Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains
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This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains © 2020 the Author(s) Published version Xu, Haiqing Xu, H. (2021). Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains. Journal of Geometric Analysis, 31(3), 2296-2330. https://doi.org/10.1007/s12220019-00340-x 2021
The Journal of Geometric Analysis https://doi.org/10.1007/s12220-019-00340-x Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains Haiqing Xu1 Received: 23 September 2019 © The Author(s) 2020 Abstract The conformal mapping f(z)=(z+1)2from Donto the standard cardioid has a homeomorphic extension of finite distortion to entire R2.We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from Donto cardioid-type domains. Keywords Extensions ·Homeomorphisms of finite distortion ·Inner cusp Mathematics Subject Classification 30C20 1 Introduction The standard cardioid domain ={(x,y)∈R2:(x2+y2)2−4x(x2+y2)−4y2<0}(1.0.1) is the image of the unit disk Dunder the conformal mapping g(z)=(z+1)2.Since the origin is an inner-cusp point of ∂, the Ahlfors’ three-point property fails, and hence ∂is not a quasicircle. Therefore the preceding conformal mapping does not possess a quasiconformal extension to the entire plane. However, there is a homeomorphic extension f:R2→R2by the Schoenflies theorem, see [10, Theorem 10.4]. Recall that homeomorphisms of finite distortion form a much larger class of homeomorphisms than quasiconformal mappings. A natural question arises: can we extend gas a homeomorphism of finite distortion? If we can, how good an extension can we find? Our first result gives a rather complete answer. BHaiqing Xu [email protected] 1Department of Mathematics and Statistics, University of Jyväskylä, PO BOX 35, 40014 Jyväskylä, Finland 123
H. Xu Theorem 1.1 Let Fbe the collection of homeomorphisms f :R2→R2of finite distortion such that f (z)=(z+1)2for all z ∈D.Then F=∅.Moreover sup{p∈[1,+∞):f∈F∩W1,p loc (R2,R2)}=+∞,(1.0.2) sup{q∈(0,+∞):f∈F,Kf∈Lq loc(R2)}=2,(1.0.3) sup{q∈(0,+∞):f∈F∩W1,p loc (R2,R2)for a fixed p >1and K f∈Lq loc(R2)} =1,(1.0.4) sup{p∈[1,+∞):f∈F,f−1∈W1,p loc (R2,R2)}=5 2(1.0.5) and sup{q∈(0,+∞):f∈F,Kf−1∈Lq loc(R2)}=5.(1.0.6) The cardioid curve ∂ contains an inner-cusp point of asymptotic polynomial degree 3/2.Motivated by this, we introduce a family of cardioid-type domains s with degree s>1,see (2.3.2). Our second result is an analog of Theorem 1.1. Theorem 1.2 Let g be a conformal map from Donto s,where sis defined in (2.3.2) and s >1.Suppose that Fs(g)is the collection of homeomorphisms f :R2→R2of finite distortion such that f |D=g.Then Fs(g)=∅.Moreover sup{p∈[1,+∞):f∈Fs(g)∩W1,p loc (R2,R2)}=+∞,(1.0.7) sup{q∈(0,+∞):f∈Fs(g), Kf∈Lq loc(R2)}=max 1 s−1,1,(1.0.8) sup{q∈(0,+∞):f∈Fs(g)∩W1,p loc (R2,R2)for a fixed p >1and K f∈Lq loc(R2)} =max 1 s−1,3p (2s−1)p+4−2s,(1.0.9) sup{p∈[1,+∞):f∈Fs(g), f−1∈W1,p loc (R2,R2)}=2(s+1) 2s−1(1.0.10) and sup{q∈(0,+∞):f∈Fs(g), Kf−1∈Lq loc(R2)}=s+1 s−1.(1.0.11) Let us recall previous extension results. In [3,4], sufficient conditions on are introduced to guarantee that a conformal mapping g:D→has a homeomorphic extension of locally exponentially integrable distortion to the whole plane. Specially, when is a Jordan domain with an outer-cusp point on its boundary, the authors from [8] established the optimal exponential regularity of distortion of homeomorphic extensions. In Sect. 2, we recall some basic definitions and facts. We also introduce auxiliary mappings and domains. In Sect. 3, we give upper bounds for integrability degrees of potential extensions. Section 4is devoted to the proof of Theorem 1.2. In Sect. 5we prove Theorem 1.1. 123
Optimal Extensions of Conformal Mappings from the Unit Disk 2 Preliminaries 2.1 Notation By s1 and t1 we mean that sis sufficiently large and tis sufficiently small, respectively. By fgwe mean that there exists a constant M>0 such that f(x)≤ Mg(x)for every x. We write f≈gif both fgand gfhold. By L2(respectively L1) we mean the 2-dimensional (1-dimensional) Lebesgue measure. Furthermore we refer to the disk with center Pand radius rby B(P,r), and S(P,r)=∂B(P,r). For asetE⊂R2we denote by Ethe closure of E.If A∈R2×2is a matrix, adj A is the adjoint matrix of A. 2.2 Basic Definitions and Facts Definition 2.1 Let ⊂R2and ⊂R2be domains. A homeomorphism f:→ is called K-quasiconformal if f∈W1,2 loc (, )and if there is a constant K≥1 such that |Df(z)|2≤KJf(z) holds for L2-a.e. z∈. Definition 2.2 Let ⊂R2be a domain. We say that a mapping f:→R2has finite distortion if f∈W1,1 loc (, R2), Jf∈L1 loc() and |Df(z)|2≤Kf(z)Jf(z)L2-a.e. z∈, (2.2.1) where Kf(z)=|Df(z)|2 Jf(z)for all z∈{Jf>0}, 1 for all z∈{Jf=0}. Note that a necessary condition in Definition 2.2 is that Jf(z)≥0forL2-a.e. z∈. When Jf(z)≤0forL2-a.e. z∈, we also define mappings of finite distortion. Modification on (2.2.1)isthat|Df(z)|2≤−Kf(z)Jf(z)for L2-a.e. z∈with Kf(z)=|Df(z)|2 −Jf(z)for all z∈{Jf<0}, 1 for all z∈{Jf=0}. Analogous explanation is applied to Definition 2.1. Definition 2.3 Given A⊂R2,amap f:A→R2is called an (l,L)-bi-Lipschitz mapping if 0 <l≤L<∞and l|x−y|≤|f(x)−f(y)|≤L|x−y| 123
H. Xu for all x,y∈A. If ⊂R2is a domain and f:→R2is an orientation-preserving bi-Lipschitz mapping, then fis quasiconformal. Definition 2.4 Given a function ϕdefined on set A⊂R2,its modulus of continuity is defined as ω(δ) ≡ω(δ,ϕ, A)=sup{|ϕ(z1)−ϕ(z2)|:z1,z2∈A,|z1−z2|≤δ} for δ≥0.Then ϕis called Dini-continuous if π 0 ω(t) tdt <∞, where the integration bound πcan be replaced by any positive constant. We say that a curve Cis Dini-smooth if it has a parametrization α(t)for t∈[0,2π] so that α(t)= 0 for all t∈[0,2π]and αis Dini-continuous. Definition 2.5 Let ⊂R2be open and f:→R2be a mapping. We say that f satisfies the Lusin (N) condition if L2(f(E)) =0 for any E⊂with L2(E)=0. Similarly, fsatisfies the Lusin (N−1) condition if L2(f−1(E)) =0 for any E⊂f() with L2(E)=0. Lemma 2.1 ([6, Theorem A.35]) Let ⊂R2be open and f ∈W1,1 loc (, R2). Suppose that ηis a nonnegative Borel measurable function on R2.Then η( f(x))|Jf(x)|dx≤f() η(y)N(f,,y)dy,(2.2.2) where the multiplicity function N(f,,y)of f is defined as the number of preimages of y under f in . Moreover (2.2.2)is an equality if we assume in addition that f satisfies the Lusin (N) condition. Let ⊂R2be open. Via Lemma 2.1, we have that if fis a W1,1 loc (, R2)homeomorphism, then Jf∈L1 loc(). (2.2.3) Lemma 2.2 ([6, Lemma A.28]) Suppose that f :R2→R2is a homeomorphism which belongs to W 1,1 loc (R2,R2). Then f is differentiable L2-a.e. on R2. Lemma 2.2 and a simple computation show that max θ∈[0,2π]|∂θf(z)|=Kf(z)min θ∈[0,2π]|∂θf(z)|L2-a.e. z∈R2(2.2.4) when f:R2→R2is a homeomorphism of finite distortion. Here ∂θf(z)= cos(θ) fx(z)+sin(θ) fy(z)for θ∈[0,2π]. 123
Optimal Extensions of Conformal Mappings from the Unit Disk Lemma 2.3 ([5, Theorem 1.2], [6, Theorem 1.6]) Let ⊂R2be a domain and f:→R2be a homeomorphism of finite distortion. Then f −1:f() →is also a homeomorphism of finite distortion. Moreover |Df−1(y)|2≤Kf−1(y)Jf−1(y)L2-a.e. y ∈f(). (2.2.5) Lemma 2.4 ([14, Theorem 2.1.11]) Let all ⊂R2, 1⊂R2and 2⊂R2be open, and T ∈Lip(1, 2). Suppose that both f ∈W1,p loc (, 1)and T ◦f∈Lp loc(, 2) hold for some p with 1≤p≤∞.Then T ◦f∈W1,p loc (, 2)and D(T◦f)(z)=DT(f(z))Df(z)L2-a.e. z ∈. Definition 2.6 A rectifiable Jordan curve in the plane is a chord-arc curve if there is a constant C>0 such that (z1,z2)≤C|z1−z2| for all z1,z2∈, where (z1,z2)is the length of the shorter arc of joining z1and z2. It is a well-known fact that a chord-arc curve is the image of the unit circle under a bi-Lipschitz mappings of the plane, see [7]. Thus chord-arc curves form a special class of quasicircles. The connections between chord-arc curves and quasiconformal theory can be found in [1,12]. 2.3 Definition of Cardioid-Type Domains Let s>1.We introduce a class of cardioid-type domains swhose boundaries contain internal polynomial cusps of order s, see Fig. 1. For technical reasons we do this in the following manner. Denote 1(s)={(u,v)∈R2:u∈[−1,0],v=(−u)s} and 2(s)={(u,v)∈R2:u∈[−1,0],v=−(−u)s}. Write 1(s)and 2(s)in the polar coordinate system as 1(s)={Rei:R=(−u)(1+(−u)2(s−1))1 2 and =π−arctan((−u)s−1)for u∈[−1,0]} and 2(s)={Rei:R=(−u)(1+(−u)2(s−1))1 2 123
H. Xu Fig. 1 Msand s and =−π+arctan((−u)s−1)for u∈[−1,0]}. Take the branch of complex-valued function z=w1/2with 11/2=1.Denote by m 1(s) and m 2(s)the images of 1(s)and 2(s)under the preceding z=w1/2,respectively. Then we can write m 1(s)and m 2(s)in the polar coordinate system as m 1(s)={reiθ:r=√−u(1+(−u)2(s−1))1 4 and θ=π−arctan((−u)s−1) 2for u∈[−1,0]} (2.3.1) and m 2(s)={reiθ:r=√−u(1+(−u)2(s−1))1 4 and θ=−π+arctan((−u)s−1) 2for u∈[−1,0]}. Denote by z1and z2the end points of m 1(s)∪m 2(s). Notice that there is a unique circle sharing both the tangent of m 1(s)at z1and the one of m 2(s)at z2.This circle is divided into two arcs by z1and z2.Concatenating m 1(s)∪m 2(s)with the arc located on the right-hand side of the line through z1and z2, we then obtain a Jordan curve m(s). Denote by (s)the image of m(s)under z2.Let Msand sbe the interior domains of m(s)and (s), respectively. (2.3.2) Then sis the desired cardioid-type domain with degree s. Moreover m(s), (s), Ms and sare symmetric with respect to the real axis. By the Riemann mapping theorem, there is a conformal mapping from D∩R2 + onto Ms∩R2 +such that D∩Ris mapped onto Ms∩R.It follows from the Schwarz reflection principle that there is a conformal mapping gs:D→Ms.(2.3.3) such that gs(¯z)=gs(z)for all z∈D.Moreover by the Osgood–Carathéodory theorem gshas a homeomorphic extension from Donto Ms,still denoted gs. 123
Optimal Extensions of Conformal Mappings from the Unit Disk Lemma 2.5 Let Msand gsbe as in (2.3.2)and (2.3.3)with s >1.Then gsis a bi-Lipschitz mapping on D. Proof If ∂Mswere a Dini-smooth Jordan curve, from [11, Theorem 3.3.5] it would follow that g sis continuous on Dand g s(z)= 0 for all z∈D.Since Msis convex, the mean value theorem would then yield that gsis a bi-Lipschitz map from Donto Ms. In order to prove that ∂Msis a Dini-smooth Jordan curve, we first analyze ∂Msin a neighborhood of the origin. For any point in m 1with Euclidean coordinate (x,y), we have x=rcos θand y=rsin θ. (2.3.4) where both rand θshare the expression in (2.3.1). We then obtain that r≈√−u,θ≈π 2,∂r ∂u≈−1 √−uand ∂θ ∂u≈(−u)s−2(2.3.5) whenever |u|1.Therefore from (2.3.4) and (2.3.5), it follows that x≈(−u)s−1 2,y≈(−u)1 2,∂x ∂u≈−(−u)s−3 2and ∂y ∂u≈−(−u)−1 2. Together with symmetry of ∂Ms,we conclude that ∂x ∂y≈|y|2(s−1)whenever |y|1. Next, notice that the part of ∂Msaway from the origin is piecewise smooth. By parametrizing ∂Msas α(y)=(x(y), y), we then obtain that the modulus of continuity of αsatisfies ω(δ,α,∂Ms)≤max{δ2(s−1),δ}∀δ1. Consequently αis Dini-continuous. Therefore ∂Msis a Dini-smooth Jordan curve. Remark 2.1 Since gs:S1→∂Msis a bi-Lipschitz map by Lemma 2.5,via[13, Theorem A] there is a bi-Lipschitz mapping gc s:Dc→Mc ssuch that gc s|S1=gs.Let Gs(z)=gs(z)∀z∈D, gc s(z)∀z∈Dc.(2.3.6) Then Gsis an orientation-preserving bi-Lipschitz mapping. Lemma 2.6 Let h1:R2→R2be a homeomorphism of finite distortion, and h2: R2→R2be an (l,L)-bi-Lipschitz, orientation-preserving mapping. Then h1◦h2is a homeomorphism of finite distortion. Proof Since h2is an orientation-preserving bi-Lipschitz mapping, we have that h2is quasiconformal. From [2, Corollary 3.7.6] it then follows that 123
H. Xu h2satisfies Lusin (N)and (N−1)condition, (2.3.7) Jh2>0L2-a.e. on R2.(2.3.8) By Lemma 2.2 we have both h1and h2are differentiable L2-a.e. on R2.(2.3.9) From (2.3.9) and (2.3.7) it therefore follows that h1◦h2is differentiable L2-a.e. on R2,and D(h1◦h2)(z)=Dh1(h2(z))Dh2(z)L2-a.e. z∈R2.(2.3.10) From (2.3.10) and the distortion inequalities for h1and h2it follows that |D(h1◦h2)(z)|2≤|Dh1(h2(z))|2|Dh2(z)|2≤Kh1(h2(z))Kh2(z)Jh1(h2(z))Jh2(z) =Kh1(h2(z))Kh2(z)Jh1◦h2(z)(2.3.11) for L2-a.e. z∈R2. To prove that h1◦h2is a homeomorphism of finite distortion, via (2.2.3) and (2.3.11) it is sufficient to prove that h1◦h2∈W1,1 loc (R2,R2). Since h2is an (l,L)-bi-Lipschitz orientation-preserving mapping, by (2.3.9) and (2.2.4) we then have that l≤|Dh2(z)|≤Land 1 ≤Kh2(z)≤L lL2-a.e. z∈R2.(2.3.12) From(2.3.8), (2.3.12), and (2.2.1) it then follows that l3 L≤Jh2(z)≤L2L2-a.e. z∈R2.(2.3.13) By (2.3.10), (2.3.12), (2.3.13), and Lemma 2.1, we therefore have M|D(h1◦h2)(z)|dz≤M|Dh1(h2(z))||Dh2(z)| Jh2(z)Jh2(z)dz ≈M|Dh1(h2(z))|Jh2(z)dz =h2(M)|Dh1(w)|dw<∞ for any compact set M⊂R2,where the last inequality is from h1∈W1,1 loc (R2,R2). 3 Bounds for Integrability Degrees For a given s>1,let Msas in (2.3.2). Define 123
Optimal Extensions of Conformal Mappings from the Unit Disk Lemma 3.5 Let sbe as in (2.3.2)with s >1.Suppose that f :R2→R2is a homeomorphism of finite distortion such that f maps Dconformally onto s.We have that (1) if f −1∈W1,p loc (R2,R2)for some p ≥1then p <2(s+1)/(2s−1), (2) if K f−1∈Lq loc(R2)for some q ≥1then q <(s+1)/(s−1), (3) if K f∈Lq loc(R2)for some q ≥1then q <max{1,1/(s−1)}, (4) if s >2,f∈W1,p loc (R2,R2)for some p >1and K f∈Lq loc for some q ∈(0,1), then q <3p/((2s−1)p+4−2s). Proof Let gsbe as in (2.3.3), and hs=z2◦gs.Since hs:D→sis conformal, there is a Möbius transformation ms(z)=eiθz−a 1−¯az where θ∈[0,2π]and |a|<1 such that f(z)=hs◦ms(z)for all z∈D.Since ms:S1→S1is a bi-Lipschitz mapping, by [13, Theorem A] there is a bi-Lipschitz mapping mc s:Dc→c ssuch that mc s|S1=ms.Define Ms(z)=ms(z)z∈D, mc s(z)z∈Dc.(3.0.30) Then Ms:R2→R2is a bi-Lipschitz, orientation-preserving mapping. Let Gsbe as in (2.3.6). Define E=f◦M−1 s◦G−1 s:R2→R2. Lemma 2.6 implies that E∈Es,where Esis from (3.0.1). From Lemmas 2.2 and 2.3, it follows that both f−1and E−1are differentiable L2-a.e. on R2.(3.0.31) Since f−1(z1)−f−1(z2) z1−z2 =E−1(z1)−E−1(z2) z1−z2 (G−1 s(E−1(z1)) −(G−1 s(E−1(z2)) E−1(z1)−E−1(z2) × ×M−1 s(G−1 s◦E−1(z1)) −M−1 s(G−1 s◦E−1(z2)) G−1 s◦E−1(z1)−G−1 s◦E−1(z2) for all z1,z2∈R2with z1= z2,by (3.0.31) and the bi-Lipschitz properties of G−1 s and M−1 swe have that Df−1(z)≈DE−1(z),(3.0.32) 123
H. Xu max θ∈[0,2π]∂θf−1(z)≈max θ∈[0,2π]∂θE−1(z), min θ∈[0,2π]∂θf−1(z)≈min θ∈[0,2π]∂θE−1(z)(3.0.33) for L2-a.e. z∈R2.If f−1∈W1,p loc for some p≥1,Lemma 3.2 together with (3.0.34) gives p<2(s+1)/(2s−1). By (3.0.33) and (2.2.4) we have that Kf−1(z)≈KE−1(z)L2-a.e. z∈R2.(3.0.34) If Kf−1∈Lq loc(R2)for some q≥1,combining (3.0.32) and Lemma 3.1 then yields q<(s+1)/(s−1). By Lemma 2.2 and 2.6, we have that both fand Eare differentiable L2-a.e. on R2.(3.0.35) From [2, Corollary 3.7.6], Gs◦Mssatisfies Lusin (N) and (N−1)conditions. Since f(z1)−f(z2) z1−z2 =E(Gs◦Ms(z1)) −E(Gs◦Ms(z2)) Gs◦Ms(z1)−Gs◦Ms(z2) |Gs(Ms(z1)) −Gs(Ms(z2)) Ms(z1)−Ms(z2) × ×Ms(z1)−Ms(z2) z1−z2 for all z1,z2∈R2with z1= z2,from (3.0.35) and the bi-Lipschitz properties of Gs and Mswe have that |Df(z)|≈|DE(Gs◦Ms(z))|,(3.0.36) max θ∈[0,2π]|∂θf(z)|≈ max θ∈[0,2π]|∂θE(Gs◦Ms(z))|,(3.0.37) min θ∈[0,2π]|∂θf(z)|≈ min θ∈[0,2π]|∂θE(Gs◦Ms(z))|(3.0.38) for L2-a.e. z∈R2.By (2.2.4), (3.0.37), and (3.0.38), we have that Kf(z)≈KE(Gs◦Ms(z)) L2-a.e. z∈R2.(3.0.39) Via the same reasons as for (2.3.13), we have that JGs◦Ms(z)≈1L2-a.e. z∈R2.(3.0.40) By (3.0.40) and Lemma 2.1, we derive from (3.0.39) that 123
Optimal Extensions of Conformal Mappings from the Unit Disk A Kq f(z)dz=A Kq E(Gs◦Ms(z)) JGs◦Ms(z) JGs◦Ms(z) dz ≈A Kq E(Gs◦Ms(z))JGs◦Ms(z)dz=Gs◦Ms(A) Kq E(w) dw (3.0.41) for any q≥0 and any compact set A⊂R2.By (3.0.36) and Lemma 2.1, we obtain that A|Df(z)|p=A|DE(Gs◦Ms(z))|pJGs◦Ms(z) JGs◦Ms(z)dz ≈A|DE(Gs◦Ms(z))|pJGs◦Ms(z)dz=Gs◦Ms(A)|DE|p(w) dw (3.0.42) for any p≥0.If Kf∈Lq loc(R2)for some q≥1,Lemma 3.3 together with (3.0.41) gives that q<max{1,1/(s−1)}.If f∈W1,p loc and Kf∈Lq loc for some p>1 and some q∈(0,1), combining Lemma 3.4 with (3.0.42) then implies q<3p/((2s− 1)p+4−2s). Under a more general assumption that fin Lemma 3.5 is K-quasiconformal from D onto s,authors from [4, Theorem 4.4] showed a result analogous to Lemma 3.5 (3). 4 Proof of Theorem 1.2 4.1 Prove that the Class Fs(g)from Theorem 1.2 is Nonempty Proof Let gbe as in Theorem 1.2. The beginning of proof for Lemma 3.4 shows that g=z2◦gs◦ms, where ms:D→Dis a Möbius transformation and gs:D→Msfrom (2.3.3)isa conformal mapping. Recall that ms(or gs) has a bi-Lipschitz extension Ms:R2→R2 (or Gs:R2→R2)asin(3.0.30)(or(2.3.6)). Via Lemma 2.6, it suffices to prove that z2:Ms→shas a homeomorphic extension E:R2→R2of finite distortion. Then f:= E◦Gs◦Ms∈Fs(g). (4.1.1) We divide the construction of Einto two steps. Step 1: we construct E1in a neighborhood of the cusp point, see Fig. 2. To be precise, we define f1, ..., f4and let E1be the sum of compositions of f1, ... f4. 123
H. Xu Fig. 2 The construction f−1 3◦f−1 4◦f2◦f−1 1:Qt→˜ Qt Aim 1: to define f1and f2.Fixs>1,and define η(x)=√x(1+x2(s−1))1 4for all x>0.(4.1.2) Then η(x)=(1+x2(s−1))1 4 2√x1+(s−1)x2s−2 1+x2(s−1).(4.1.3) For a given t1,let L1 t=η((t/2)2), L2 t=η(t2), and σt=L2 t−L1 t.(4.1.4) Then L1 t≈t/2,L2 t≈tand σt≈t/2 whenever t1.Set Qt=B(0,L2 t)\(B(0,L1 t)∪Ms), and f1(x,y)=xeiy ∀x≥0 and y∈[0,2π]. (4.1.5) Let (r)be the length of f−1 1(Qt)∩{(x,y)∈R2:x=r}.Define 123
Optimal Extensions of Conformal Mappings from the Unit Disk f2(r,θ)=r,σt (r)(π −θ)∀(r,θ)∈f−1 1(Qt). (4.1.6) Since ∂Msis mapped onto ∂sby z2,we have that (r)=π+arctan τ2(s−1)and r=η(τ2)(4.1.7) for all τ∈(t/2,t). Then (r)≈πand r≈τwhenever τ1.From (4.1.3), it follows that ∂r ∂τ ≈1.Together with ∂ ∂τ ≈τ2s−3,we have that ∂(r) ∂r≈r2s−3for all r1.(4.1.8) Denote Rt=f2◦f−1 1(Qt). Then Rt=[L1 t,L2 t]×[−σt/2,σ t/2].Combining (4.1.5) with (4.1.6) implies f1◦f−1 2(x,y)=−xcos (x)y σt ,xsin (x)y σt∀(x,y)∈Rt. Therefore Df1◦f−1 2(x,y)=−cos (x)y σt+xy(x) σtsin (x)y σt x(x) σtsin (x)y σt sin (x)y σt+xy(x) σtcos (x)y σt x(x) σtcos (x)y σt.(4.1.9) By (4.1.4), (4.1.7), and (4.1.8), we deduce from (4.1.9) that |Df1◦f−1 2(x,y)|1 and Jf1◦f−1 2(x,y)=−x(x) σ≈−1 (4.1.10) for all t1 and each (x,y)∈Rt.Since Kf1◦f−1 2≥1,from (4.1.10)wehave Kf1◦f−1 2≈1.(4.1.11) By (4.1.10) again we have that |Df2◦f−1 1|=|adjDf1◦f−1 2| |Jf1◦f−1 2|≈|Df1◦f−1 2|1 and Jf2◦f−1 1=1 Jf1◦f−1 2≈−1. (4.1.12) Analogously to (4.1.11), we have that Kf2◦f−1 1(x,y)≈1∀t1 and ∀(x,y)∈Qt.(4.1.13) Aim 2: to define f3:˜ Qt→˜ Rt.Let ˜ Qt={(x,y)∈R2:x∈[−t2,−(t/2)2],|y|≤|x|s}. 123
H. Xu Define f3(u,v)=−u,t2s (−u)sv∀(u,v)∈˜ Qt. Then f3is diffeomorphic and Df3(u,v)=−10 st2s (−u)s+1vt2s (−u)s.(4.1.14) From (4.1.14) we have that |Df3|1 and Jf3≈−1∀(u,v)∈˜ Qt.(4.1.15) Analogously to (4.1.11), we have that Kf3(u,v)≈1∀t1 and ∀(u,v)∈˜ Qt.(4.1.16) Let ˜ Rt=f3(˜ Qt). Then ˜ Rt=[(t/2)2,t2]×[−t2s,t2s].The same reasons as for (4.1.12) and (4.1.13) imply that |Df−1 3(x,y)|1,Jf−1 3(x,y)≈−1 and Kf−1 3(x,y)≈1 (4.1.17) for all t1 and (x,y)∈˜ Rt. Aim 3: to define f4:˜ Rt→Rt. Denote by P1,P2,P3,P4and ˜ P1,˜ P2,˜ P3,˜ P4the four vertices of ˜ Rtand Rt,respectively. Then P1=L1 t,σt 2,P2=L2 t,σt 2,P3=L2 t,−σt 2,P4=L1 t,−σt 2 and ˜ P1=(t/2)2,t2s,˜ P2=(t2,t2s), ˜ P3=(t2,−t2s), ˜ P4=((t/2)2,−t2s). Since ∂Msis mapped onto ∂sby z2,the line segment ˜ P1˜ P2is mapped onto P1P2 by (u,t2s)→ η(u), σt 2∀u∈[(t/2)2,t2], and the line segment ˜ P4˜ P3is mapped onto P4P3by (u,−t2s)→ η(u), −σt 2∀u∈[(t/2)2,t2]. 123
Optimal Extensions of Conformal Mappings from the Unit Disk Define f4(u,v)=η(u), σt 2t2sv∀(u,v)∈˜ Rt.(4.1.18) Then f4is a diffeomorphism from ˜ Rtonto Rtand Df4(u,v)=η(u)0 0σt 2t2s.(4.1.19) By (4.1.3) and (4.1.4) we have that η(u)≈t−1and σt 2t2s≈t1−2swhenever t1 and (u,v)∈˜ Rt.It follows from (4.1.19) that |Df4(u,v)|≈t1−2sand Jf4(u,v)≈t−2s(4.1.20) for all t1 and all (u,v)∈˜ Rt.Then Kf4(u,v)=|Df4(u,v)|2 Jf4(u,v) ≈t2−2s∀t1 and (u,v)∈˜ Rt.(4.1.21) The same reasons as for (4.1.12) and (4.1.13) imply that Df−1 4(x,y)≈t,Jf−1 4(x,y)≈t2sand Kf−1 4(x,y)≈t2−2s(4.1.22) for all t1 and all (x,y)∈Rt. Aim 4: to define E1. Set Ft=f−1 3◦f−1 4◦f2◦f−1 1. Then Ftis a diffeomorphism from Qtonto ˜ Qt.Therefore DF t(z)=Df−1 3(f−1 4◦f2◦f−1 1(z))Df−1 4(f2◦f−1 1(z))D(f2◦f−1 1)(z) for all z∈Qt.From (4.1.17), (4.1.22), and (4.1.12) it then follows that Qt|DF t|pdz≤QtDf−1 3(f−1 4◦f2◦f−1 1) pDf−1 4(f2◦f−1 1) pDf2◦f−1 1 p dz tpL2(Qt)≈t2+p(4.1.23) for all p≥0.For a fixed large j0,we now consider the set Qtwith t=2−jfor all j≥j0.Define E1=+∞ j=j0 F2−jχQ2−j.(4.1.24) 123
H. Xu Denote 1=∪ +∞ j=j0Q2−jand ˜ 1=∪ +∞ j=j0˜ Q2−j.Then E1is a homeomorphism from 1onto ˜ 1,and satisfies (2.2.1)forE1on L2-a.e. 1.In order to prove that E1has finite distortion on 1,via (2.2.3) it thus suffices to prove that E1∈W1,1 loc (1,˜ 1). Actually, from (4.1.23) we have that 1|DE1|p=+∞ j=j0Q2−j|DF 2−j(z)|pdz+∞ j=j0 2−j(2+p)<∞(4.1.25) for all p≥1. Step 2: we construct E2on the domain away from the cusp point. Denote 2=Mc s\1and ˜ 2=c s\˜ 1. Notice that both ∂2and ∂˜ 2are piecewise smooth Jordan curves with nonzero angles at the two corners. Therefore both ∂2and ∂˜ 2are chord-arc curves. By [7] there are bi-Lipschitz mappings H1:R2→R2and H2:R2→R2(4.1.26) such that H1(S1)=∂2and H2(S1)=∂˜ 2.Define h(z)=E1(z)∀z∈∂2∩∂1, z2∀z∈∂2∩∂Ms. Then his a bi-Lipschitz mapping in terms of the arc lengths. By the chord-arc properties of both ∂2and ∂˜ 2,we have that his also a bi-Lipschitz mapping with respect to the Euclidean distances. Taking (4.1.26) into account, we conclude that H−1 2◦h◦H1: S1→S1is a bi-Lipschitz mapping. By [13, Theorem A] there is then a bi-Lipschitz mapping H:R2→R2(4.1.27) such that H|S1=H−1 2◦h◦H1.Define E2=H2◦H◦H−1 1.(4.1.28) By (4.1.26) and (4.1.27), we have that E2is a bi-Lipschitz extension of h.Furthermore since degMs(h,w) =1,we obtain that E2is orientation-preserving. Hence E2is a quasiconformal mapping. The same reasons as for (2.3.12) and (2.3.13)imply |DE2(z)|,KE2(z), and JE2(z)are bounded from both above and below (4.1.29) 123
Optimal Extensions of Conformal Mappings from the Unit Disk for L2-a.e. z∈R2,and DE−1 2(w),K−1 E2(w) and J−1 E2(w) are bounded from both above and below (4.1.30) for L2-a.e. w∈R2. Via (4.1.24) and (4.1.28), we set E(x,y)=⎧ ⎪ ⎨ ⎪ ⎩ E1(x,y)for all (x,y)∈1, E2(x,y)for all (x,y)∈2, (x2−y2,2xy)for all (x,y)∈Ms. (4.1.31) By the properties of E1and E2,we conclude that E∈Es. 4.2 Proof of (1.0.7), (1.0.10), and (1.0.11)inTheorem1.2 Proof of (1.0.7)Letgbe as in Theorem 1.2. It suffices to check that there is f∈Fs(g) satisfying that f∈W1,p loc (R2,R2)for all p≥1.Let fbe as in (4.1.1) and Ebe as in (4.1.31). By (4.1.25), (4.1.29), and the fact that E(z)=z2for all z∈Ms,we obtain that E∈W1,p loc (R2,R2)for all p≥1.By (3.0.42)f∈W1,p loc (R2,R2)for all p≥1. Proof of (1.0.10)Letgbe as in Theorem 1.2. By Lemma 3.5 (1) it suffices to construct af∈Fs(g)satisfying that f−1∈W1,p loc (R2,R2)for all p<2(s+1)/(2s−1). Let fbe as in (4.1.1) and Ebe as in (4.1.31). Via (3.0.32) it suffices to check that E−1∈W1,p loc (R2,R2)for all p<2(s+1)/(2s−1). By (4.1.15), (4.1.20), and (4.1.10), we have that DF−1 2−j(w)≤Df1◦f−1 2(f4◦f3(w))Df4(f3(w))Df3(w)2j(2s−1) for all j≥j0and L2-a.e. w∈˜ Q2−j.Together with L2(˜ Q2−j)≈2−2j(s+1),we hence obtain that ˜ 1DE−1 1 p=+∞ j=j0˜ Q2−jDF−1 2−j p+∞ j=j0 2−j(2(s+1)+p(1−2s)) <∞(4.2.1) for all p<2(s+1)/(2s−1). Since DE−1(u,v) (u2+v2)−1/4∀(u,v)∈s,(4.2.2) by a change of variables we have that sDE−1(w) p dw2π 01 0 r1−p 2drdθ≈1 0 r1−p 2dr<∞(4.2.3) 123
H. Xu for all p<2(s+1)/(2s−1). By (4.1.30), (4.2.1), and (4.2.3), we conclude that E−1∈W1,p loc (R2,R2)for all p<2(s+1)/(2s−1). Proof of (1.0.11)LetEbe as in (4.1.31). Analogously to the proof of (1.0.10), it suffices to check that KE−1∈Lq loc(R2)for all q<(s+1)/(s−1). Note that Lemma 3.5 (2) and (3.0.34)playgamenow.From(4.1.11), (4.1.21), and (4.1.16), we have that KF−1 2−j (w) =Kf1◦f−1 2(f4◦f3(w))Kf4(f3(w))Kf3(w) ≈2j(2s−2) for all j≥j0and L2-a.e. w∈˜ Q2−j.Together with L2(˜ Q2−j)≈2−j2(s+1),we then obtain that ˜ 1 Kq E−1=+∞ j=j0˜ Q2−j Kq F−1 2−j +∞ j=j0 22j[(s−1)q−(s+1)]<∞(4.2.4) for all q<(s+1)/(s−1). By (4.1.30), (4.2.4), and the fact that Eis conformal on Ms,we conclude that KE−1∈Lq loc(R2)for all q<(s+1)/(s−1). 4.3 Proof of (1.0.8)inTheorem1.2 Proof Analogously to the proof of (1.0.10) in Sect. 4.2, via Lemma 3.5 (3) and (3.0.41) it suffices to construct a E∈Essatisfying that KE∈Lq loc for all q<max{1,1/(s−1)}. The construction is divided into two cases. Case 1: s∈(1,2).Let Ebe as in (4.1.31). From (4.1.17), (4.1.22), and (4.1.13), it follows that KF2−j(z)=Kf−1 3(f−1 4◦f2◦f−1 1(z))Kf−1 4(f2◦f−1 1(z))Kf2◦f−1 1(z)≈22j(s−1) for all j≥j0and L2-a.e. z∈Q2−j.Together with L2(Q2−j)≈2−2jwe then have that 1 Kq E=+∞ j=j0Q2−j Kq F2−j≈+∞ j=j0 22j(q(s−1)−1)<∞(4.3.1) for all q<1/(s−1). By (4.3.1), (4.1.29), and the fact that Eis conformal on Ms, we conclude that KE∈Lq loc(R2)for all q<1/(s−1). Therefore we have proved (1.0.8) whenever s∈.(1,2). Case 2: s∈[2,∞).Except for redefining f−1 4:Rt→˜ Rtas in (4.1.18), we follow all processes in Sect. 4.1 to define a new E,see Fig. 3. To redefine f−1 4,we should define mappings A,B,C. We begin with notation. Let αtand βtbe the length of sides of ˜ Rt,and γtbe the length of a side of Rt.Whenever t1,we have that αt=t2−(t/2)2≈t2,β t=2t2sand γt=η(t2)−η((t/2)2)≈t.(4.3.2) 123
Optimal Extensions of Conformal Mappings from the Unit Disk From (4.1.12), (4.1.17), (4.3.14), (4.3.22), and (4.3.26), we have that |DF 2−j(z)|2−j δ2−j∀z∈f1◦f−1 2(∪4 k=1Tk), 2j(1−2s)∀z∈f1◦f−1 2(T0), (4.4.2) for all j≥j0.It follows from (4.4.2) and (4.3.29) that Q2−jDF 2−j p= 4 k=0f1◦f−1 2(Tk)DF 2−j pδ1−p 2−j2−j(2+p)+2j(p(1−2s)−2). Therefore 1|DE|p=+∞ j=j0Q2−jDF 2−j p+∞ j=j0 1 jp++∞ j=j0 2−j(p(2s−1)+2)<∞. (4.4.3) By (4.4.3), (4.1.29), and the fact that E(z)=z2for all z∈Ms,we conclude that E∈W1,p loc (R2,R2). By (4.1.12), (4.1.13), Lemma 2.1, and (4.1.17), we have f1◦f−1 2(T1) Kq F2−j≈f1◦f−1 2(T1) Kq f−1 3 (f−1 4◦f2◦f−1 1)Kq f−1 4 (f2◦f−1 1)Kq f2◦f−1 1Jf2◦f−1 1 ≤T1 Kq f−1 3 (f−1 4)Kq f−1 4 T1 Kq f−1 4 (4.4.4) for all q≥0 and all j≥j0.Notice ˜ (γ2−j/2)=α2−jand ˜ (γ2−j(1 2−δ2−j)) =β2−j/2 for all j≥1.By Fubini’s theorem, (4.3.16), (4.3.6), and (4.3.2), we then have T1 Kq f−1 4 γ2−j 2 γ2−j(1 2−δ2−j)xp+(y) xp2j(2s−4) δ2−j˜ (y)q dxdy ≈2jq(2s−4)γ2−j δq 2−j γ2−j 2 γ2−j(1 2−δ2−j) 1 ˜ q(y)dy =2jq(2s−4)γ2−j (1−q)δq 2−j 2δ2−jγ2−j 2α2−j−β2−j˜ 1−q(γ2−j 2)−˜ 1−q(γ2−j(1 2−δ2−j)) δ1−q 2−j2−2j[1+q(1−s)] 1−M(p,s)(4.4.5) for any fixed q∈(0,M(p,s)). Combining (4.4.4) with (4.4.5)impliesthat f1◦f−1 2(T1) Kq F2−jδ1−q 2−j2−2j[1+q(1−s)]∀j≥j0.(4.4.6) 123
H. Xu By symmetry of f−1 4between T1and T3,it follows from (4.4.6) that f1◦f−1 2(T3) Kq F2−j=f1◦f−1 2(T1) Kq F2−jδ1−q 2−j2−2j[1+q(1−s)](4.4.7) for all j≥j0.By (4.3.31) and (4.3.29), we have that f1◦f−1 2(T0) Kq F2−j2−2j(4.4.8) and f1◦f−1 2(T2∪T4) Kq F2−jδ2−j2−2j22j(s−1) δ2−jq =δ1−q 2−j22j[q(s−1)−1](4.4.9) for all j≥j0.From (4.4.6), (4.4.7), (4.4.8), and (4.4.9), we conclude that 1 Kq E=+∞ j=j0Q2−j Kq F2−j=+∞ j=j0 4 k=0f1◦f−1 2(Tk) Kq F2−j +∞ j=j0 2−2j+2−j(p+2)(1−q) p−1+2[1+q(1−s)]log p(1−q) p−12j.(4.4.10) Note that (p+2)(1−q) p−1+2[1+q(1−s)]>0⇔q<M(p,s). It from (4.4.10) follows that 1Kq E<∞for all q∈(0,M(p,s)). Together with (4.1.29) and the fact that Eis conformal on Ms,we conclude that KE∈Lq loc(R2)for all q∈(0,M(p,s)). 5 Proof of Theorem 1.1 Proof Let be as in (1.0.1). The representation of ∂ in Cartesian coordinates is (x2+y2)2−4x(x2+y2)−4y2=0. Hence we can parametrize ∂ in a neighborhood of the origin as ˜ 0={(x,y)∈R2:x∈[−2−j0,0],y2=d(x)}, 123
Optimal Extensions of Conformal Mappings from the Unit Disk M E Ωd Ω1 ˜ Ω1 Δ ˜ Ωd Fig. 4 The existence of an extension where j01 and d(x)=−x3(4−x) 2−x2+2x+√1+2x.Since d(x)≈|x|3for all |x|1,there are c1>0,c2>0 such that −c1x3≤d(x)≤−c2x3∀x∈[−2−j0,0]. Denote ˜ 1={(x,y)∈R2:x∈[−2−j0,0],y2=−c1x3}, ˜ 2={(x,y)∈R2:x∈[−2−j0,0],y2=−c2x3}, ˜ 3={(x,y)∈R2:x=−2−j0,y2∈[c1(2−j0)3,d(−2−j0)}, ˜ 4={(x,y)∈R2:x=−2−j0,y2∈[d(−2−j0), c2(2−j0)3]}. Let ˜ uand ˜ dbe the domains bounded by ˜ 0∪˜ 2∪˜ 4and ˜ 0∪˜ 1∪˜ 3,respectively. Denote by u, dand kfor k=0, ..., 4 the images of ˜ u,˜ dand ˜ kunder the branch of complex-valued function z1/2with 11/2=1,respectively. We first prove the existence of an extension, see Fig. 4. Let r=(2−2j0+c12−3j0)1/4.Denote M={(x+1,y)∈R2:(x,y)∈D}, 1=B(0,r)\(M∪d), 2=R2\(1∪d∪M), ˜ 1={(x,y)∈R2:x∈[−2−j0,0],y2≤c1|x|3}and ˜ 2=R2\(˜ 1∪˜ d∪). Analogously to the arguments in Sect. 4.1, we define E1:1→˜ 1and E2:2→ ˜ 2.Here η(x)=√x(1+c1x)1/4and s=3/2.Define E(x,y)=⎧ ⎪ ⎨ ⎪ ⎩ E1(x,y)∀(x,y)∈1, E2(x,y)∀(x,y)∈2, (x2−y2,2xy)∀(x,y)∈M∪d, (5.0.1) and f0(x,y)=E(x+1,y). By the analogous arguments as in Sect. 4.1, we have that f0∈F. 123
H. Xu We next prove (1.0.3). Suppose f∈F.Then ˆ f(u,v) =f(u−1,v)is a homeomorphism of finite distortion on R2and ˆ f(M\u)=\˜ u.By Remark 3.1,we have that if Kˆ f∈Lq loc(R2)then q<2.Therefore if Kf∈Lq loc(R2)then q<2. In order to prove (1.0.3), it then suffices to construct a mapping f0∈Fsuch that Kf0∈Lq loc(R2)for all q<2.Let Ebe as in (5.0.1) and f0(x,y)=E(x+1,y). Then f0∈F.The same arguments as for the case s∈(1,2)in Sect. 4.3 show that KE∈Lq loc(R2)for all q<2.Therefore Kf0∈Lq loc(R2)for all q<2. The strategies to prove (1.0.2), (1.0.4), (1.0.5), and (1.0.6) are same as the one to prove (1.0.3). We leave the details to the interested reader. Acknowledgements Open access funding provided by University of Jyväskylä (JYU). The author has been supported by China Scholarship Council (project No.201706340060) and Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Grant No. 307333). This paper is a part of the author’s doctoral thesis. The author thanks his advisor Professor Pekka Koskela for posing this question and for valuable discussions. The author thanks Zheng Zhu for comments on the earlier draft. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Astala, K., González, M.: Chord-arc curves and the Beurling transform. Invent. Math. 205(1), 57–81 (2016) 2. Astala, K., Iwaniec, T., Martin, G.J.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, vol. 48. Princeton University Press, Princeton, NJ, pp. xviii+677 (2009) 3. Guo, C.-Y.: Generalized quasidisks and conformality II. Proc. Am. Math. Soc. 143(8), 3505–3517 (2015) 4. Guo, C.-Y., Koskela, P., Takkinen, J.: Generalized quasidisks and conformality. Publ. Mat. 58(1), 193–212 (2014) 5. Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180(1), 75–95 (2006) 6. Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion. Lecture Notes in Mathematics, 2096. Springer, Cham, pp. xii+176 (2014) 7. Jerison, D., Kenig, C.: Hardy spaces, A∞, and singular integrals on chord-arc domains. Math. Scand. 50(2), 221–247 (1982) 8. Koskela, P., Takkinen, J.: Mappings of finite distortion: formation of cusps. Publ. Mat. 51(1), 223–242 (2007) 9. Koskela, P., Takkinen, J.: Mappings of finite distortion: formation of cusps. III. Acta Math. Sin. 26(5), 817–824 (2010) 10. Moise, E.: Geometric Topology in Dimensions 2 and 3. Graduate Texts in Mathematics, Vol. 47. Springer-Verlag, New York-Heidelberg, pp. x+262 (1977) 11. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), 299. Springer-Verlag, Berlin, pp. x+300 (1992) 12. Semmes, S.: Quasiconformal mappings and chord-arc curves. Trans. Am. Math. Soc. 306(1), 233–263 (1988) 123
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