关于beurling-ahlfors扩张推广

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1、eTh(x)+$DmS({2Th(x+t)−h(x)ρ(x,t)=x∈R,t∈(0,∞).h(x)−h(x−t)XoR

2、7Q.CB#(e>ρ(x,t)T$2ρ,

3、ρ.7"vH7:2(r2+(1−w)2)1K≤max{,}ρ+o(ρ).r2r2(r2+(1−w)2)1ρ+o(ρ)r2+(1−w)2≥,=r411ρ+o(ρ)r2+(1−w)2<,2r412(r2+(1−w)2)22v3r>0,w∈[0,1],>r+(1−w)≥4,^2rla,,1>r=2,w=1",K≤ρ+o(ρ),v3^21la,.>ρ(

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5、dsbAEDB#,~h(x)T(.,>D2:D(z)3_1q:17511yyyyD(x+iy)≤((+ρ(x,y)))(ρ(x+,)+ρ(x−,)+2).322ρ(x,y)22222ρ(x,t)[Gρ(t)-mS({2−1ρ(t)≤ρ(x,t)≤ρ(t),x∈R,t∈(0,∞),t∗Sρ(t)=sup{ρ(s),s∈[2,t]},5

6、dsbAEDB#,~h(x)T(.,>D:D(z)3_1q∗D(x+iy)≤ρ(y0)+3.mzU:jkat/wV&2"IABSTRACTSupposeh(x)isa

7、monotonichomeomorphismofrealaxisRontoitselfinthecomplexplaneandsatisfiesh(±∞)=±∞,itsquasi-symmetricfunctionish(x+t)−h(x)ρ(x,t)=x∈R,t∈(0,∞).h(x)−h(x−t)InthispaperanotherclassofQ.Cextensionisconstructed,whenρ(x,t)isaconstantρandρbigenough,weprovethatthemaximaldilatationKsatisfies:2(r2+(1−w)2)1K≤max

8、{,}ρ+o(ρ).r2r2(r2+(1−w)2)1ρ+o(ρ)r2+(1−w)2≥,=r411ρ+o(ρ)r2+(1−w)2<,2r412(r2+(1−w)2)22wherer>0,w∈[0,1].Whenr+(1−w)≥,thecoefficient4r1cannotbeimproved.Whenr=,w=1,wehaveK≤ρ+o(ρ)andthecoefficient211cannotbeimproved.Whenρ(x,t)isnotaconstant,letr=,w=1,then2thereexistsaextensionf(x,y)ofhalfplaneont

9、oitself,f(x,0)=h(x).LetD(z)bethedilatationoff(x,y),then17511yyyyD(x+iy)≤((+ρ(x,y)))(ρ(x+,)+ρ(x−,)+2).322ρ(x,y)22222Ifρ(x,t)satisfiesρ(t)-quasi-symmetricfunction−1ρ(t)≤ρ(x,t)≤ρ(t),x∈R,t∈(0,∞),letρ∗(t)=sup{ρ(s),s∈[t,t]},thenthereexistsaextensionofhalfplaneonto2itselff(x,y),f(x,0)=h(x),letD(z)bethe

10、dilatationoff(x),then∗D(x+iy)≤ρ(y)+3.Keywords:Quasi-conformalextension;Quasi-symmetricfunction;Dilata-tionIIG5W?AT)'8e<*DxWYX+D/_,ry0i<
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