1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.4. UNIFORM DOMAINS ARE RIGID 129 LEMMA 9.4.10. If Eis a convex set in R^2 and if f: E-+R^2 is locally L-lipschitz in E, then f ...

130 9. SECOND SERIES OF IMPLICATIONS To establish (9.4.14), we show first that there is a unique continuous function g defined o ...

9.4. UNIFORM DOMAINS ARE RIGID 131 PROOF. By hypothesis, there exists a constant c 2: 1 such that each pair of points z 1 , z2 E ...

132 9. SECOND SERIES OF IMPLICATIONS by (9.4.21). Then F(z) = E j(z) - Ej+ 1 (z) is linear in x and y where z = x + iy and F(B*) ...

9.5. RIGID DOMAINS ARE LINEARLY LOCALLY CONNECTED 133 LEMMA 9.5.l. Suppose that ¢(t) is a real-valued function definedforO < ...

134 9. SECOND SERIES OF IMPLICATIONS in which case g(z ) = f(z) in U. Hence g is lo cally L-bilipschitz in D and hence injective ...

9.7. MIN-MAX PROPERTY AND LOCAL CONNECTIVITY PROOF. Fix z E "(and w €J_ D. Then while I I z - w < ---------'----"----'------' ...

136 9. SECOND SERIES OF IMPLICATIONS Then min lz - Zjl ::::; min lzj - wl + lz -wl ::::; (a+ l)dist(z, 8D) J=l,2 J=l,2 and (9.7. ...

CHAPTER 10 Third series of implications We establish here three more characterizations for quasidisks making use of the followin ...

138 10. THIRD SERIES OF IMPLICATIONS 10.1. Quasidisks and BMO-extension We shall need the following result due to H. M. Reimann ...

10.2. BMO-EXTENSION AND THE HYPERBOLIC METRIC PROOF. Choose z 1 , z 2 ED with ri = dist(z1, aD) S dist(z 2 , aD) = r 2 and let t ...

140 10. THIRD SERIES OF IMPLICATIONS for j = 1, 2 by (5.1.3) in the proof of Lemma 5.1.2. Thus hD(z1, z2) = lu(z1) - u(z2)I :S i ...

10.3. HYPERBOLIC METRIC AND HYPERBOLIC SEGMENTS We shall consider the cases where (10.3.7) (10.3.8) separately. r < max(dist( ...

142 10. THIRD SERIES OF IMPLICATIONS We shall show first that (10.3.12) length('Y(zj, Wj)) ::::; b dist( Wj, oD), length(t(zj, z ...

10.3. HYPERBOLIC METRIC AND HYPERBOLIC SEGMENTS 143 This is the first inequality. For the second, if z E 'Y( z 1 , w 1 ), then z ...

144 10. THIRD SERIES OF IMPLICATIONS by Lemma 10.3.1 and, as in (10.3.9), length(r(w 1 , w 2 )):::; e 2b dist(w1,8D)^1 d. ( ds o ...

10.4. APOLLONIAN METRIC IN A QUASIDISK 145 PROOF. Set Then f'(u) = K(cu)K-^1 g(u) where Hence g(u):::; g(l):::; 0, J'(u):::; 0, ...

146 10. THIRD SERIES OF IMPLICATIONS THEOREM 10.4.9. If DC R^2 is a K-quasidisk, then (10.4.10) hD (z1, z2) :'S: K^2 aD (z1, z2) ...

10 .4. APOLLONIAN METRIC IN A QUASIDISK 147 by (10.4.15). Hence m < r/2. Next C = 8D C {z: r - 1 ::=::: lz l ::=::: r + l}. a ...

148 10. THIRD SERIES OF IMPLICATIONS PROOF. Fix z 1 , z 2 ED. Then hn(z1, z2)::::; >..(K) an(z1, z2) by Theorem 10.4.14 if wh ...