Extension of boundary homeomorphisms to mappings of finite distortion
EEXTENSION OF BOUNDARY HOMEOMORPHISMS TOMAPPINGS OF FINITE DISTORTION
CHRISTINA KARAFYLLIA AND DIMITRIOS NTALAMPEKOS
Abstract.
We provide sufficient conditions so that a homeomorphism of thereal line or of the circle admits an extension to a mapping of finite distortion inthe upper half-plane or the disk, respectively. Moreover, we can ensure that thequasiconformal dilatation of the extension satisfies certain integrability condi-tions, such as p -integrability or exponential integrability. Mappings satisfyingthe latter integrability condition are also known as David homeomorphisms.Our extension operator is the same as the one used by Beurling and Ahlforsin their celebrated work. We prove an optimal bound for the quasiconfor-mal dilatation of the Beurling–Ahlfors extension of a homeomorphism of thereal line, in terms of its symmetric distortion function. More specifically, thequasiconformal dilatation is bounded above by an average of the symmetricdistortion function and below by the symmetric distortion function itself. Asa consequence, the quasiconformal dilatation of the Beurling–Ahlfors exten-sion of a homeomorphism of the real line is (sub)exponentially integrable, is p -integrable, or has a BMO majorant if and only if the symmetric distortion is(sub)exponentially integrable, is p -integrable, or has a BMO majorant, respec-tively. These theorems are all new and reconcile several sufficient extensionconditions that have been established in the past. Introduction
The goal of this work is to provide extension theorems for homeomorphisms ofthe real line or of the circle whose regularity is beyond the quasisymmetric class(defined below). While quasisymmetric homeomorphisms are suitable for studyingself-similar sets, or sets with uniform geometry, they are not sufficient for the studyof fractals with non-uniform geometry. Such fractals appear often in the field ofComplex Dynamics as Julia sets of non-hyperbolic rational maps. Hence, extensiontheorems for homeomorphisms beyond the quasisymmetric class provide valuabletools for studying non-hyperbolic dynamical systems.Several recent works in the field are based on extensions of homeomorphisms ofthe circle to David homeomorphisms of the disk (defined below). More specifically,David extensions are useful for turning a hyperbolic dynamical system into a para-bolic one, an observation that was originally made by Ha¨ıssinsky. Thus, parabolicsystems can be studied in terms of hyperbolic systems, which are much better un-derstood. In some instances these extensions have been constructed “by hand”; see
Date : February 2, 2021.2020
Mathematics Subject Classification.
Primary 30C62, 30C65; Secondary 37F31, 46E35.
Key words and phrases. quasiconformal mapping, mapping of finite distortion, Beurling–Ahlfors extension, distortion, dilatation, exponentially integrable distortion, David homeomor-phism,
BMO -quasiconformal mapping.The second author is partially supported by NSF Grant DMS-2000096. a r X i v : . [ m a t h . C V ] J a n CHRISTINA KARAFYLLIA AND DIMITRIOS NTALAMPEKOS [Ha¨ı98, PZ04] and also [BF14, Chapter 9]. Later Zakeri [Zak08] studied systemati-cally extension problems and provided a useful criterion for David extensions, basedon the work of J. Chen, Z. Chen, and He [CCH96]. We will discuss these resultslater in detail. We also cite the recent works [LLMM19, LMMN20], where Davidextensions of circle homeomorphisms have been used successfully in the study ofgeometrically finite rational maps and Kleinian groups. Our results in this paperare stronger than the existing extension theory. Hence, we expect that they willprovide useful tools for further developments in Complex Dynamics and they willbroaden the understanding of mappings of finite distortion (defined below).Let h : R → R be an increasing homeomorphism. For x ∈ R and t > symmetric distortion function ρ h ( x, t ) = max (cid:26) | h ( x + t ) − h ( x ) || h ( x ) − h ( x − t ) | , | h ( x ) − h ( x − t ) || h ( x + t ) − h ( x ) | (cid:27) . The symmetric distortion function measures how far the homeomorphism h is frommapping adjacent intervals of equal length to adjacent intervals of equal length.If there exists (cid:37) > ρ h ( x, t ) ≤ (cid:37) for all x ∈ R , t >
0, then h is called quasisymmetric . Beurling and Ahlfors proved in [BA56] that if h is quasisymmetric,then there exists a quasiconformal extension of h to the upper half plane. Recallthat a homeomorphism H : U → V between two open sets U, V ⊂ R is quasi-conformal if H is orientation-preserving, H lies in the Sobolev space W , ( U ), theJacobian J H lies in L ( U ), and the quasiconformal dilatation K H ( x, y ) = inf { K ≥ (cid:107) DH ( x, y ) (cid:107) ≤ KJ H ( x, y ) } of H lies in L ∞ , where (cid:107) DH (cid:107) denotes the operator norm of the differential matrixof H .If one relaxes the assumption that K H ∈ L ∞ to merely K H < ∞ a.e., then we saythat H is a mapping of finite distortion . See [Kos10] for an enlightening survey and[HK14] for a treatise on the general theory of these mappings. Among mappings offinite distortion, of particular interest are the mappings of exponentially integrabledistortion , or else David homeomorphisms , because of their increased regularity andof the fact that they provide a substitute for quasiconformal maps in many caseswhen the use of the latter is not possible, such as in the framework of ComplexDynamics mentioned above. These maps were introduced by David in [Dav88] andtheir defining condition is that (cid:90) U e pK H ( x,y ) dσ ( x, y ) < ∞ for some p >
0, where σ denotes the spherical measure on the Riemann sphere (cid:98) C and U ⊂ (cid:98) C is an open set. We prove the following result, which provides extensionsof boundary homeomorphisms to mappings that have exponentially integrable dis-tortion. We also refer to these extensions as David extensions . We denote by H theupper half-plane and by D the unit disk in the plane. Theorem 1.1.
Let h : R → R be an increasing homeomorphism such that (cid:90) H e qρ h ( x,y ) dσ ( x, y ) < ∞ for some q > . Then there exists an extension of h to a homeomorphism of H thathas exponentially integrable distortion. XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 3
An analogous theorem can also be formulated for homeomorphisms of the circle.For a, b ∈ S we denote by (cid:96) ( a, b ) the length of the arc of the circle that connects a to b in the positive orientation. If h : S → S is an orientation-preservinghomeomorphism, we define the circular symmetric distortion function ρ ch ( θ, t ) = max (cid:26) (cid:96) ( h ( e iθ ) , h ( e i ( θ + t ) )) (cid:96) ( h ( e i ( θ − t ) ) , h ( e iθ )) , (cid:96) ( h ( e i ( θ − t ) ) , h ( e iθ )) (cid:96) ( h ( e iθ ) , h ( e i ( θ + t ) )) (cid:27) for θ ∈ [0 , π ] and t ∈ (0 , π/ ρ ch for all small t . Theorem 1.2.
Let h : S → S be an orientation-preserving homeomorphism suchthat (cid:90) π (cid:90) π/ e qρ ch ( θ,t ) dtdθ < ∞ for some q > . Then there exists an extension of h to a homeomorphism of D thathas exponentially integrable distortion. Corollary 1.3.
Let h : S → S be an orientation-preserving homeomorphism andsuppose that there exists a non-negative function g ∈ L ([0 , π ]) such that ρ ch ( θ, t ) = O (cid:18) log (cid:18) g ( θ ) t (cid:19)(cid:19) as t → . Then there exists an extension of h to a homeomorphism of D that hasexponentially integrable distortion. This extension result can be applied in order to turn hyperbolic dynamical sys-tems into parabolic ones with global David homeomorphisms of the sphere. Infact, the main result of [LMMN20] relies on a weaker version of this corollary from[Zak08]. Theorem 1.2 follows from the following more general result.
Theorem 1.4.
There exists a uniform constant C > such that the followingholds. Let Φ : [0 , ∞ ) → [0 , ∞ ) be an increasing convex function and suppose that h : S → S is an orientation-preserving homeomorphism such that (cid:90) π (cid:90) π/ Φ( qρ ch ( θ, t )) dtdθ < ∞ for some q > . Then there exists an extension of h to a homeomorphism H of D that has finite distortion and (cid:90) D Φ( qC − K H ( x, y )) dxdy < ∞ . The constant C is the same as the constant of Theorem 1.6 below. We proveTheorem 1.4 in Section 3. For Φ( x ) = e x , this theorem implies Theorem 1.2.Moreover, if Φ( x ) = x q , q ≥
1, then we obtain extensions having q -integrabledistortion and if Φ( x ) = e x/ log( e + x ) then we obtain extensions of subexponentiallyintegrable distortion . The latter class of mappings is slightly weaker than Davidhomeomorphisms, but their general theory has no essential differences. Moreover,the condition of subexponentially integrable distortion is very close to the optimalsufficient condition for obtaining solutions to the Beltrami equation ; see [AIM09,
CHRISTINA KARAFYLLIA AND DIMITRIOS NTALAMPEKOS
Section 20.5, p. 570] for more background. We remark that Theorem 1.4 generalizesthe result of Zakeri [Zak08, Theorem B], which uses sup t> ρ ch ( x, t ) in place of ρ ch .We do not know whether the assumption of Theorem 1.4 is also necessary forextensions that have q -integrable or exponentially integrable distortion. In fact, sofar there exists a necessary and sufficient condition only for mappings of 1-integrabledistortion, due to Astala, Iwaniec, Martin, and Onninen [AIMO05, Theorem 11.1].Namely, a homeomorphism h : S → S extends to a homeomorphism H of the diskwith K H ∈ L ( D ) if and only if (cid:90) π (cid:90) π (cid:12)(cid:12) log | h ( e iθ ) − h ( e iφ ) | (cid:12)(cid:12) dθdφ < ∞ . We pose the following question.
Question . Is the sufficient condition (cid:90) π (cid:90) π/ e qρ ch ( θ,t ) dtdθ < ∞ (cid:32) resp. (cid:90) π (cid:90) π/ ( ρ ch ( θ, t )) q dtdθ < ∞ (cid:33) also necessary for obtaining an extension of exponentially integrable distortion(resp. q -integrable distortion, q ≥ ρ ch ( θ, t ) = O ( t − α )as t → α >
0, which is much weaker than the condition in question.Next, we discuss the main theorem that leads to all the mentioned results. Let h : R → R be an increasing homeomorphism. Beurling and Ahlfors constructed in[BA56] an operator that extends h to a C -diffeomorphism of the upper half-plane.We denote by K h the quasiconformal dilatation of the extension. They showedthat if ρ h ( x, t ) ≤ (cid:37) for some (cid:37) >
0, then the extension of h is quasiconformal andmoreover K h ≤ (cid:37) . This bound was later improved by Reed [Ree66] to K h ≤ (cid:37) ,and by Li [Li83], who proved that K h ≤ . (cid:37) . Later Lehtinen [Leh83] improvedthis bound to 2 (cid:37) , which is currently the best known bound; see also [Leh84] and[Tan87].It is crucial for all these results to assume that ρ h ( x, t ) ≤ (cid:37) for all x ∈ R , t > cannot obtain in general any bound of the form K h ( x, y ) ≤ Cρ h ( x, y ) , (1.1)which would be an ideal bound for the extension problem. This was observed by Z.Chen and He [Che01, CH06], who gave examples of homeomorphisms h such that K h (0 , y ) ρ h (0 , y ) − → ∞ as y →
0. Z. Chen [Che01] also established, under nofurther assumptions on h , a bound of the form K h ( x, y ) ≤ Cρ h ( x, y )( ρ h ( x + y/ , y/
2) + ρ h ( x − y/ , y/ , which is, roughly speaking, of the form K h = O ( ρ h ). Nevertheless, this is a weakbound and does not imply sufficient integrability of K h for practical purposes.Therefore, in previous works, in order to obtain favorable bounds for K h in the spiritof (1.1), further assumptions were imposed on the symmetric distortion function ρ h . XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 5
For instance, J. Chen, Z. Chen, and He [CCH96] proved that there exists auniform constant
C > ρ h ( x, t ) ≤ (cid:37) ( t ) for some decreasing function (cid:37) ( t ), then K h ( x, y ) ≤ C(cid:37) ( y/ x ∈ R , y >
0. In fact, they claim this inequality with (cid:37) ( y ) in place of (cid:37) ( y/ (cid:37) ( y ) = O (log(1 /y ))as y →
0; the error mentioned above does not affect this result. We remark thatTheorems 1.1–1.2 and Corollary 1.3 are stronger. Moreover, under the assumptionsthat h ( x + 1) = h ( x ) + 1 for x ∈ R and exp(sup y> ρ h ( · , y )) ∈ L q ([0 , q >
0, Zakeri obtains a David extension by proving the bound K h ( x, y ) ≤ (cid:26) sup y> ρ h ( x, y ) , C ( q ) log (cid:18) C ( h ) y (cid:19)(cid:27) , where the constant C ( h ) depends on the L q norm of exp(sup y> ρ h ( x, y )). De Faria[dF11] remarked that this inequality is not optimal, by constructing homeomor-phisms h of R that extend to David homeomorphisms of H , but sup y> ρ h ( x, y ) = ∞ for a.e. x ∈ R . His examples, though, satisfy the sufficient condition ρ h ( x, y ) = O (log(1 /y )).Under no assumptions whatsoever on the homeomorphism h we prove, as ourmain theorem, the following optimal bounds for the quasiconformal dilatation K h of the Beurling–Ahlfors extension. These bounds are close to the ideal bound (1.1),but as we will see, for many practical purposes they are as good as that. Theorem 1.6.
There exists a uniform constant C > such that the followingholds. Let h : R → R be an increasing homeomorphism. Then ρ h ( x, y )4 ≤ K h ( x, y ) ≤ C max (cid:40) ρ h ( x, y ) , y (cid:90) y/ − y/ ρ h ( x + z, y − | z | ) dz (cid:41) for all x ∈ R and y > . This result implies the bounds of Chen et al. and Zakeri with possibly differentconstants. One can take C = 50, but we have not attempted to optimize thevalue of the constant C . The integral in the right-hand side is the average of ρ h on the two segments, from the point ( x, y ) to ( x − y/ , y/
4) and from ( x, y )to ( x + y/ , y/ K h and ρ h satisfy essentially the same integrability conditions. Theorem 1.7.
Let
Φ : [0 , ∞ ) → [0 , ∞ ) be an increasing convex function and sup-pose that h : R → R is an increasing homeomorphism. Then for all q > wehave (cid:90) H Φ( q − C − ρ h ) dσ ≤ (cid:90) H Φ( qC − K h ) dσ ≤ C (cid:90) H Φ( qρ h ) dσ, where C is the constant from Theorem 1.6 and C > is a uniform constant. CHRISTINA KARAFYLLIA AND DIMITRIOS NTALAMPEKOS RH • ( x, y ) • ( x − y/ , y/ • ( x + y/ , y/ Figure 1.
The segments on which ρ h is averaged in Theorem 1.6.Note that Theorem 1.1 follows from Theorem 1.7. We prove Theorem 1.6 inSection 2 and Theorem 1.7 in Section 3.Another class of well-studied generalizations of quasiconformal maps are BM O -quasiconformal maps. A homeomorphism H : U → V is BM O -quasiconformal if H is a mapping of finite distortion and there exists Q ∈ BM O ( U ) such that K H ( x, y ) ≤ Q ( x, y )a.e. in U . In other words, the quasiconformal dilatation of H has a BM O majorant (in U ). This condition is locally slightly stronger than the condition of exponentiallyintegrable distortion. These mappings were studied by Ryazanov, Srebro, andYakubov [RSY01].Sastry [Sas02, Theorem 3.1] established a sufficient condition for a homeomor-phism h : R → R to admit a BM O -quasiconformal extension to H and Zakeri[Zak08, Theorem C] proved a stronger result for homeomorphisms of R that com-mute with x (cid:55)→ x + 1. Namely, he proved that if h commutes with x (cid:55)→ x + 1and ρ h ( x, y ) ≤ y (cid:90) x + yx − y A ( t ) dt, where A ∈ BM O ( R ) and A is 1-periodic, then h has a BM O -quasiconformalextension to H . His proof is based on the John–Nirenberg inequality [JN61] and thedeep result of Bennett, DeVore, and Sharpley [BDS81], that the maximal function ofa BM O function is either identically equal to infinity or it lies in
BM O . Using theestimate of the main Theorem 1.6, we prove an even stronger result with elementarymeans.For a function A ∈ L ( H ) and for z ∈ H we define (cid:98) A ( z ) = 1 | B z | (cid:90) B z A, where B z is the ball B ( z, Im( z ) /
2) and | · | denotes the Lebesgue measure. We alsodefine q A ( z ) = 1 | Q z | (cid:90) Q z A = 12 y (cid:90) y/ y/ (cid:90) x + yx − y A, XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 7 where Q z is the 2 y × y rectangle, centered at z = ( x, y ). We prove in Section 3.2the elementary fact that (cid:98) · and q · are bounded operators from BM O into itself. Thecombination of this fact with the main Theorem 1.6 leads to the following result,proved in Section 3.2. Here, (cid:107) A (cid:107) ∗ denotes the BM O semi-norm of A . Theorem 1.8.
There exists a uniform constant
C > such that the followingholds. Let h : R → R be an increasing homeomorphism and A ∈ BM O ( H ) .If ρ h ≤ (cid:98) A in H then K h ≤ C (cid:98) A + C (cid:107) A (cid:107) ∗ in H , andif K h ≤ (cid:98) A in H then ρ h ≤ (cid:98) A in H . The same conclusions hold with q A in place of (cid:98) A . In particular, under any of theseconditions, the Beurling–Ahlfors extension of h to the upper half-plane is BM O -quasiconformal.
We pose some questions for further study. It is proved in [LMMN20, Proposition2.5] that David homeomorphisms of the unit disk are invariant under compositionwith quasiconformal homeomorphisms of the disk. Since the boundary maps ofquasiconformal homeomorphisms of the disk are precisely quasisymmetric home-omorphisms of S , it follows that the circle homeomorphisms that have a Davidextension in the disk are invariant under composition with quasisymmetric maps.We pose, therefore, the following question. Question . Let h : S → S be an orientation-preserving homeomorphism suchthat e ρ ch ∈ L q ([0 , π ] × [0 , π/ q >
0. Is is true that the pre- and post-compositions of h with quasisymmetric homeomorphisms of S also have the sameproperty (with a possibly different q )?If the answer to Question 1.5 is positive, then the answer to this question wouldalso be positive.Another natural problem is to characterize welding homeomorphisms of Davidcircles , i.e., Jordan curves that arise as the image of the unit circle under a globalDavid homeomorphism of (cid:98) C . A welding homeomorphism is a homeomorphism of thecircle that arises as the composition of the conformal map from the unit disk ontothe interior region of a Jordan curve with a conformal map from the exterior of thisJordan curve onto the exterior of the unit disk. The existence of a David extensionof a circle homeomorphism h inside the disk, as in the conclusion of Theorem 1.2,implies that h is a welding homeomorphism of a David circle. This follows fromstandard arguments; see, for instance, the discussion in [LMMN20, Section 5]. Question . What is a characterization of welding homeomorphisms of Davidcircles?For quasicircles, i.e., images of the unit circle under global quasiconformal maps,the characterization is known. Namely, quasisymmetric maps of the circle areprecisely the welding homeomorphisms of quasicircles. However, we do not expectthat the answer to the above question is the exponential integrability of ρ ch . Thereason is that the inverse of a welding homeomorphism is a welding homeomorphismtrivially. However, the inverse of a David homeomorphism is not necessarily aDavid map and likewise, we do not expect that the exponential integrability of ρ ch is equivalent to the exponential integrability of ρ ch − . CHRISTINA KARAFYLLIA AND DIMITRIOS NTALAMPEKOS Proof of the main theorem
In this section we prove the main Theorem 1.6. We first recall some basic factsand collect some properties of the Beurling–Ahlfors extension in Section 2.1, andthen we give the proof of the theorem in Section 2.2. Our proof is self-containedfor the convenience of the reader.2.1.
The Beurling–Ahlfors extension.
We recall the definition of the exten-sion operator of Beurling and Ahlfors [BA56]. Let h : R → R be an increasinghomeomorphism. The Beurling–Ahlfors extension H : H → H of h is defined by H ( x, y ) = u ( x, y ) + iv ( x, y ) , where u ( x, y ) = 12 y (cid:90) x + yx − y h ( t ) dt and v ( x, y ) = 12 y (cid:18)(cid:90) x + yx h ( t ) dt − (cid:90) xx − y h ( t ) dt (cid:19) for x ∈ R and y >
0. Moreover, we define H | R = h . Beurling and Ahlfors provedin [BA56, p. 135] that H : H → H is a homeomorphism. Indeed, H is proper,continuous, and locally injective and thus it is a covering map from H onto itself.Since H is simply connected, it follows that H : H → H is a homeomorphism. Thisin conjunction with the fact that H | R = h implies that H is continuous and bijectiveon H . But H − is also continuous on H and hence H : H → H is a homeomorphism.By general properties (see e.g. [AIM09, (21.1), p. 587]), the quasiconformal di-latation K h of H satisfies K h + 1 K h = u x + u y + v x + v y u x v y − u y v x , (2.1)where in our case u x ( x, y ) = 12 y ( h ( x + y ) − h ( x − y )) ,u y ( x, y ) = 12 y (cid:18) h ( x + y ) + h ( x − y ) − y (cid:90) x + yx h ( t ) dt − y (cid:90) xx − y h ( t ) dt (cid:19) ,v x ( x, y ) = 12 y ( h ( x + y ) + h ( x − y ) − h ( x )) , and v y ( x, y ) = 12 y (cid:18) h ( x + y ) − h ( x − y ) − y (cid:90) x + yx h ( t ) dt + 1 y (cid:90) xx − y h ( t ) dt (cid:19) . We list some transformation properties of the Beurling–Ahlfors extension. Welet a > b ∈ R . The extension of a function h is denoted by H and theextension of a function h ∗ is denoted by H ∗ .(BA1) If h ∗ ( t ) = ah ( t ) + b , then ρ h ∗ ( x, t ) = ρ h ( x, t ), H ∗ ( x, y ) = aH ( x, y ) + b , and K h ∗ ( x, y ) = K h ( x, y ).(BA2) If h ∗ ( t ) = h ( at + b ), then ρ h ∗ ( x, t ) = ρ h ( ax + b, at ), H ∗ ( x, y ) = H ( ax + b, ay ),and K h ∗ ( x, y ) = K h ( ax + b, ay ).(BA3) If h ∗ ( t ) = − h ( − t ), then ρ h ∗ ( x, t ) = ρ h ( − x, t ), H ∗ ( x, y ) = − H ( − x, y ), and K h ∗ ( x, y ) = K h ( − x, y ); here H ( − x, y ) denotes the complex conjugate of H ( − x, y ). XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 9
In all properties, the transformation of the extension H follows immediately fromthe definition of the Beurling–Ahlfors extension. Moreover, the transformation ofthe symmetric distortion function ρ h is also immediate from the definition. Thetransformation of the quasiconformal dilatation follows only from the transforma-tion of H and does not depend on the properties of the Beurling–Ahlfors extension.If h is a normalized homeomorphism with h (0) = 0 and h (1) = 1, then u x (0 ,
1) = 12 (1 − h ( − , u y (0 ,
1) = 12 (cid:18) h ( − − (cid:90) h ( t ) dt − (cid:90) − h ( t ) dt (cid:19) and v x (0 ,
1) = 12 (1 + h ( − , v y (0 ,
1) = 12 (cid:18) − h ( − − (cid:90) h ( t ) dt + (cid:90) − h ( t ) dt (cid:19) . If we set β = − h ( − , ξ = 1 − (cid:90) h ( t ) dt, and η = 1 + 1 β (cid:90) − h ( t ) dt, then u x (0 ,
1) = (1 + β ), u y (0 ,
1) = ( ξ − βη ), v x (0 ,
1) = (1 − β ), and v y (0 ,
1) = ( ξ + βη ). Therefore, by (2.1) we derive that K h (0 ,
1) + 1 K h (0 ,
1) = 1 ξ + η (cid:18) β (1 + η ) + 1 β (1 + ξ ) (cid:19) . (2.2)By our normalization on h , it is clear that ξ, η ∈ (0 , F ( ξ, η ) the right-hand side of (2.2), where β is treated as a con-stant. Beurling and Ahlfors [BA56, pp. 137–138] observed that F ( ξ, η ) is a convexfunction for ξ, η >
0. Indeed, we set F ( ξ, η ) = 1 + η ξ + η , F ( ξ, η ) = 1 + ξ ξ + η , and observe that F ( ξ, η ) = βF ( ξ, η ) + (1 /β ) F ( ξ, η ). Hence, it suffices to see that F , F are convex. A direct calculation shows thatHess( F ) = Hess( F ) = 2 (1 + η ) / ( ξ + η ) (1 − ηξ ) / ( ξ + η ) (1 − ηξ ) / ( ξ + η ) (1 + ξ ) / ( ξ + η ) . The determinant is equal to 4 / ( ξ + η ) > η ) / ( ξ + η ) > ξ, η >
0, we conclude that F and F are convex; see [Roc70, Theorem 4.5].2.2. Proof of Theorem 1.6.
Throughout the proof we fix x ∈ R and y > R h ∗ ( t ) = h ( x + yt ) − h ( x ) h ( x + y ) − h ( x )with h ∗ (0) = 0 and h ∗ (1) = 1. By properties (BA1) and (BA2), it follows that ρ h ∗ ( s, t ) = ρ h ( x + ys, yt ) and K h ∗ ( s, t ) = K h ( x + ys, yt )for s ∈ R and t >
0. For s = 0 and t = 1, by (2.2) we have K h ( x, y ) + 1 K h ( x, y ) = 1 ξ + η (cid:18) β (1 + η ) + 1 β (1 + ξ ) (cid:19) , (2.3) where β = − h ∗ ( − , ξ = 1 − (cid:90) h ∗ ( t ) dt, and η = 1 + 1 β (cid:90) − h ∗ ( t ) dt. (2.4)We first establish the lower estimate in the statement of the theorem. Since0 < ξ, η < K h ≥
1, by (2.3) we have2 K h ( x, y ) ≥ K h ( x, y ) + 1 K h ( x, y ) ≥ (cid:18) β + 1 β (cid:19) . Note that β is equal to either ρ h ∗ (0 ,
1) = ρ h ( x, y ) or 1 /ρ h ∗ (0 ,
1) = 1 /ρ h ( x, y ). Thus,12 (cid:18) β + 1 β (cid:19) = 12 (cid:18) ρ h ( x, y ) + 1 ρ h ( x, y ) (cid:19) ≥ ρ h ( x, y )2 . Therefore, K h ( x, y ) ≥ ρ h ( x, y )4 . This completes the proof of the lower estimate.In order to prove the upper estimate for K h ( x, y ), by (2.3), it suffices to provethe required estimate for the function F ( ξ, η ) = 1 ξ + η (cid:18) β (1 + η ) + 1 β (1 + ξ ) (cid:19) , which dominates K h . We consider two main cases: β ≥ β < Case 1.
Suppose that β ≥
1. In this case, we have β = ρ h ( x, y ). To simplify theproof we take two further subcases. Case 1(a).
Suppose that h ∗ ( − / ≤ − β/ h ∗ ( − / . (2.5)Essentially, this is the main non-trivial case and we will treat it in full detail. Theremaining cases are either trivial or symmetric to this one. We split the proof inseveral steps for the convenience of the reader. Step 1:
We will show that (cid:90) / h ∗ ( t ) dt ≤
14 + h ∗ ( − / A ∗ , where A ∗ = 4 (cid:90) / ρ h ∗ ( t, − t ) dt = 4 (cid:90) / ρ h ( x + ty, (1 − t ) y ) dt = 4 y (cid:90) y/ ρ h ( x + z, y − z ) dz. For 0 ≤ t ≤ /
4, we have 2 t − ≤ − /
2, and since h ∗ is increasing, it followsthat h ∗ (2 t − ≤ h ∗ ( − / ρ h ∗ ( t, − t ),we infer that h ∗ ( t ) − h ∗ ( − / h ∗ (1) − h ∗ ( t ) ≤ h ∗ ( t ) − h ∗ (2 t − h ∗ (1) − h ∗ ( t ) ≤ ρ h ∗ ( t, − t ) =: ρ ∗ XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 11 for 0 ≤ t ≤ /
4. Thus, h ∗ ( t ) − h ∗ ( − / ≤ (1 − h ∗ ( t )) ρ ∗ , which implies that h ∗ ( t ) ≤ ρ ∗ ρ ∗ + h ∗ ( − / ρ ∗ ≤ h ∗ ( − / ρ ∗ . Integrating over t ∈ [0 , / (cid:90) / h ∗ ( t ) dt ≤
14 + h ∗ ( − / (cid:90) /
41 + ρ ∗ dt. Finally, by Jensen’s inequality, (cid:90) /
41 + ρ ∗ dt ≥
11 + 4 (cid:82) / ρ ∗ dt = 11 + A ∗ . Since h ∗ ( − / <
0, the desired conclusion follows.
Step 2:
We will show that 4 ξ + 23 β A ∗ η ≥ β A ∗ . (2.6)Geometrically, this inequality says that the point ( ξ, η ) in the plane lies above acertain line with slope − A ∗ ) β − ; see Figure 2.Using the estimate from Step 1, we obtain the estimates4 ξ = 4 (cid:18) − (cid:90) h ∗ ( t ) dt (cid:19) = 4 − (cid:32)(cid:90) / h ∗ ( t ) dt + (cid:90) / h ∗ ( t ) dt (cid:33) ≥ − (cid:18)
14 + h ∗ ( − / A ∗ + 34 · (cid:19) = − h ∗ ( − / A ∗ . (2.7)Moreover, by the main assumption (2.5) of Case 1(a), we have β A ∗ η = β A ∗ (cid:18) β (cid:90) − h ∗ ( t ) dt (cid:19) = β A ∗ + 11 + A ∗ (cid:32)(cid:90) − / − h ∗ ( t ) dt + (cid:90) − / h ∗ ( t ) dt (cid:33) ≥ β A ∗ + 11 + A ∗ h ∗ ( −
1) + h ∗ ( − / ≥ β A ∗ + 32 h ∗ ( − / A ∗ . Equivalently, 23 β A ∗ η ≥ β A ∗ + h ∗ ( − / A ∗ Adding this inequality to (2.7) leads to the claimed inequality.
Step 3:
We finally estimate F ( ξ, η ) from above, under the restrictions 0 < ξ, η < β A ∗ ) is less than 1.Suppose first that β A ∗ ) < . Q β A ∗ ) β A ∗ ) η η ηξ ξ ξ Figure 2.
Convex polygons containing the point ( ξ, η ) in Case 1.We deduce that the point ( ξ, η ) lies in the convex quadrilateral Q (see Figure 2)bounded by the lines ξ = 1 , η = 0 , η = 1 , and 4 ξ + 23 β A ∗ η = 23 β A ∗ . Since F is convex, its maximum in Q is attained at one of the vertices (0 , , , β/ (6(1 + A ∗ )) , F at these vertices, we have F (0 ,
1) = 2 β + 1 β ≤ β + 1 ≤ β = 3 ρ h ( x, y ) ,F (1 ,
1) = β + 1 β ≤ F (0 , ,F (1 ,
0) = β + 2 β ≤ F (0 , , and F (cid:18) β A ∗ ) , (cid:19) = 6(1 + A ∗ ) (cid:18) β (cid:19) + 16(1 + A ∗ ) ≤ A ∗ ) + 112 ≤ A ∗ , because β, A ∗ ≥
1. Thus, F ( ξ, η ) ≤
25 max { ρ h ( x, y ) , A ∗ } . (2.8)If, instead, β A ∗ ) ≥ , then ( ξ, η ) lies in the triangle (see Figure 2) bounded by the lines ξ = 1 , η = 1 , and 4 ξ + 23 β A ∗ η = 23 β A ∗ , which is contained in the triangle with vertices (0 , , , F in this triangle is attained at one of the vertices. From the previousestimates, F is bounded above by 3 ρ h ( x, y ). Thus, (2.8) also holds in this case. Case 1(b).
Suppose that h ∗ ( − / > − β/ XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 13
Then η = 1 + 1 β (cid:90) − h ∗ ( t ) dt = 1 + 1 β (cid:32)(cid:90) − / − h ∗ ( t ) dt + (cid:90) − − / h ∗ ( t ) dt (cid:33) ≥ β ( h ∗ ( −
1) + h ∗ ( − / ≥ − β + β/ β = 14 . Since 0 < ξ < / ≤ η <
1, the point ( ξ, η ) lies in the rectangle (see Figure2) bounded by the lines ξ = 0 , ξ = 1 , η = 1 / , and η = 1 . Hence, the function F reaches its maximum at one of the vertices (0 , / , / , , F (0 , /
4) = 17 β β ≤ β ≤ β = 9 ρ h ( x, y ) , and F (1 , /
4) = 17 β
20 + 85 β ≤ β = 3 ρ h ( x, y ) , the relation (2.8) is still true.Thus, in both Cases 1(a) and 1(b), we derive that F ( ξ, η ) ≤
25 max { ρ h ( x, y ) , A ∗ } . This in conjunction with (2.3) gives K h ( x, y ) ≤
25 max (cid:40) ρ h ( x, y ) , y (cid:90) y/ ρ h ( x + z, y − z ) dz (cid:41) . (2.9) Case 2.
Suppose that β <
1. This case is symmetric to Case 1.In this case, we have β = 1 /ρ h ( x, y ). We consider the normalized increasing home-omorphism (cid:101) h ( t ) = − h ∗ ( − t ) /β with (cid:101) h (0) = 0 and (cid:101) h (1) = 1. By properties (BA1)and (BA3), it follows that ρ (cid:101) h ( s, t ) = ρ h ∗ ( − s, t ) = ρ h ( x − ys, yt ) and K (cid:101) h ( s, t ) = K h ∗ ( − s, t ) = K h ( x − ys, yt ) . So, if in (2.3) and (2.4) we replace β , ξ , η , and h ∗ by (cid:101) β , (cid:101) ξ , (cid:101) η , and (cid:101) h , respectively,then we have K h ( x, y ) = K (cid:101) h (0 , ≤ (cid:101) ξ + (cid:101) η (cid:18) (cid:101) β (1 + (cid:101) η ) + 1 (cid:101) β (1 + (cid:101) ξ ) (cid:19) =: (cid:101) F ( (cid:101) ξ, (cid:101) η )and (cid:101) β = − (cid:101) h ( −
1) = 1 β = ρ h ( x, y ) > . This reduces Case 2 to Case 1. Hence, we obtain the conclusion (cid:101) F ( (cid:101) ξ, (cid:101) η ) ≤
25 max { ρ h ( x, y ) , (cid:101) A } , where (cid:101) A = 4 (cid:90) / ρ (cid:101) h ( t, − t ) dt = 4 (cid:90) / ρ h ( x − ty, (1 − t ) y ) dt = 4 y (cid:90) − y/ ρ h ( x + z, y + z ) dz. We deduce that K h ( x, y ) ≤
25 max (cid:40) ρ h ( x, y ) , y (cid:90) − y/ ρ h ( x + z, y + z ) dz (cid:41) . (2.10)Combining (2.9) from Case 1 and (2.10) from Case 2, we finally have K h ( x, y ) ≤
50 max (cid:40) ρ h ( x, y ) , y (cid:90) y/ − y/ ρ h ( x + z, y − | z | ) dz (cid:41) . This completes the proof. (cid:3)
Remark . Our proof above follows some ideas from the proof of Theorem 3 in[CCH96]. However, there is an error in that proof. Namely, the last inequality in(3.10) is incorrect, since h (2 t −
1) is a negative number for t ∈ (0 , / ρ isassumed to be a decreasing function, in order to obtain a correct estimate one wouldhave to replace ρ ( y ) by ρ ( y /
2) in the last displayed formula of (3.10). This altersthe conclusion of the theorem, inequality (3.4), to the inequality D ( x + iy ) ≤ ρ ( y /
2) + C. Consequences of the main theorem
In this section we establish the consequences of the main theorem; that is, The-orem 1.7, which provides integrability conditions for homeomorphisms of the realline, Theorem 1.4, which provides integrability conditions for homeomorphisms ofthe circle, and Theorem 1.8, regarding the
BM O majorants.3.1.
Integrability conditions.
Proof of Theorem 1.7.
Let Φ : [0 , ∞ ) → [0 , ∞ ) be an increasing convex function.The first inequality of Theorem 1.6 immediately implies the first inequality of The-orem 1.7.For the second inequality, we multiply the second inequality of Theorem 1.6 with qC − , and then apply the function Φ. Using Jensen’s inequality we obtainΦ( qC − K h ( x, y )) ≤ max (cid:40) Φ( qρ h ( x, y )) , y (cid:90) y/ − y/ Φ( qρ h ( x + z, y − | z | )) dz (cid:41) , for all ( x, y ) ∈ H . In order to obtain the conclusion, it suffices to show that (cid:90) R (cid:90) ∞ y (cid:90) y/ − y/ Φ( qρ h ( x + z, y − | z | )) dz dydx (1 + x + y ) ≤ C (cid:90) H Φ( qρ h ) dσ for some uniform constant C >
0. For simplicity, we define f = Φ( qρ h ) and let g ( x, y ) be the spherical density 4 / (1 + x + y ) . XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 15
We break the inner integral into two integrals, from 0 to y/ − y/ (cid:90) R (cid:90) ∞ y (cid:90) y/ f ( x + z, y − z ) g ( x, y ) dzdydx = (cid:90) R (cid:90) ∞ y (cid:90) y y/ f ( x + y − u, u ) g ( x, y ) dudydx = (cid:90) ∞ y (cid:90) y y/ (cid:90) R f ( x + y − u, u ) g ( x, y ) dxdudy = (cid:90) ∞ y (cid:90) y y/ (cid:90) R f ( w, u ) g ( w + u − y, y ) dwdudy. Now we claim that g ( w + u − y, y ) ≤ g ( w, u ) for 0 < u < y and w ∈ R . Indeed,1 + ( w + u − y ) + y ≥ w − ( y − u )) + ( y − u ) + u ≥ w − ( y − u )) + ( y − u ) u ≥ w u ≥ w + u . The claim follows immediately. Therefore, it suffices to bound the integral of f ( w, u ) g ( w, u ), instead of f ( w, u ) g ( w + u − y, y ). We have (cid:90) ∞ y (cid:90) y y/ (cid:90) R f ( w, u ) g ( w, u ) dwdudy = (cid:90) R (cid:90) ∞ f ( w, u ) g ( w, u ) (cid:90) u/ u y dydudw. Finally, we observe that (cid:90) u/ u y dy = 2 log(4 / u > (cid:3) Next, we prove Theorem 1.4, which is a transportation of Theorem 1.7 to home-omorphisms of the circle. Our proof follows from an adaptation of the argument ofZakeri [Zak08, p. 243].
Proof of Theorem 1.4.
Let h : S → S be an orientation-preserving homeomor-phism of the circle. If h (1) = e iθ (cid:54) = 1, we consider the homeomorphism h · e − iθ .If we extend this homeomorphism to a homeomorphism H of the disk with thedesired integrability properties for K H , then H · e iθ will be an extension of h withthe desired properties. Therefore, it suffices to prove the theorem assuming that h (1) = 1.We lift the homeomorphism h to the real line, under the universal covering map ψ ( z ) = e πiz . We thus obtain an increasing homeomorphism (cid:101) h of the real linewith (cid:101) h (0) = 0, (cid:101) h (1) = 1, and (cid:101) h ( x + 1) = (cid:101) h ( x ) + 1 for all x ∈ R . Consider theBeurling–Ahlfors extension (cid:101) H of (cid:101) h in the upper half-plane. By properties (BA1)and (BA2), we have (cid:101) H ( z + 1) = (cid:101) H ( z ) + 1 for all z ∈ H . It follows that (cid:101) H descendsto a homeomorphism H of the unit disk that extends h . Since the circular symmetric distortion ρ ch is defined using arclength, we have ρ ch ( θ, t ) = ρ (cid:101) h ( θ/ π, t/ π ) for all θ ∈ [0 , π ] and t ∈ (0 , π/ qρ ch ) ∈ L ([0 , π ] × (0 , π/ qρ (cid:101) h ) ∈ L ([0 , × (0 , / ρ (cid:101) h on [0 , × [1 / ,
1] implies that Φ( qρ (cid:101) h ) ∈ L ([0 , × (0 , (cid:101) h commutes with x (cid:55)→ x + 1, we have that ρ (cid:101) h is bounded on [0 , × [1 , ∞ ). Therefore,Φ( qρ (cid:101) h ) ∈ L ([0 , × (0 , ∞ ); dσ ). Again, since (cid:101) h commutes with x (cid:55)→ x + 1, weconclude that Φ( qρ (cid:101) h ) ∈ L ( H ; dσ ).By Theorem 1.7, the Beurling–Ahlfors extension (cid:101) H of (cid:101) h satisfies Φ( qC − K (cid:101) H ) ∈ L ( H ; dσ ). Since ψ ◦ (cid:101) H = H ◦ ψ and ψ is locally conformal, we have K (cid:101) H = K H ◦ ψ .By changing coordinates under the conformal map ψ | (0 , × (0 , ∞ ) we obtain (cid:90) D \ [0 , ×{ } Φ( qC − K H ) J σψ − dσ = (cid:90) (0 , × (0 , ∞ ) Φ( qC − K (cid:101) H ) dσ < ∞ , (3.1)where J σψ − denotes the spherical Jacobian of ψ − ( z ) = log( z )2 πi , with a branch cutalong the non-negative real axis. We have J σψ − ( z ) = J ψ − ( z ) (1 + | z | ) (1 + | ψ − ( z ) | ) (cid:39) | z | (1 + | z | ) (1 + log | z | ) (cid:38) z ∈ D \ [0 , × { } . Since dσ (cid:39) dxdy for ( x, y ) ∈ D , by (3.1) we have (cid:90) D Φ( qC − K H ) dxdy < ∞ . This completes the proof. (cid:3)
Functions of bounded mean oscillation.
We first recall the definition ofa function of bounded mean oscillation. Let U ⊂ R n , n ≥
1, be an open set and A ∈ L ( U ). The function A lies in BM O ( U ) if (cid:107) A (cid:107) ∗ := sup B ⊂ U | B | (cid:90) B | A − A B | < ∞ , where A B = | B | (cid:82) B A , and the supremum is taken over all closed balls B ⊂ U .For A ∈ L ( U ), we define (cid:98) A ( z ) = 1 | B z | (cid:90) B z A, where B z = B ( z, dist( z, ∂U ) / Lemma 3.1.
There exists a uniform constant C = C ( n ) > such that if A ∈ BM O ( U ) , then (cid:98) A ∈ BM O ( U ) and (cid:107) (cid:98) A (cid:107) ∗ ≤ C (cid:107) A (cid:107) ∗ .Proof. Suppose that A ∈ BM O ( U ) and consider a ball B = B ( z , r ) ⊂ B ( z , r ) ⊂ U . We will show that there exists a uniform constant C = C ( n ) > c ∈ R depending on B such that (cid:90) B | (cid:98) A − c | ≤ C (cid:107) A (cid:107) ∗ | B | . (3.2)This will imply that (cid:98) A ∈ BM O ( U ) and (cid:107) (cid:98) A (cid:107) ∗ ≤ C (cid:107) A (cid:107) ∗ ; see [WZ15, Lemma 14.49,p. 445]. XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 17
Let z ∈ B and consider a chain of points z , z , . . . , z N = z lying on the segmentbetween z and z such that | z i − z i − | = 2 − i r for i ∈ { , . . . , N − } and | z N − z N − | ≤ − N r . We fix i ∈ { , . . . , N } . Recall that B z i = B ( z i , d i / d i = dist( z i , ∂U ). If d i ≤ d i − , then we define B ( w i , R i ) = B ( z i − , d i − / | z i − z i − | ) , while if d i − < d i , then we define B ( w i , R i ) = B ( z i , d i / | z i − z i − | ) . In both cases we have B z i ∪ B z i − ⊂ B ( w i , R i ) . We observe that d i − ≥ d − | z i − − z | > r − r (2 − + · · · + 2 − i +1 ) = 2 − i +1 r ≥ | z i − z i − | . It follows that R i = max { d i − , d i } / | z i − z i − | < max { d i − , d i } (3.3)and thus B ( w i , R i ) ⊂ B ( w i , R i ) ⊂ U . Since | d i − d i − | ≤ | z i − z i − | < d i − / d i − < d i and d i < d i − /
2. Therefore, max { d i − , d i } < { d i − , d i } .This, in conjunction with (3.3), gives R i / < min { d i , d i − } ≤ max { d i , d i − } < R i . (3.4)Since B z i ⊂ B ( w i , R i ) and these balls have comparable radii, we have | A B zi − A B ( w i ,R i ) | ≤ | B z i | (cid:90) B zi | A − A B ( w i ,R i ) |≤ C | B ( w i , R i ) | (cid:90) B ( w i ,R i ) | A − A B ( w i ,R i ) | , for a uniform constant C = C ( n ) >
0. The fact that the closure of B ( w i , R i )is contained in U and the assumption that A ∈ BM O ( U ) imply that the latteraverage is bounded by (cid:107) A (cid:107) ∗ . Hence, | A B zi − A B ( w i ,R i ) | ≤ C (cid:107) A (cid:107) ∗ and similarly | A B zi − − A B ( w i ,R i ) | ≤ C (cid:107) A (cid:107) ∗ for a uniform constant C = C ( n ) >
0. Therefore, we have | (cid:98) A ( z ) − (cid:98) A ( z ) | ≤ N (cid:88) i =1 | (cid:98) A ( z i ) − (cid:98) A ( z i − ) |≤ N (cid:88) i =1 ( | A B zi − A B ( w i ,R i ) | + | A B zi − − A B ( w i ,R i ) | ) ≤ CN (cid:107) A (cid:107) ∗ . Note that | z − z | ≥ (cid:80) N − i =1 | z i − z i − | = r (cid:80) N − i =1 − i = r (1 − − N +1 ). Therefore, N ≤ − log (cid:18) − | z − z | /r (cid:19) . Finally, by integrating over B we have (cid:90) B | (cid:98) A ( z ) − (cid:98) A ( z ) | dz ≤ C (cid:48) | B |(cid:107) A (cid:107) ∗ + C (cid:48) (cid:107) A (cid:107) ∗ (cid:90) B log (cid:18) − | z − z | /r (cid:19) dz, where C (cid:48) = C (cid:48) ( n ) > C (cid:48)(cid:48) | B | for a constant C (cid:48)(cid:48) = C (cid:48)(cid:48) ( n ) > c = (cid:98) A ( z ). (cid:3) Recall that for A ∈ L ( H ) we have defined q A ( z ) = 1 | Q z | (cid:90) Q z A = 12 y (cid:90) y/ y/ (cid:90) x + yx − y A, where Q z is the 2 y × y open rectangle, centered at z = ( x, y ). Lemma 3.2.
There exists a uniform constant
C > such that if A ∈ BM O ( H ) ,then (i) | q A − (cid:98) A | ≤ C (cid:107) A (cid:107) ∗ in H , and (ii) q A ∈ BM O ( H ) with (cid:107) q A (cid:107) ∗ ≤ C (cid:107) A (cid:107) ∗ . Of course, the particular choice of the dimensions of the rectangle Q z is not ofimportance, as long as the lengths of the sides of Q z are comparable to the distanceof Q z to the boundary of H . Proof.
For any z = ( x, y ) ∈ H there exists a ball B ⊂ B ⊂ H with radiuscomparable to y , such that B ⊃ B z = B ( z, y/
2) and B ⊃ Q z . For example, onecan take the center to be ( x, y ) and the radius to be y √ /
2. We now have | q A ( z ) − (cid:98) A ( z ) | ≤ | q A ( z ) − A B | + | A B − (cid:98) A ( z ) |≤ | Q z | (cid:90) Q z | A − A B | + 1 | B z | (cid:90) B z | A − A B |≤ C (cid:48) | B | (cid:90) B | A − A B | ≤ C (cid:48) (cid:107) A (cid:107) ∗ for a uniform constant C (cid:48) >
0, since A ∈ BM O ( H ).Upon integration, it follows that | ( q A ) B − ( (cid:98) A ) B | ≤ C (cid:48) (cid:107) A (cid:107) ∗ for any ball B ⊂ B ⊂ H . Therefore, by the above and Lemma 3.1 we have1 | B | (cid:90) B | q A − ( q A ) B | ≤ | B | (cid:90) B | q A − (cid:98) A | + 1 | B | (cid:90) B | (cid:98) A − ( (cid:98) A ) B | + 1 | B | (cid:90) B | ( (cid:98) A ) B − ( q A ) B |≤ C (cid:48) (cid:107) A (cid:107) ∗ + C (cid:107) A (cid:107) ∗ + C (cid:48) (cid:107) A (cid:107) ∗ = (2 C (cid:48) + C ) (cid:107) A (cid:107) ∗ , where C is the constant from Lemma 3.1. This completes the proof. (cid:3) Proof of Theorem 1.8. If K h ≤ (cid:98) A , then ρ h ≤ (cid:98) A by the first inequality of Theorem1.6. The same claim holds with q A in place of (cid:98) A .Conversely, suppose that ρ h ≤ (cid:98) A . By the second inequality of Theorem 1.6, wehave K h ( x, y ) ≤ C (cid:98) A ( x, y ) + C y (cid:90) y/ − y/ (cid:98) A ( x + z, y − | z | ) dz, Hence, it suffices to show that2 y (cid:90) y/ − y/ (cid:98) A ( x + z, y − | z | ) dz ≤ (cid:98) A ( x, y ) + C (cid:107) A (cid:107) ∗ XTENSION OF BOUNDARY MAPS TO MAPPINGS OF FINITE DISTORTION 19 for a uniform constant
C >
0. We fix z ∈ ( − y/ , y/
4) and consider the ball B = B (( x, y ) , R ), where R = y √ / y/ < y , so that B ⊂ H , B (( x, y ) , y/ ⊂ B, and B (( x + z, y − | z | ) , ( y − | z | ) / ⊂ B. We now estimate | (cid:98) A ( x + z, y − | z | ) − (cid:98) A ( x, y ) | ≤ | (cid:98) A ( x + z, y − | z | ) − A B | + | A B − (cid:98) A ( x, y ) |≤ C | B | (cid:90) B | A − A B | ≤ C (cid:107) A (cid:107) ∗ for a uniform constant C >
0. Therefore, (cid:98) A ( x + z, y − | z | ) ≤ (cid:98) A ( x, y ) + C (cid:107) A (cid:107) ∗ for all z ∈ ( − y/ , y/ ρ h ≤ q A , then by Lemma 3.2 (i) we have ρ h ≤ (cid:98) A + C (cid:48) (cid:107) A (cid:107) ∗ = (cid:92) ( A + C (cid:48) (cid:107) A (cid:107) ∗ )for a uniform constant C (cid:48) >
0. By the previous case, we have K h ≤ C (cid:48)(cid:48) ( (cid:92) ( A + C (cid:48) (cid:107) A (cid:107) ∗ ) + (cid:107) A + C (cid:48) (cid:107) A (cid:107) ∗ (cid:107) ∗ ) = C (cid:48)(cid:48) (cid:98) A + C (cid:48)(cid:48) ( C (cid:48) + 1) (cid:107) A (cid:107) ∗ for some uniform constant C (cid:48)(cid:48) >
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