers p > 0 wth n p = 1 and dsjont dsks. Then for each K quasconformal mappng f : for whch f(0) = 0 and we have the nequalty f n 1 s conformal, (15) p log f p 1 K p log p + C(K) (16) where C(K) = O(K 1) depends only on K. n ) In fact, choosng p = f /( f shows that (16) generalzes Lemma 3.1. Somewhat curously, the varatonal nequalty (16) s not true for general quasconformal mappngs, for mappngs whch do not satsfy (15). We shall return to ths n Secton 5 where t wll have mplcatons on the estmates of the L log L norm of the eurlng-ahlfors operator. To prove the complementary case n the area dstorton nequalty we use therefore a dfferent method. We shall apply here the approach due to Gehrng and Rech [GR] based on a parametrc representaton. 11
Lemma 3.3 Let f : be K quasconformal wth f(0) = 0. If E s closed and f s conformal outsde E, then fe b(k) E where b(k) = 1 + O(K 1) depends only on K. Proof: As n [GR] defne the eltram coeffcents ν t (z) = sgn(µ(z)) tanh( t T arctanh µ(z) ), t R +, where µ s the complex dlataton of f, T = log K and sgn(w) = w w f w 0 wth sgn(0) = 0. y the measurable Remann mappng theorem we can fnd ν t quasconformal h t : wth h t (0) = 0. If A(t) = h t E, then Gehrng and Rech show that d dt A(t) = φs(χ ) dxdy + c(t) h ht E te (17) where S s the eurlng-ahlfors operator and c(t) s unformly bounded. The functon φ depends only on the famly {h t }, not on E, and from [GR, (2.6) and (3.6)] we conclude that φ 1 and that φ(w) = 0 whenever µ(h 1 t (w)) = 0. Suppose now that f s conformal outsde the compact subset E. Then µ 0 n \E and, n partcular, we obtan φ(z) χ ht E (z). ut S : L 2 L 2 s an sometry and therefore for any set F C, ( ) Sχ F dxdy F 1 2 Sχ F 2 1 dxdy 2 = F. (18) Thus F C d dt A(t) C 0A(t), 0 < t <, and an ntegraton gves h t E = A(t) e C 0t A(0) = e C 0t E. Takng t = log K shows that fe e C 0 log K E = b(k) E, where b(k) = 1 + O(K 1). Remark. On the other hand, as kndly ponted out by the referee, f one consders n C the normal solutons f λ (z) = z + O( z 1 ) of the eltram equaton f z = µf z wth µ supported on E, then n that stuaton the followng argument gves a drect proof for a very precse estmate f(e) K E. Namely, for ω = f z we have f z = 1 + Sω wth ω = µ(1 + Sµ + SµSµ +...). In vew of S 2 = 1 we obtan f(e) = 1 + Sω 2 ω 2 E + 2R Sω, E E 12
where for the k th terate E SµSµ... Sµ µ k E as n (18). Thus f(e) E + 2 µ E + 2 µ 2 E +... = E ( 1 + 2 1 µ ) = K E. The area dstorton nequalty s now an mmedate corollary of the two prevous lemmas 3.1 and 3.3. Proof of Theorem 1.1: Suppose that f : s K quasconformal and f(0) = 0. In provng the estmate fe M E 1/K t suffces to study sets of the type E = n 1, where the are subdsks of wth parwse dsjont closures. The general case follows then from Vtal s coverng theorem. To factor f we fnd by the measurable Remann mappng theorem a K quasconformal mappng g :, g(0) = 0, wth complex dlataton µ g = χ \E µ f. Then g s conformal n E and f = h g, where h : s also K quasconformal, h(0) = 0, but now h s conformal outsde ge. Snce quasconformal mappngs preserve sets of zero area, h( ge) = ge = 0, and then Lemmas 3.1 and 3.3 mply fe = h(ge) b(k) ge b(k)c(k) E 1 K, where M(K) b(k)c(k) = 1 + O(K 1) as requred. 2 One of the equvalent formulatons of Theorem 1.1 s the statement that for a K quasconformal f the Jacoban J f belongs to the class weak-l p ( ), p = K K 1. Corollary 3.4 If f : s K quasconformal, f(0) = 0, then for all s > 0, ( M {z : J f (z) s} s ) K K 1, where M depends only on K. Moreover the exponent p = K K 1 Proof: If E s = {z : J f (z) s}, then by Theorem 1.1 s E s J f dm = fe s M(K) E s E s No p larger than 1 K. s the best possble. K K 1 wll do, snce E s = π(ks) K 1 K 1 for f(z) = z z K 1. Proof of Corollary 1.2: If D s a compact dsk n the doman Ω and f : Ω Ω s K quasconformal, choose conformal ψ, φ whch map negbourhoods of D and f D, respectvely, onto the unt dsk. As ψ D and φ fd are blpschtz, applyng Corollary 3.4 to φ f ψ 1 proves that J f L p K loc (Ω) for all p < K 1. 2 Davd Hamlton has nformed us that the same methods can also be used to obtan good bounds for the constant M n fe M E 1/K f one consders nstead of the case f : those mappngs f whch are conformal outsde wth f(z) z = O(1/ z ). 13
4 Dstorton of dmenson In the prevous secton we determned the quasconformal area dstorton from the propertes of the pressure P ( 2 log g λ ). Smlarly Corollary 2.3, or the varatonal nequalty (16) wth a sutable choce of the probabltes p, also admts a geometrc nterpretaton: If f : s K quasconformal wth f(0) = 0 and f, n addton, f s conformal n the unon of the dsks, 1 n, then f tk 1+t(K 1) ( C(K) t) 1 1+t(K 1), 0 < t 1, where the constant C(K) depends only on K. Snce the complementary lemma 3.3 fals for exponents t < 2, n the general case we content wth slghtly weaker nequaltes. Lemma 4.1 If 0 < t < 1, f : s K quasconformal, f(0) = 0 and { } n 1 are parwse dsjont sets n, then whenever (1 + t(k 1)) 1 tk < p 1. ( f p C K (t, p) t) 1 1+t(K 1) Proof: We use the ntegrablty of the Jacoban J f as n [GV]. Snce p(1 + t(k 1)) > tk we can choose an exponent 1 < p 0 < K K 1 such that 1 K q 1 + t(k 1) 0 < p, (19) tk where q 0 = p 0 s the conjugate exponent. Then usng Hölder s nequalty twce one p 0 1 obtans f p = ( ) p ( ) p p J f dm p J 0 p q f dm 0 0 ( ) p p J 0 f dm p 0 ( p p 0 ) q 0 p 0 p p 0 p p 0 ( ) p p J 0 f dm p 0 ( p p 0 ) q 0 p 0 p p 0 p p 0. K 1 On the other hand, as p 0 K < 1 < p 1+t(K 1) tk, t follows that > 1 + t(k 1). Combnng ths wth Corollary 3.4 (or 1.2) yelds ( f p M p q 0 (1+t(K 1))) 1 1+t(K 1), p 0 p 0 p where M depends only on p 0 and K. Snce by (19) also t < p q 0 (1 + t(k 1)), the clam follows. 14
Proof of Corollary 1.3: If f : Ω Ω s K quasconformal, let E Ω be a compact subset wth dm(e) < 2. Choose also a number 1 2dm(E) < t 1 and cover E by squares wth parwse dsjont nterors. Accordng to [LV], Theorems III.8.1 and III.9.1, da(f ) 2 C 0 f, where the constant C 0 depends only on K, E and Ω. Hence we conclude from Lemma 4.1 that ( da(f ) δ C 1 da( ) 2t) 1 1+t(K 1), δ > 2tK 1 + t(k 1). Wth a proper choce of the coverng { } the sum on the rght hand sde can be made arbtrarly small and thus dm(f E) δ. Consequently, dm(fe) 2 K dm(e) 2 + (K 1) dm(e) (20) whch proves the corollary. In the specal case of K quascrcles Γ, the mages of S 1 under global K quasconformal mappngs, Corollary 1.3 reads as dm(γ) 1 + ( K 1 ) = 2 2 K + 1 K + 1. Ths sharpens recent results due to Jones-Makarov [JM] and ecker-pommerenke [P]. On the other hand, ecker and Pommerenke showed that f the dlataton K 1, then ) 2 ( 1 + 0.09( K 1 K+1 dm(γ) 1 + 37 K 1 2. K+1) These results suggest the followng Queston 4.2 If Γ s a K quascrcle, s t true that dm(γ) 1 + In the postve case, s the bound sharp? ( K 1 ) 2. K + 1 Let us next show that the equalty can occur n (20) for any value of K and dm(e). Note frst that n terms of the holomorphc motons Corollary 1.3 obtans the followng form. Corollary 4.3. Let Φ : E C be a holomorphc moton of a set E C and wrte d(λ) = dm(φ λ (E)). Then d(λ) 2d(0) (2 d(0)) 1 λ (21) 1+ λ + d(0). Proof: y Slodkowsk s extended λ lemma Φ λ s a restrcton of a K quasconformal mappng of C, K 1+ λ 1 λ, and hence the clam follows from 1.3. For the converse, we start by constructng holomorphc motons of Cantor sets such that the equalty holds n (21) up to a gven ε > 0. Thus for each, say, n 10 fnd 15
dsjont dsks (z, r) all of the same radus r = r n, such that 1 2 nr2 1. If 0 < t < 2, let also β(t) = log(n 1/t r). (22) For n large enough, β(t) > 0 and t 2 log r log r β(t) t + ɛ. (23) 2 Set then a t (λ) = exp( β(t) 1 λ 1+λ ). Clearly a t s holomorphc n wth a t ( ) = \{0}. Therefore we can consder the holomorphc famly of smlartes γ,λ (z) = ra t (λ)z + z. Snce the dsks γ,λ (z, r) are dsjont, the smlartes γ,λ generate Cantor sets (g λ, J(g λ )) as n Secton 2. Furthermore, the dervatves γ,λ do not depend on and so the dmenson d(λ) = dm(j(g λ )) s determned from the equaton n(r a t (λ) ) d(λ) = 1. y (22) n(r a t (0) ) t = 1 and therefore d(0) = t. Smlarly, f 0 < λ < 1, t follows from (23) that d(0) d(λ) = log(r a t(λ) ) log(r a t (0) ) d(0) 2 + ( 1 d(0) 2 = log r β(t) 1 λ 1+λ log r β(t) ) 1 λ + ε. (24) 1 + λ Proof of Theorem 1.4: Choose a countable collecton { k } 1 of parwse dsjont subdsks of and defne, usng the argument above, n each dsk k a holomorphc moton Φ of a Cantor set J k wth Φ λ (J k ) k. If d(λ) = dm(φ λ (J k )), we may assume that d(0) = t and that for each k (24) holds wth ε = 1 k. Clearly ths constructon determnes a holomorphc moton Ψ of the unon J = k J k. Wrtng stll d(λ) = dm(ψ λ (J)) we have d(λ) = 2d(0) (2 d(0)) 1 λ 0 λ < 1. 1+λ + d(0), Now Slodkowsk s generalzed λ lemma apples and Ψ extends to a K quasconformal mappng f of C, where K = 1+λ 1 λ, 0 λ < 1. In other words, f E = J, then dm(e) = t and dm(fe) = (2Kdm(E))/(2 + (K 1)dm(E)). Fnally, Corollary 1.5 s an mmedate consequence of 1.3 and 1.4 snce K quasregular mappngs f can be factored as f = φ g, where φ s holomorphc and g K quasconformal; for holomorphc φ sets E wth dm(e) < 1 are removable by Panlevé s theorem whle those wth dm(e) > 1 are never removable [Ga, III. 4.5]. 16
Therefore n consderng the removablty questons for K quasregular mappngs, the dmenson d K = 2 K+1 s the border-lne case and there we have the Iwanec-Martn conjecture that all sets of zero Hausdorff d K measure are removable. More generally, t s natural to ask whether the precse bound on the dmenson dm(fe) 2 K dm(e) 2+(K 1) dm(e) gven by Corollary 1.3 s stll correct on the level of measures. 2Kτ 2+τ(K 1). If f s a planar K quas- Queston 4.4 Let 0 < τ < 2 and δ = δ K (τ) = conformal mappng, s t true that H τ (E) = 0 H δ (fe) = 0. If not, what s the optmal Hausdorff measure H h or measure functon h such that f H h H τ? 5 Estmates for the eurlng-ahlfors operator As we saw earler quasconformal mappngs have mportant connectons to the sngular ntegrals and n partcular to the eurlng-ahlfors operator, the complex Hlbert transform Sω(z) = 1 ω(ζ) dm(ζ) π (ζ z) 2. C There are even hgher dmensonal counterparts, see [IM1] and the references there. In fact, many propertes of the S operator can be reduced to the dstorton results of quasconformal mappngs. We shall here consder only the operaton of S on the functon space L log L and refer to the work of Iwanec and Koseck [IK] for further results. In case of the characterstc functons ω = χ E we have then by Corollary 1.7 that Sχ E dm E log α E for all orel subsets E of a dsk C; the constant α does not depend on E or. Ths translates also to the L settng: Corollary 5.1 Let C be a dsk. If ω s a measurable functon such that ω(z) χ (z) a.e. then {z : R Sω(z) > t} 2α e t. (26) Proof: Let E + = {z : R Sω > t}. Snce S has a symmetrc kernel, t E + R Sω dm = R ωsχ E+ E + dm E + log α E + by (25). Thus E + α e t and snce by the same argument E = {z : R Sω < t} satsfes E α e t, the nequalty (26) follows. (25) 17
The estmate (26) s sharp snce for ω = (z/z)χ (z) we have Sω = (1 + 2 log z )χ (z). For the modulus Sω Iwanec and Koseck [IK, proposton 12] have shown that (25) mples {z : Sω(z) > t} α(1 + 19t) e t. (27) It remans open f the lnear term 19t can be replaced by a constant. Corollary 5.2 For each δ > 1 there s a constant M(δ) < such that Sv dm δ whenever the rght hand sde s fnte. v(z) log (1 + M(δ) v(z) ) dm(z) v Proof: Let ω be a functon, unmodular n and vanshng n C\, such that Sv dm = ωsv dm = v Sω v dm. v We apply then the elementary nequalty ab a log(1+a)+exp(b) 1. Snce accordng to (27) e Sω /δ α 19δ 1 dm (1 + δ 1 δ 1 ) = M 1(δ), t follows that Sv dm M 1 (δ) v dm + δ v log (1 + δ v ) dm. v Defne now E 0 = {z : v(z) < 1 e v }. As t t log 1 t s ncreasng on (0, 1 e ), e E 0 v log( v v ) dm v E 0 v dm, where we use the conventon 0 log 0 = 0. Thus ( v ) v log (1 + δ v ) dm v \E 0 v log v (e + δ) dm + where M 2 = e 2 + eδ. In concluson, f M = M 2 exp(m 1 (δ)), Sv dm δ whch completes the estmaton. E 0 v log(1 + δ e ) dm v ) v log (M 2 dm, (28) v v log (M v ) dm δ v v log (1 + M v ) dm, v 18
Snce the varatonal nequalty (16), p log f p 1 K p log p + C(K) wth C(K) = O(K 1) and f conformal, was the key n the area dstorton Theorem 1.1 t s of nterest to know whether the nequalty s vald wthout any conformalty assumptons. Another natural queston s whether Corollary 5.2 stll holds at δ = 1; for characterstc functons ths s true and (25) wth [IK, proposton 19] mples that for nonnegatve functons v, \E Sv dm E v(z) log (1 + α v(z) ) dm(z), f supp(v) E. v Indeed, t can be shown that these two questons are equvalent (f v 0 n Corollary 5.2). However, t turns out that the answer to them s the negatve. We omt here the proof of the equvalence; nstead we gve frst a smple counterexample to the general varatonal nequalty and then show how ths reflects n the L log L estmates of the complex Hlbert transform. Example 5.3 Choose 0 < ρ < 1 and for 1 n consder the dsjont dsks = (ρ, aρ ) where 0 < a < 1 ρ 1+ρ. Let also p = 1 n and f 0(z) = z z 1 K 1. Then whle p log p = (n + 1) log ρ + log n + log πa 2 p log f 0 n + 1 p K log ρ + log n + C 0 = 1 K p log p + K 1 K log n + C 1, where C 0, C 1 depend only on K and a. Lettng n shows that the varatonal nequalty fals for f 0. Proposton 5.4 For each M < there s an ε > 0 and a nonnegatve functon v L log L( ) such that Sv dm > (1 + ε) v(z) log (1 + M v(z) ) dm(z). v Proof: y nequalty (28) t suffces to show that for no M < does Sv dm hold for all nonnegatve functons v L log L( ). v(z) log (M v(z) ) dm(z) (29) v 19
We argue by contradcton. Hence consder frst the mappng f(z) = z z K 1 and mbedd t to a one parameter famly of quasconformal mappngs h t :, as n the proof of Lemma 3.3. Thus for t = log K, h t = f. Suppose next that we have dsjont open sets {D } n 1 and numbers p > 0 wth n1 p = 1. Set then p v t (z) = h t D χ h t D (z). Clearly v t dm = 1 and f ψ(t) = v t log(mv t ) dm then by Jensen s nequalty ψ(t) 0 for M π. Furthermore, we can deduce from the Gehrng-Rech dentty (17) that ψ p d (t) = h t D dt h td = φsv dm c(t) where c(t) s unformly bounded. Thus f (29) holds, then ψ (t) ψ(t) c(t) and after ntegraton ψ(t) e t ψ(0) e t t 0 c(s)e s ds = e t ψ(0) + c 1 (t). Takng t = log K we obtan ( p ) ( p ) p log K p log + C(K), fd D where C(K) = (K 1) log M + c 1 (log K). Fnally, f, p are as n the prevous example wth f 1 (z) = f 0 (z) = z z 1 K 1, we can choose D = f 0. ut ths would mean that p log f 0 p 1 K p log p C(K) K, contradctng Example 5.3. Therefore (29) cannot hold and so the estmate of Corollary 5.2 s sharp. REFERENCES [A] Ahlfors, L., On quasconformal mappngs. J. Analyse Math., 3 (1954), 1-58. [A] Ahlfors, L. & ers L., Remann s mappng theorem for varable metrcs. Ann. Math., 72 (1960), 385-404. [AM] Astala, K. & Martn, G., Holomorphc Motons. Preprnt, 1992. [P] ecker, J. & Pommerenke Chr., On the Hausdorff Dmenson of Quascrcles. Ann. Acad. Sc. Fenn. Ser. A I Math., 12 (1987), 329-333. [j] ojarsk,., Generalzed solutons of a system of dfferental equatons of frst order and ellptc type wth dscontnuous coeffcents. Math. Sb., 85 (1957), 451-503. [w] owen, R., Equlbrum States and the Ergodc Theory of Anosov Dffeomorphsms. Lecture Notes n Math., 470. Sprnger-Verlag, New York-Hedelberg, 1975. [DH] Douady, A. & Hubbard J., On the dynamcs of polynomal-lke mappngs. Ann. Sc. Ec. Norm. Sup., 18 (1985), 287-345. 20