ABSTRACT
From[8,Chapter11,Section6]itfollowsthat,forallpairs(U,V)of elementsinC~(X,Bq),onehas
b0(U,V)=LlJp(U,V) P""l
insuchamannerthat,foreachintegerp2:1,lJ/U,V)isthesumofa numberap:s::p-1oftermsoftheforma;·l]~(U,V),1:s::s:s::p-1, witha;:s::1/p,andwithl]~(U,V)consistingofamultibracketcontaining stimesUandp-stimesV. Lemma5.LetIIIIbeanadmissibleseminormIIIIonS!ll(X,C§)= C~(X,C§),i.e.,suchthat,forallU,VinS!ll(X,C§),
PROOF.LetU,VinC~(X,B,);fromtheabm·ediscussionitfollows that,foreachp2:Iand1:s::s:s::p-I:
lllJ~(U,V)ll:s::IIUII'·IIVIIP-' andthenlllJp(U,\l)ll:s::<IIUII+IIVIIY.:\1oreowr
assoonasIIUII+IIVII:s::!.0 Lemma6.Let(k,)1beanincreasingsequenceofpositivenumbers\Vith lim 1 ~+xk1=+oo.ForallU,VinC~(X,Bq)withIUik,,i:S::~andIVIk,,i:S:: !,onehas
PROOF.FromProposition1,withoutlossofgenerality,wemay supposethat,forallU,VinC~(X,C§),I[U,V]lk,,i:s::IUlk,,i·IVIk,,i;the assertionfollowsthenfromLemma5.o Lemma7.Letk>(l/2)dim(X).ForallU,VinC~(X,Bq),withIlL%*>:s:: ~andIIVII~*>:s::i,onehas
PROOF.Hereagain,fromProposition3,wemaysupposewithout lossofgeneralitythat,ask>(l/2)dim(X),II[U,VJII~*>:s::IIUII~k)·IIVII~k)for
Let 9ll(X, G) be the set of all C' - mappings g: X~ G which have a compact support, and for all compact subset K of X, let 9ll~~.{X, G) = {g E 9ll(X, G)i[g] ~ A} One easily sees that 9ll(X, G) is a group for the pointwise product coming from the product in G, for \vhich the mapping £ : ~ E(x) = e, V x EX is the unit, and in \Vhich 9ll~~.{X, G) is an invariant subgroup.
