1549055259-Ubiquitous_Quasidisk__The__Gehring_

7.2. COMPARABLE DIRICHLET INTEGRALS 89 7.2. Comparable Dirichlet integrals We observed in Theorem 5.2.3 that if D C R^2 is a sim ...

90 7. MISCELLANEOUS PROPERTIES DEFINITION 7.2.6. A Jordan domain D has the comparable Dirichlet integral property if there exist ...

7.4. HOMOGENEITY 91 A set E C R 2 is homogeneous with respect to a family F of mappings if for each z 1 , z2 E E there exists a ...

92 7. MISCELLANEOUS PROPERTIES FIGURE 7.1 Hjelle shows that D is the union of an increasing sequence of simply connected domains ...

h o g 1 (E) ho g 1 o h-^1 (E) 7.5. FAMILY OF ALL QUASICIRCLES h(E) FIGURE 7 .2 7.5. Family of all quasicircles 93 We construct n ...

94 7. MISCELLANEOUS PROPERTIES Then let S 2 be any of the 24 polygons obtained by replacing each of the four sides 'Yj of S 1 by ...

7.6. QUASICONFORMAL EQUIVALENCE OF R^3 \ 75 AND B^3 95 To see this, we may assume without loss of generality that D = S(a) where ...

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Part 2 Some proofs of these properties ...

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CHAPTER 8 First series of implications In the preceding chapters we have presented many different ways to view a quasi disk D in ...

100 8. FIRST SERIES OF IMPLICATIONS 8.1. Quasidisks and hyperbolic segments We begin with the following elementary observation. ...

8 .1. QUASIDISKS AND HYPERBOLIC SEGMENTS H iyj iy f 'lYJ+l 0 FIGURE 8.2 by Corollary 1.3.7. This yields y log-^1 - :::; (k + 1) ...

102 8. FIRST SERIES OF IMPLICATIONS f B FIGURE 8.3 for z E 1, we see that it is sufficient to establish (8.1.6) and (8.1.7) for ...

8.1. QUASIDISKS AND HYPERBOLIC SEGMENTS 103 :oo H · W g f h 0 0 FIGURE 8.4 and we obtain (8.1.6) with c = c 2. Finally, since Di ...

104 8. FIRST SERIES OF IMPLICATIONS 8.2. Hyperbolic segments and uniform domains The following is an immediate consequence of th ...

8.3. UNIFORM DOMAINS AND LINEAR LOCAL CONNECTIVITY 105 r FIGURE 8.7 PROOF. Fix zo E R^2 , 0 < r < oo, and suppose that z 1 ...

106 8. FIRST SERIES OF IMPLICATIONS FIGURE 8.8 before, uniform. Thus conditions (3.5.2) and (3.5.3) in Definition 3.5.1 for a un ...

8.5. THE THREE-POINT CONDITION AND QUADRILATERALS 107 Chooses with r < s < t. Since z1, z2 E 8D n B(zo, s), we can find fo ...

108 8. FIRST SERIES OF IMPLICATIONS Next let /3 1 ,/3 2 denote the components of 8D \ (a 1 U a2), labeled so that /3j c 'Yj and ...