PETER HÄSTÖ, RIKU KLÉN, SWADESH KUMAR SAHOO, AND MATTI VUORINEN

Transcription

1 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS PETER HÄSTÖ, RIKU KLÉN, SWADESH KUMAR SAHOO, AND MATTI VUORINEN Abstract. We consider proper subdomains G of R n and their images G = fg under quasiconformal mappings f of R n. We compare the distance ratio metrics of G and G ; as an application we show that ϕ-uniform domains are preserved under quasiconformal mappings of R n. A sufficient condition for ϕ-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If G R n is ϕ-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar ϕ-uniform domain whose complement is not ψ-uniform for any ψ. 1. Introduction and Main Results Classes of subdomains of the Euclidean n-space R n, n, occur often in geometric function theory and modern mapping theory. For instance, the boundary regularity of a conformal mapping of the unit disk onto a domain D depends on the properties of D at its boundary. Similar results have been established for various classes of functions such as quasiconformal mappings and mappings with finite distortion. In such applications, uniform domains and their generalizations occur [Ge99, GO79, GH1, Va71, Va88, Va91, Va98, Vu88]; ϕ-uniform domains have been recently studied in [KLSV14]. Let γ : [0, 1] G R n be a path, i.e. a continuous function. All the paths γ are assumed to be rectifiable, that is, to have finite Euclidean length notation-wise we write lγ <. Let G R n be a domain and x, y G. We denote by δ G x, the Euclidean distance from x to the boundary G of G. When the domain is clear, we use the notation δx. The j G metric also called the distance ratio metric [Vu85] is defined by j G x, y := log 1 + x y δx δy where a b = min{a, b}. A slightly different form of this metric was studied in [GO79]. The quasihyperbolic metric of G is defined by the quasihyperbolic-length-minimizing property k G x, y = inf l dz kγ, l k γ = γ Γx,y δz, where l k γ is the quasihyperbolic length of γ cf. [GP76] and Γx, y is the set of all rectifiable curves joining x and y in G. For a given pair of points x, y G, the infimum is always attained [GO79], i.e., there always exists a quasihyperbolic geodesic J G [x, y] which minimizes the above integral, k G x, y = l k J G [x, y] and furthermore with the property that File: hksv tex, printed: , Mathematics Subject Classification. Primary 30F45; Secondary 30C65, 30L05, 30L10. Key words and phrases. The distance ratio metric, the quasihyperbolic metric, uniform and ϕ-uniform domains, quasi-convex domains, John domains, c-joinable domains, quasiconformal and quasisymmetric mappings. 1, γ

2 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN the distance is additive on the geodesic: k G x, y = k G x, z + k G z, y for all z J G [x, y]. It also satisfies the monotonicity property: k G1 x, y k G x, y for all x, y G G 1. If the domain G is emphasized we call J G [x, y] a k G -geodesic. Note that for all domains G, 1.1 j G x, y k G x, y for all x, y G [GP76]. In 1979, uniform domains were introduced by Martio and Sarvas [MS79]. A domain G R n is said to be uniform if there exists C 1 such that for each pair of points x, y G there is a path γ G with i lγ C x y ; and ii δz 1 [lγ[x, z] lγ[z, y]] for all z γ. C Subsequently, Gehring and Osgood [GO79] characterized uniform domains in terms of an upper bound for the quasihyperbolic metric as follows: a domain G is uniform if and only if there exists a constant C 1 such that k G x, y Cj G x, y for all x, y G. As a matter of fact, the above inequality appeared in [GO79] in a form with an additive constant on the right hand side; it was shown by Vuorinen [Vu85,.50] that the additive constant can be chosen to be 0. This observation leads to the definition of ϕ-uniform domains introduced in [Vu85]. Definition 1.. Let ϕ : [0, [0, be a strictly increasing homeomorphism with ϕ0 = 0. A domain G R n is said to be ϕ-uniform if x y k G x, y ϕ δx δy for all x, y G. An example of a ϕ-uniform domain which is not uniform is given in Section 4. That domain has the property that its complement is not ψ-uniform for any ψ. Väisälä has also investigated the class of ϕ-domains [Va91] see also [Va98] and references therein and pointed out that ϕ-uniform domains are nothing but uniform under the condition that ϕ is a slow function, i.e. ϕt/t 0 as t. In the above definition, uniform domains are characterized by the quasi-convexity i and twisted-cone ii conditions. In Section 3, we show that the former can be replaced by ϕ- uniformity, which may in some situations be easier to establish. Theorem 1.3. If a domain G R n is ϕ-uniform and satisfies twisted cone condition, then it is uniform. Let G R n be a domain and f : G fg R n be a homeomorphism. The linear dilatation of f at x G is defined by Hf, x := lim sup r 0 sup{ fx fy : x y = r} inf{ fx fz : x z = r}. A homeomorphism f : G fg R n is said to be K-quasiconformal if sup x G Hf, x K.

3 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 3 In Section we study ϕ-uniform domains in relation to quasiconformal and quasisymmetric mappings. Gehring and Osgood [GO79, Theorem 3 and Corollary 3], proved that uniform domains are invariant under quasiconformal mappings of R n. Our next theorem extends this result to the case of ϕ-uniform domains. Theorem 1.4. Suppose that G R n is a ϕ-uniform domain and f is a quasiconformal mapping of R n which maps G onto G R n. Then G is ϕ 1 -uniform for some ϕ 1.. Quasiconformal and quasi-symmetric mappings In general, quasiconformal mappings of a uniform domain do not map onto a uniform domain. For example, by the Riemann Mapping Theorem, there exists a conformal mapping of the unit disk D = {z R : z < 1} onto the simply connected domain D\[0, 1. Note that the unit disk D is ϕ-uniform whereas the domain D \ [0, 1 is not. However, this changes if we consider quasiconformal mappings of the whole space R n : uniform domains are invariant under quasiconformal mappings of R n [GO79]. In this section we provide the analogue for ϕ-uniform domains. We notice that the quasihyperbolic metric and the distance ratio metric have similar natures in several senses. For instance, if f : R n R n is a Möbius mapping that takes a domain onto another, then f is -bilipschitz with respect to the quasihyperbolic metric [GP76]. Counterpart of this fact with respect to the distance ratio metric can be obtained from the proof of [GO79, Theorem 4] with the bilipschitz constant. Lemma.1. [GO79, Theorem 3] If f : G G is a K-quasiconformal mapping of G R n onto G R n, then there exists a constant c depending only on n and K such that for all x, y G, where α = K 1/1 n. k G fx, fy c max{k G x, y, k G x, y α } We obtain an analogue of Lemma.1 for j G with the help of the following result of Gehring s and Osgood s: Lemma.. [GO79, Theorem 4] If f : G G is a K-quasiconformal mapping of R n which maps G R n onto G R n, then there exist constants c 1 and d 1 depending only on n and K such that for all x, y G. j G fx, fy c 1 j G x, y + d 1 In order to investigate the quasiconformal invariance property of ϕ-uniform domains we reformulate Lemma. in the form of the following lemma. We make use of both the above lemmas in the reformulation. Lemma.3. If f : G G is a K-quasiconformal mapping of R n which maps G onto G, then there exists a constant C depending only on n and K such that for all x, y G, where α = K 1/1 n. j G fx, fy C max{j G x, y, j G x, y α }

4 4 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN Proof. Without loss of generality we assume that δx δy for x, y G. Suppose first that y G \ B n x, δx/. Since x y δx/, it follows that j G x, y log3/. By Lemma., we obtain j G fx, fy c 1 + d 1 j G x, y. log3/ Suppose then that y B n x, δx/. By [Vu88, Lemma 3.7 ], k G x, y j G x, y. Hence we obtain that j G fx, fy k G fx, fy c max{k G x, y, k G x, y α } c max{j G x, y, j G x, y α }, where the first inequality always holds by 1.1 and the second inequality is due to Lemma.1. As a consequence of Lemmas.1 and.3, we prove our main result Theorem 1.4 about the invariance property of ϕ-uniform domains under quasiconformal mappings of R n. Proof of Theorem 1.4. By Lemma.3, there exists a constant C such that.4 j G x, y C max{j G fx, fy, j G fx, fy α } for all x, y G. Define ψt := ϕe t 1. Then k G fx, fy c max{k G x, y, k G x, y α } c max{ψj G x, y, ψj G x, y α } c max{ψc max{j G fx, fy, j G fx, fy α }, ψc max{j G fx, fy, j G fx, fy α } α }, where the first inequality is due to Lemma.1, the second inequality holds by hypothesis, and the last inequality is due to.4. Thus, G is ϕ 1 -uniform with ϕ 1 t = c max{ψc max{log1 + t, log1 + t α }, ψc max{log1 + t, log1 + t α } α }, where α = K 1/1 n. A mapping f : X 1, d 1 X, d is said to be η-quasi-symmetric η-qs if there exists a strictly increasing homeomorphism η : [0, [0, with η0 = 0 such that d fx, fy d fy, fz η d1 x, y d 1 y, z for all x, y, z X 1 with x y z. Note that L-bilipschitz mappings are η-qs with ηt = L t and η-qs mappings are K- quasiconformal with K = η1. It is pointed out in [Ko09] that quasiconformal mappings are locally quasi-symmetric. The following result gives a sufficient condition for G to be a ϕ-uniform domain. Proposition.5. If the identity mapping id : G, j G G, k G is η-qs, then G is ϕ- uniform for some ϕ depending on η only.

5 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 5 Proof. By hypothesis we have k G x, y k G y, z η jg x, y j G y, z for all x, y, z with x y z. Choose z y such that δz = e 1 δy. Then y z δy j G y, z = k G y, z = log 1 + = log = 1. δz δz Hence, by the hypothesis we conclude that k G x, y ηj G x, y. This shows that G is ϕ-uniform with ϕt = ηlog1 + t. Question.6. Is the converse of Proposition.5 true? 3. The ϕ-uniform domains which are uniform A domain G R n is said to satisfy the twisted cone condition, if for every x, y G there exists a rectifiable path γ G joining x and y such that 3.1 min{lγ[x, z], lγ[z, y]} c δz for all z γ and for some constant c > 0. Sometimes we call the path γ a twisted path. Domains satisfying the twisted cone condition are also called John domains see for instance [GHM89, He99, KL98, NV91]. If, in addition, lγ c x y holds then the domain G is uniform. Note that the path γ in the definition of the twisted cone condition may be replaced by a quasihyperbolic geodesic see [GHM89]. We observe from Section 1 that a ϕ-uniform domain need not be uniform or quasi-convex. Nevertheless, a ϕ-uniform domain satisfying the twisted-cone condition is uniform. Proof of Theorem 1.3. Assume that G is ϕ-uniform and satisfies the twisted cone condition 3.1. Let x, y G be arbitrary and γ be a twisted path in G joining x and y. Choose x, y γ such that lγ[x, x ] = lγ[y, y ] = 1 x y. 10 Now, by the cone condition, we have δx 1 c min{lγ[x, x ], lγ[x, y]} and δy 1 c min{lγ[x, y ], lγ[y, y]}. By the choice of x and y, on one hand, we see that lγ[x, y] x y x y x x 9 x y. 10 On the other hand, lγ[x, x ] = 1 x y. The same holds for x and y interchanged. Thus, min{δx, δy } 1 x y. 10c To complete the proof, our idea is to prove the following three inequalities: k G x, x a 1 j G x, x b 1 j G x, y; 3.3 k G x, y b j G x, y; k G y, y a 3 j G y, y b 3 j G x, y for some constants a i, b i, i = 1,, 3. Finally, the inequality k G x, y k G x, x + k G x, y + k G y, y c j G x, y

6 6 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN with c = b 1 + b + b 3 would conclude the proof of the theorem. It is sufficient to show the first two lines in 3.3, as the third is analogous to the first. We start with a general observation: if j G z, w < log 3, then z lies in the ball Bw, 1δw, and we can connect the points by the segment [z, w] G. Furthermore, due to [Vu88, Lemma 3.7 ] k G z, w j G z, w. Thus in each inequality between the k and j metrics, we may assume that j G z, w log 3 for all z, w G. First we prove the second line of 3.3. Since G is ϕ-uniform and ϕ is an increasing homeomorphism, k G x, y x y ϕ min{δx, δy } ϕ1c, where the triangle inequality x y x x + x y + y y and the relation 3. are applied to obtain the second inequality. On the other hand, j G x, y log 3. Thus k G x, y b j G x, y with b = ϕ1c/ log 3. Then we consider the first line of 3.3: k G x, x a 1 j G x, x b 1 j G x, y. The second inequality is easy to prove. Indeed, we have j G x, x = log 1 + x x min{δx, δx } < log c x y 1 + c j G x, y, δx where the first inequality holds since x x 1 10 x y and min{δx, δx } δx/1 + c. Fix a point z γ with lγ[x, z] = 1 δx and denote γ 1 = γ[x, z]. Assume for the time being that x γ 1 Clearly, k G x, x k G γ 1 + k G γ, where γ = γ[z, x ]. For w γ 1 we have δw 1 δx, and for w γ, the twisted cone condition and the fact lγ[w, y] 9lγ[x, w] together give δw 1/c lγ[x, w]. Thus we find that Furthermore, k G γ k G γ 1 lγ 1 1 δx = 1 1 log 3 j G x, x 1 log cj G x, y. lγ[x,x ] lγ[x,z] c dt t = c log lγ[x, x ] lγ[x, z] = c log 1 x y 10 1 δx cj G x, y. So the inequality is proved in this case. If, on the other hand, x γ 1, then we set z = x and repeat the argument of this paragraph for γ 1, since γ is empty in this case. This completes the proof of our theorem. 4. Complement of ϕ-uniform domains In [KLSV14, Section 3] the following question was posed: Are there any bounded planar ϕ-uniform domains whose complementary domains are not ϕ-uniform? In this section we show that the answer is yes. Proposition 4.1. There exists a bounded ϕ-uniform Jordan domain D R such that R \D is not ψ-uniform for any ψ.

7 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 7 Proof. Fix 0 < u < t < v < 1. Let Rk = xk, xk + uk [0, v k ] be the rectangle, k 1. The parameter xk is chosen such that x1 = 0 and xk+1 = xk + uk + tk. At the top of each rectangle Rk we place a semi-disc Ck with radius uk / and center on the midpoint of the top side of Rk. Set s = u/1 u + t/1 t. With these elements we define D, shown in Figure 1, by D := 0, s, 0 k Rk k Ck. Ck Rk Ck+1 Rk+1 Figure 1. The ϕ-uniform domain D constructed in the proof of Proposition 4.1. Let us show that D is ϕ-uniform. For x Rk and y Rl, k > l, let d := min{δx, δy}. We choose a polygonal path γ as follows: from x the shortest line segment to the medial axis of Rk, then horizontally at y = d and finally from the medial axis of Rl to y along the shortest line segment see again Figure 1. The lengths of the vertical and horizontal parts k l are at most v k + d, v l + d and x y + u +u. The line segments joining x and y to the k k medial axis have length at most u / and v /. The whole curve is at distance at least d from the boundary. Thus kd x, y kd γ `γ 3v k + 3v l + d + x y 8v k + x y. d d d On the other hand, tk x y. δx δy d x y Let ϕτ := τ + 8τ α, where α is such that tα = v. Then kd x, y ϕ δx δy. The case when x, y Rk or in the base rectangle are handled similarly, although they are simpler. Thus we conclude that D is ϕ-uniform.

8 8 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN We show then that R \ D is not ψ-uniform for any ψ. We choose z k = x k+1 t k /, t k in the gap between R k and R k+1. Then z k z k+1 δz k δz k+1 tk / + u k+1 + t k+1 / + t k t k+1 = 3 t u k+1 3 t t + 3. On the other hand, a curve connecting these points has length at least v k t k /, so that k R \D z k, z k+1 v k t k / t k / dx x = log vk t k / t k / as k. Therefore, it is not possible to find ψ such that k R \D z k, z k+1 ψ z k z k+1, δz k δz k+1 as claimed. References [Ge99] F. W. Gehring, Characterizations of quasidisks, Quasiconformal geometry and dynamics, Banach center publications, Vol 48, Polish Academy of Sciences, Warszawa [GH1] F. W. Gehring and K. Hag, The Ubiquitous Quasidisk, Mathematical Surveys and Monographs 184, Amer. Math. Soc., Providence, RI, 01. [GHM89] F. W. Gehring, K. Hag, and O. Martio, Quasihyperbolic geodesics in John domains, Math. Scand., , [GO79] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math , [GP76] F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Anal. Math , [He99] D. A. Herron, John domains and the quasihyperbolic metric, Complex Variables, , [KL98] K. Kim and N. Langmeyer, Harmonic measure and hyperbolic distance in John disks, Math. Scand., , [KLSV14] R. Klén, Y. Li, S.K. Sahoo, and M. Vuorinen, On the stability of ϕ-uniform domains, Monatshefte für Mathematik, , [Ko09] P. Koskela, Lectures on Quasiconformal and Quasisymmetric Mappings, Lecture note, December 009. [MS79] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Math , [NV91] R. Näkki and J. Väisälä, John disks, Expo. Math., , [Va71] J. Väisälä, Lectures On n-dimensional Quasiconformal Mappings, Lecture Notes in Mathematics 9, Springer, [Va88] J. Väisälä, Uniform domains, Tohoku Math. J , [Va91] J. Väisälä, Free quasiconformality in Banach spaces II, Ann. Acad. Sci. Fenn. Math , [Va98] J. Väisälä, Relatively and inner uniform domains, Conformal Geometry and Dynamics, 1998, [Vu85] M. Vuorinen, Conformal invariants and quasiregular mappings, J. Anal. Math , [Vu88] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics 1319, Springer-Verlag, Berlin Heidelberg New York, 1988.

9 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 9 Department of Mathematical Sciences, P.O. Box 3000, University of Oulu, Finland address: peter.hasto@oulu.fi Department of Mathematics and Statistics, University of Turku, FIN-0014 Turku, Finland address: riku.klen@utu.fi Department of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore-45 00, India address: swadesh@iiti.ac.in Department of Mathematics and Statistics, University of Turku, FIN-0014 Turku, Finland address: vuorinen@utu.fi