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

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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 = f(g) 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 quasi-symmetric mappings. We give two geometric conditions for uniformity: (1) If G R 2 and R 2 \ G are quasiconvex, then G is uniform. (2) 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 2, 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, GH12, 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 l(γ) < ). 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 δ(x). The j G metric (also called the distance ratio metric) [Vu85] is defined by j G (x, y) := log ( 1 + x y δ(x) δ(y) where a b = min{a, b}. In 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 k(γ), l k (γ) = γ Γ(x,y) δ(z), where l k (γ) is the quasihyperbolic length of γ (cf. [GP76]). 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 the distance is additive on the geodesic: k G (x, y) = k G (x, z) + k G (z, y) File: hksv151020b.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 2 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN for all z J G [x, y]. It also satisfies the monotonicity property: k G1 (x, y) k G2 (x, y) for all x, y G 2 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) l(γ) C x y ; and (ii) δ(z) 1 [l(γ[x, z]) l(γ[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, 2.50] that the additive constant can be chosen to be 0. This observation leads to the definition of ϕ-uniform domains introduced in [Vu85]. Definition 1.2. Let ϕ : [0, ) [0, ) be a strictly increasing homeomorphism with ϕ(0) = 0. A domain G R n is said to be ϕ-uniform if ( ) x y k G (x, y) ϕ δ(x) δ(y) 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. ϕ(t)/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. Along the way, we show that for a planar Jordan domain assumption (ii) can replaced by the quasi-convexity of the domain s complement. Theorem 1.4. Let D R 2 be a Jordan domain. If D and its complement R 2 \ D are quasi-convex then D is uniform (and hence ϕ-uniform).

3 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 3 Let G R n be a domain and f : G f(g) R n be a homeomorphism. The linear dilatation of f at x G is defined by H(f, x) := lim sup r 0 sup{ f(x) f(y) : x y = r} inf{ f(x) f(z) : x z = r}. A homeomorphism f : G f(g) R n is said to be K-quasiconformal if sup x G H(f, x) K. In Section 2 we study ϕ-uniform domains in relation to quasiconformal and quasisymmetric mappings. This is a natural question to ask and in fact motivation comes from [GO79, Theorem 3 and Corollary 3], which prove invariance of ϕ-uniform domains under quasiconformal mappings in terms of the distance ratio metric j G. Theorem 1.5. 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 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 2 : 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 2-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 2. Lemma 2.1. [GO79, Theorem 3] If f 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 (f(x), f(y)) c max{k G (x, y), k G (x, y) α } We obtain an analogue of Lemma 2.1 for j G with the help of the following result of Gehring s and Osgood s: Lemma 2.2. [GO79, Theorem 4] If f 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 j G (f(x), f(y)) c 1 j G (x, y) + d 1 for all x, y G. In order to investigate the quasiconformal invariance property of ϕ-uniform domains we reformulate Lemma 2.2 in the form of the following lemma. We make use of both the above lemmas in the reformulation.

4 4 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN Lemma 2.3. If f 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 (f(x), f(y)) C max{j G (x, y), j G (x, y) α } Proof. Without loss of generality we assume that δ(x) δ(y) for x, y G. Suppose first that y G \ B n (x, δ(x)/2). Since x y δ(x)/2, it follows that j G (x, y) log(3/2). By Lemma 2.2, we obtain j G (f(x), f(y)) ( c 1 + ) d 1 j G (x, y). log(3/2) Suppose then that y B n (x, δ(x)/2). By [Vu88, Lemma 3.7 (2)], k G (x, y) 2j G (x, y). Hence we obtain that j G (f(x), f(y)) k G (f(x), f(y)) c max{k G (x, y), k G (x, y) α } 2c max{j G (x, y), j G (x, y) α }, where the first inequality always holds (1.1) and the second inequality is due to Lemma 2.1. As a consequence of Lemmas 2.1 and 2.3, we prove our main result Theorem 1.5 about the invariance property of ϕ-uniform domains under quasiconformal mappings of R n. Proof of Theorem 1.5. By Lemma 2.3, there exists a constant C such that (2.4) j G (x, y) C max{j G (f(x), f(y)), j G (f(x), f(y)) α } for all x, y G. Define ψ(t) := ϕ(e t 1). Then k G (f(x), f(y)) c max{k G (x, y), k G (x, y) α } c max{ψ(j G (x, y)), ψ(j G (x, y)) α } c max{ψ(c max{j G (f(x), f(y)), j G (f(x), f(y)) α }), ψ(c max{j G (f(x), f(y)), j G (f(x), f(y)) α }) α }, where the first inequality is due to Lemma 2.1, the second inequality holds by hypothesis, and the last inequality is due to (2.4). Thus, G is ϕ 1 -uniform with ϕ 1 (t) = c max{ψ(c max{log(1 + t), log(1 + t) α }), ψ(c max{log(1 + t), log(1 + t) α }) α }, where α = K 1/(1 n). A mapping f : (X 1, d 1 ) (X 2, d 2 ) is said to be η-quasi-symmetric (η-qs) if there exists a strictly increasing homeomorphism η : [0, ) [0, ) with η(0) = 0 such that ( ) d 2 (f(x), f(y)) d 2 (f(y), f(z)) 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 η(t) = L 2 t and η-qs mappings are K- quasiconformal with K = η(1). 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.

5 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 5 Proposition 2.5. 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. 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 δ(z) = e 1 δ(y). Then ( ) ( ) y z δ(y) j G (y, z) = k G (y, z) = log 1 + = log = 1. δ(z) δ(z) Hence, by the hypothesis we conclude that k G (x, y) η(j G (x, y)). This shows that G is ϕ-uniform with ϕ(t) = η(log(1 + t)). Question 2.6. Is the converse of Proposition 2.5 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{l(γ[x, z]), l(γ[z, y])} c δ(z) 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, l(γ) 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 use the following characterization of uniform domains which can be easily formulated from [Ahl63, MS79] (see also [Ge99] on characterization of quasidisks). Lemma 3.2. A Jordan domain D R 2 is uniform if and only if there exists a constant c > 0 such that for each pair of points a, b D \ { } we have min diam(γ j) c a b j=1,2 where γ 1, γ 2 are the components of G \ {a, b}. The next lemma shows that quasi-convexity extends from the domain to its boundary. Lemma 3.3. Let D R n be a quasi-convex domain. Then any pair of points a, b D can be joined by a rectifiable path γ D such that l(γ) c a b for some constant c > 0. Proof. Let x i, y i D, x i a and y i b. Let γ i D with l(γ i ) c x i y i join x i and y i. Since (x i ), (y i ) and (l(γ i )) are bounded sequences, there exists R > 0 such that γ i B(0, R) for every i. Then γ i is a set of curves of uniformly bounded length in the compact set D B(0, R). Hence we find a subsequence converging to γ D which connects a and b. Furthermore, as required. l(γ) = lim l(γ i ) lim c x i y i = c a b,

6 6 P. HÄSTÖ, R. KLÉN, S. K. SAHOO, AND M. VUORINEN We can now show that a domain which is quasi-convex both from the inside and the outside is uniform. Proof of Theorem 1.4. Consider the arbitrary points a, b D \ { }. By Lemma 3.3 there exist ˆγ 1 and ˆγ 2 joining a and b in D and R 2 \ D, respectively, with l(ˆγ 1 ), l(ˆγ 2 ) c a b. Then the closed curve C := {a} ˆγ 1 {b} ˆγ 2 encloses one of the components of D \ {a, b}, whose diameter is consequently bounded by diam C. Furthermore, diam C c a b so D is uniform by Lemma 3.2. 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 l(γ[x, x ]) = l(γ[y, y ]) = 1 x y. 10 Now, by the cone condition, we have δ(x ) 1 c min{l(γ[x, x ]), l(γ[x, y])} and δ(y ) 1 c min{l(γ[x, y ]), l(γ[y, y])}. By the choice of x and y, on one hand, we see that l(γ[x, y]) x y x y x x 9 x y. 10 On the other hand, l(γ[x, x ]) = 1 x y. The same holds for x and y interchanged. Thus, 10 (3.4) min{δ(x ), δ(y )} 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); (3.5) k G (x, y ) b 2 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, 2, 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) with (c = b 1 + b 2 + b 3 ) would conclude the proof of the theorem. It is sufficient to show the first two lines in (3.5), 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 B(w, 1δ(w)), 2 2 and we can connect the points by the segment [z, w] G. Furthermore, k G (z, w) 2 z w /δ(w) 1 log 3 j G (z, w). 2 Thus in each inequality between the k and j metrics, we may assume that j G log 3. 2 First we prove the second line of (3.5). Since G is ϕ-uniform and ϕ is an increasing homeomorphism, ( ) k G (x, y x y ) ϕ ϕ(12c), min{δ(x ), δ(y )} where the triangle inequality x y x x + x y + y y and the relation (3.4) are applied to obtain the second inequality. On the other hand, j G (x, y) log 3. Thus 2 k G (x, y ) b 2 j G (x, y)

7 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 7 with b 2 = ϕ(12c)/ log 3 2. Then we consider the first line of (3.5): 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 < log 1 + min{δ(x), δ(x )} ) (1 + c) x y (1 + c) j G (x, y), δ(x) where the first inequality holds since x x 1 10 x y and min{δ(x), δ(x )} δ(x)/(1 + c). Fix a point z γ with l(γ[x, z]) = 1 2 δ(x) 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 2 ), where γ 2 = γ[z, x ]. For w γ 1 we have δ(w) 1 2 δ(x), and for w γ 2, δ(w) c l(γ[x, w]). Thus we find that Furthermore, k G (γ 1 ) l(γ 1) 1 δ(x) = 2 2 log 3 j G (x, x ). 2 2 k G (γ 2 ) l(γ[x,x ]) l(γ[x,z]) dt ct = 1 c log l(γ[x, x ]) l(γ[x, z]) = 1 c log 1 x y 10 1 δ(x) 1 c j G(x, x ). 2 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 2 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 2 such that R 2 \D is not ψ-uniform for any ψ. Proof. Fix 0 < u < t < v < 1. Let R k = (x k, x k + u k ) [0, v k ] be the rectangle, k 1. The parameter x k is chosen such that x 1 = 0 and x k+1 = x k +u k +t k. At the top of each rectangle R k we place a semi-disc C k with radius u k /2 and center on the midpoint of the top side of R k. Set s = u/(1 u) + t/(1 t). With these elements we define D, shown in Figure 1, by D := ((0, s) ( 2, 0)) ( k R k ) ( k C k ).

8 8 P. HA STO, R. KLE N, S. K. SAHOO, AND M. VUORINEN 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{δ(x), δ(y)}. 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. The line segments joining x and y to the are at most 2v k + d, 2v l + d and x y + u +u 2 medial axis have length at most uk /2 and v k /2. The whole curve is at distance at least d from the boundary. Thus kd (x, y) kd (γ) `(γ) 3v k + 3v l + 2d + x y 8v k + x y. d d d On the other hand, x y tk. δ(x) δ(y) d x y Let ϕ(τ ) := τ + 8τ α, where α is such that tα = v. Then kd (x, y) ϕ( δ(x) δ(y) ). 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. We show then that R2 \ D is not ψ-uniform for any ψ. We choose zk = (xk+1 tk /2, tk ) in the gap between Rk and Rk+1. Then zk zk+1 tk /2 + uk+1 + tk+1 /2 + tk u k+1 = t k+1 δ(zk ) δ(zk+1 ) t 2t 2 2t 2 On the other hand, a curve connecting these points has length at least v k tk /2, so that Z vk tk /2 dx v k tk /2 kr2 \D (zk, zk+1 ) = log x tk /2 tk /2

9 GEOMETRIC PROPERTIES OF ϕ-uniform DOMAINS 9 as k. Therefore, it is not possible to find ψ such that k R 2 \D (z k, z k+1 ) ψ( z k z k+1 ), δ(z k ) δ(z k+1 ) as claimed. References [Ahl63] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), [Ge99] F. W. Gehring, Characterizations of quasidisks, Quasiconformal geometry and dynamics, Banach center publications, Vol 48, Polish Academy of Sciences, Warszawa [GH12] F. W. Gehring and K. Hag, The Ubiquitous Quasidisk, Mathematical Surveys and Monographs 184, Amer. Math. Soc., Providence, RI, [GHM89] F. W. Gehring, K. Hag, and O. Martio, Quasihyperbolic geodesics in John domains, Math. Scand., 65 (1989), [GO79] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. Anal. Math. 36 (1979), [GP76] F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Anal. Math. 30 (1976), [He99] D. A. Herron, John domains and the quasihyperbolic metric, Complex Variables, 39 (1999), [KL98] K. Kim and N. Langmeyer, Harmonic measure and hyperbolic distance in John disks, Math. Scand., 83 (1998), [KLSV14] R. Klén, Y. Li, S.K. Sahoo, and M. Vuorinen, On the stability of ϕ-uniform domains, Monatshefte für Mathematik, 174 (2014)(2), [Ko09] P. Koskela, Lectures on Quasiconformal and Quasisymmetric Mappings, Lecture note, December [MS79] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Math. 4 (1979), [NV91] R. Näkki and J. Väisälä, John disks, Expo. Math., 9 (1991), [Va71] J. Väisälä, Lectures On n-dimensional Quasiconformal Mappings, Lecture Notes in Mathematics 229, Springer, [Va88] J. Väisälä, Uniform domains, Tohoku Math. J. 40 (1988), [Va91] J. Väisälä, Free quasiconformality in Banach spaces II, Ann. Acad. Sci. Fenn. Math. 16 (1991), [Va98] J. Väisälä, Relatively and inner uniform domains, Conformal Geometry and Dynamics, 2 (1998), [Vu85] M. Vuorinen, Conformal invariants and quasiregular mappings, J. Anal. Math. 45 (1985), [Vu88] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Mathematics 1319, Springer-Verlag, Berlin Heidelberg New York, 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 Turku, Finland address: riku.klen@utu.fi Department of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore , India address: swadesh@iiti.ac.in Department of Mathematics and Statistics, University of Turku, FIN Turku, Finland address: vuorinen@utu.fi