1549055259-Ubiquitous_Quasidisk__The__Gehring_

CHAPTER 5 Criteria for extension Suppose F denotes a certain property of functions, for example, continuity or integrability, an ...

70 5. CRITERIA FOR EXTENSION PROOF. If Bo is any disk with center z 0 and Bo c D , then iuBo - u(zo)I :S: m(~o) l 0 lhD(z, z1) - ...

5.2. SOBOLEV AND FINITE ENERGY FUNCTIONS 71 by what was proved above and the choice of k. D We turn now to a proof of the result ...

CRITERIA FOR EXTENSION EXAMPLE 5.2.1. If Dis a disk or half-plane and if u E Li(D), then u has an extension v E Lr(R^2 ) with ...

5.3, QUASICONFORMAL MAPPINGS 73 DEFINITION 5.2.4. A domain D is a Sobolev extension domain if for each p , 1 ::::; p < oo, th ...

74 5. CRITERIA FOR EXTENSION is connected, and hence w 1 is also not a cut point of fJD. 0 We turn now to the proof of the resul ...

5 .4. BILIPSCHITZ MAPPINGS 75 THEOREM 5.4.3 (Gehring [50], Tukia [160], [162]). A bounded Jordan domain D is a K -quasidisk if a ...

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CHAPTER 6 Two-sided criteria Suppose that a property of a domain D is not enough to guarantee that it is a quasidisk. Could it s ...

78 6. TWO-SIDED CRITERIA FIGURE 6.1 The corresponding result for property 2° does not follow for merely simply connected domains ...

6.2. HARDY-LITTLEWOOD PROPERTY 79 then f E Lipa(B) with II! Ila S com a where co is an absolute constant. This result can be ext ...

80 6. TWO-SIDED CRITERIA FIGURE 6.2 EXAMPLE 6.2.9. Define a Jordan domain D as 00 where 6.i is the interior of an equilateral tr ...

6.3. Lip"-EXTENSION DOMAINS 81 6.3. Lip 0 -extension domains We return to Theorem 6.2.4. This result can be viewed as a conseque ...

82 6. TWO-SIDED CRITERIA Next, assume that D is a Lip 0 -extension domain, i.e. that (6.3.2) is satisfied. Consider f analytic i ...

6.4. HARMONIC DOUBLING CONDITION 83 Finally, we have a characterization of quasidisks in terms of Lipa-extension domains paralle ...

84 6. TWO-SIDED CRITERIA where f = 'ljJ-^1 o ¢ : S^1 --+ S^1 denotes the sewing homeomorphism associated with D. We have proved ...

6.4. HARMONIC DOUBLING CONDITION 85 Next we will use the doubling condition in D*. For this we choose n :::; bas above so that d ...

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CHAPTER 7 Miscellaneous properties We conclude our account of ways to describe a quasidisk D with some seem- ingly unrelated cha ...

88 7. MISCELLANEOUS PROPERTIES and f(z) = z, then g(z) = ez is conformal in D , f(z) =log g'(z), and llfllB(D):::; 8 by what was ...