Course Hero Logo

61 function 1 2 3 4 5 theorem for n 2 there exist

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 90 - 94 out of 227 pages.

6.1.function(1)(2)(3)(4)(5)Theorem.For n > 2 there exist positive numbers dl,. • •,d4and a setc(.) in R '~ such thatc(E) = c(hE) whenever h: R'~~n is a spherical isometryand EC~n.O<3c(O) = O,AC BC I:U ~ impliesc(A)< c(B)andc(Uy=IEj)<dl ~j°~=l c(Ej) if Ej cRY.If E C ~r~ is compact, then c(E) > 0 if and only ifcapE >0.Moreoverc(R '~) < d2 < oo.c(E) >_ d3 q(E) if EC R'~is connectedand E ¢ 0.M(A(E, F)) _>d4min{ c(E), c(F) } , if E,F C R~ .Furthermore,for n ~_ 2andt E (0, 1)there exists a positive number d5 such that(6)M(A(E,F)) _~dsmin{c(E),c(F)}wheneverE,F c R'~ and q(E,F) >_ t.It should be emphasized that the main interest in Theorem 6.1 lies in the inequal-ities (5) and (6). The condition cap E > 0 in 6.1(3) is not needed in this section andits definition will be postponed until Section 7.
73We shall next give the reader some idea about the set function c(-). To this enddefine (see (5.45))(6.2)Mt(E,r,x) = M(A( Sn-t(x,t), -Bn(x,r) A E; R'~)) ,M(E, r,x)= M2,(E, r, x)whenever Ec~nxER '~,and 0<r<t.Moreover, let E -l={z/lzt2:xEE}and(6.3)a(E)= max{M(E, 1, 0), M(E -1, 1, O) }mfor E C R '~ . It follows from the results of this section that there are numbers qland "~2 depending only on the dimension n such that(6.4)?'la(E) < c(E) < "72a(E) .In what follows we shall give a construction of the set functionc(E).We remarkthat there may also be many other methods of constructingc(E):it is clear by(6.4) that any method which yields a set function differing froma(E)by at most amultiplicative constant is adequate also for constructingc(E) .For the next lemma we recall that the ballsQ(x,r) of the metric space (R n, q)were defined in (1.22).6.5. Lemma.Letr > 1 and tE (0,1)besuch that q(S '~-l,S~-l(r))>2t.Thereisa numberb(t) depending only on n, r,andt such that the following holds.If E c B '~ and G t= U=eE Q,(x,t), thenM(A(E, OGt)) < b(t)M(A(E,S'~-l(r))) .Proof.q(F1,F3) >_ t5.42(2) thatLetF1 =E,F2 =S'~-l(r),andFa =OGt = F4.Becauseandq(F2,F4) >_ tit follows from the comparison principle 5.40 and(6.6)M(r12) _> 3-nmin{ M(F13), M(F23),Dt}where rij =A(Fi, Fj).We shall first find a lower bound forM(r~3).From thechoice of t it follows that t < l/v/2 and henceq(Q(z,t))= 2t~/i -t 2 > tv~(see
741.25). It follows thatF3 contains a continuum of euclidean diameter at least tV~.Hence we deduce by 5.9 and 5.34 that2r + tv~cntlog 2D t(6.7)M(r~3) >cnlog --> --:>-2r -tv~--r--rHere nis the number in 5.42(2) and (6.6). Since d(l'~I) > q(l~]) >- t and I'll cB~(r)for ~ ~ A(Fl,F3;a,)= A and M(zX) = M(ri3)(of. (S.10)) we get by 5.~r -M(r~3) <_ U"i-~ •By (6.6) and (6.7)M(I',2) _> 3 -n min{ M(F,3), --Dt}r_>3 -"rainM(r~),n.r-+~M(r~3)=u(ri~),b(t)= 3n/rain{ 1,ntn+l/(Ct,~rn+X)} .[]6.8,(cf. (6.2))The constructionofc(E).For E C R~,0 < r < t < 1 denotef mr(E,r, x) = M(A(OQ(x, t), E n Q(x, r))) ,(6.9)/,m(E,x) = ms(E, 1/v/2, x) ; s = ½~/3.We define (see (1.16) and (1.23))f c(E,x) =max(m(E,x),m(E,x~ },(6.10)/c(E) = inf{ c(E, :~) : x c ~." }.±1~(~)-=(~1=1½V~1/v~Diagram 6.1.
756.11. Remark.By 5.18(2)lmt(E,r,x)<r['t/l~r2 ~]l-n(6.12)[m(E,x) <_ m(R'~,x)= w,~_,(logv/3) 1-.IfF c-Q(x,r),where r E (0,1/v~], by (6.12) we obtain(01 )Hencec(F,x) -+ 0 as r -* O.

Upload your study docs or become a

Course Hero member to access this document

Upload your study docs or become a

Course Hero member to access this document

End of preview. Want to read all 227 pages?

Upload your study docs or become a

Course Hero member to access this document

Term
Fall
Professor
NoProfessor
Tags
Conformal map, Quasiregular Mappings

Newly uploaded documents

Show More

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture