ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaA Global Theory of Algebrasof Generalized FunctionsM. GrosserM. KunzingerR. SteinbauerJ. Vickers

Vienna, Preprint ESI 813 (1999) December 27, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

A global theory of algebras of generalizedfunctionsM. Grosser, M. Kunzinger, R. SteinbauerUniversit�at WienInstitut f�ur MathematikJ. VickersUniversity of Southampton,Department of MathematicsAbstract. We present a geometric approach to de�ning an algebra G(M) (theColombeau algebra) of generalized functions on a smooth manifold M containingthe space D0(M) of distributions onM . Based on di�erential calculus in convenientvector spaces we achieve an intrinsic construction of G(M). G(M) is a di�erentialalgebra, its elements possessing Lie derivatives with respect to arbitrary smoothvector �elds. Moreover, we construct a canonical linear embedding of D0(M) intoG(M) that renders C1(M) a faithful subalgebra of G(M). Finally, it is shownthat this embedding commutes with Lie derivatives. Thus G(M) retains all thedistinguishing properties of the local theory in a global context.2000 Mathematics Subject Classi�cation. Primary 46F30; Secondary 46T30.Key words and phrases. Algebras of generalized functions, Colombeau algebras,distributions on manifolds, calculus on in�nite dimensional spaces.1 IntroductionColombeau's theory of algebras of generalized functions ([6], [7], [8], [23])is a well-established tool for treating nonlinear problems involving singularobjects, in particular, for studying nonlinear di�erential equations in gener-alized functions. A Colombeau algebra G() on an open subset of Rn isa di�erential algebra containing D0() as a linear subspace and C1() as afaithful subalgebra. The embedding D0() ,! G() commutes with partialdi�erentiation and the functor ! G() de�nes a �ne sheaf of di�erentialalgebras. In view of L. Schwartz's impossibility result ([26]) these propertiesof Colombeau algebras are optimal in a very precise sense (cf. [23], [24]). Inthe so-called \full" version of the theory the embedding D0() ,! G() iscanonical, as opposed to the \special" or \simpli�ed" version (cf. the remarkbelow), where the embedding depends on a particular molli�er.The main interest in the theory so far has come from the �eld of nonlin-ear partial di�erential equations (cf. e.g. [3], [4], [10], [11], [12], as well as1

[23] and the literature cited therein), whereas the development of a theoryof Colombeau algebras on manifolds proceeded at a much slower pace. [1]presents an approach which basically consists in lifting the sheaf G fromRn to a manifold M . A more re�ned sheaf-theoretic analysis of Colombeaualgebras on manifolds is given in [13]. It treats the \special" version of thealgebra in the sense of [23], p. 109f, whose elements (termed ultrafunctions bythe authors) depend on a real regularization parameter. In both approaches,as well as in the construction envisaged in [2], the canonical embeddingC1(M) ,! D0(M) ,! G(M) is lost when passing from M = � Rn to ageneral manifold (due to the fact that in these versions action of di�eomor-phisms commutes with embedding only in the sense of association but notwith equality in the algebra). Also, although Lie derivatives of elements ofG(M) are de�ned in [13] the operation of taking Lie derivatives does notcommute with the embedding (again this property only holds in the sense ofassociation).To remedy the �rst of these defects, J. F. Colombeau and A. Meril in [9](using earlier ideas of [6]) introduced an algebra of generalized functions on whose elements depend smoothly (in the sense of Silva-di�erentiability)on ('; x), where ' 2 D(Rn) and x 2 . The aim of [9] is to make theembedding of D0() (which is done basically by convolution with the ''s:u 2 D0() 7! (('; x) 7! hu(y); '(y�x)i)) commute with the action of di�eo-morphisms. However, an explicit counterexample in [19] demonstrated thatthe construction given in [9] is in fact not di�eomorphism invariant. Also in[19], J. Jel��nek gave an improved version of the theory clarifying a numberof open questions but still falling short of establishing the existence of an in-variant (local) Colombeau algebra. Finally, this aim was achieved in [16] and[17], where a complete construction of di�eomorphism invariant Colombeaualgebras on open subsets of Rn is developed. Several characterization resultsderived there will be crucial for the presentation in the following sections.Summing up, the current state of the theory is that a di�eomorphism in-variant local version (and, therefore, a version on manifolds) of Colombeaualgebras providing canonical embeddings of smooth functions and distribu-tions is available. Recent applications of Colombeau algebras to questionsof general relativity (e.g. [5], [2], [21], [27], for a survey see [28]) have under-scored the need for a theory of algebras of generalized functions on manifoldsthat enjoys two additional features: �rst, it should be geometric in the sensethat its basic objects should be de�ned intrinsically on the manifold itself.Second, the object to be constructed should be a di�erential algebra with Liederivatives commuting with the embedding of D0(M).In this paper we construct an algebra G(M) satisfying both of these re-quirements. In particular, elements of G(M) possess a Lie derivative LX2

with respect to arbitrary smooth vector �elds X on M . Moreover, each Liederivative commutes with the canonical embedding of D0(M) into G(M), sothat in fact all the distinguishing properties of Colombeau algebras on opensubsets of Rn are retained in the global case. The key concept leading to aglobal formulation of the theory is that of smoothing kernels: De�nition 3.3(i) below is in a sense the di�eomorphism invariant `essence' of the processof regularization via convolution and linear scaling on Rn while 3.3 (ii) is theinvariant formulation of the interplay between x- and y-di�erentiation in thelocal context. Finally, a number of localization results in section 4 allow oneto make full use of the well-developed local theory also in the global context.2 Notation and terminology, the local theoryThroughout this paper,M will denote an oriented paracompact C1-manifoldof dimension n. An atlas of M will usually be written in the form A =f(U�; �) : � 2 Ag. By nc (M) we mean the space of compactly supported(smooth) n-forms on M . Locally, for coordinates y1; : : : ; yn on U�, elementsof nc ( �(U�)) will be written as 'dny := 'dy1 ^ � � � ^ dyn. The pullbackof any ! 2 nc ( �(U�)) under � is written as ��(!). Then RM ��('dny) =R �(U�) 'dny for all 'dny 2 nc ( �(U�)). For U(open) � M open we willnotationally suppress the embedding of nc (U) into nc (M), and similarly forthe inclusion of D(U) (compactly supported smooth functions with supportcontained in U) into D(M). We generally use the following convention:if B � �(U�) then B := �1� (B) and if f : �(U�) ! R resp. C thenf := f � �. A similar \hat-convention" will be applied to the functionspaces to be de�ned in the following section. Moreover, since M is supposedto be oriented we can and shall identify n-forms and densities henceforth.For an open subset of Rn, the space of distributions on (i.e., the dualof the (LF)-space D()) will be denoted by D0(). For a di�eomorphism� : ~! , the pullback of any u 2 D0() under � is de�ned byh��(u); 'i = hu(y); '(��1(y)) � jdetD��1(y)ji (1)Concerning distributions on manifolds we follow the terminology of [14] and[22], so the space of distributions on M is de�ned as D0(M) = nc (M)0.Observe that in this setting test objects no longer have function characterbut are n-forms. In the context of Colombeau algebras it is natural to regardsmooth functions as regular distributions (which in the non-orientable caseenforces the use of test densities; this is in accordance with [18] but has tobe distinguished from the setting of [15]).3

Operations on distributions are de�ned as (sequentially) continuous exten-sions of classical operations on smooth functions. In particular, for X 2X(M) (the space of smooth vector �elds on M) and u 2 D0(M) the Liederivative of u with respect to X is given by hLXu; !i = �hu;LX!i. Ifu 2 D0(M), (U�; �) 2 A then the local representation of u on U� is theelement ( �1� )�(u) 2 D0( �(U�)) de�ned byh( �1� )�(u); 'i = hujU�; ��('dny)i 8' 2 D( �(U�)) (2)It should be clear from (1) and (2) that the character of test objects as n-forms is actually already built into the local theory of distributions on Rn(i.e., on the right hand side of (1) u acts exactly on the coe�cient functionof (��1)�('dny)).As in [16], [17] di�erential calculus in in�nite dimensional vector spaces willbe based on the presentation in [20]. The basic idea is that a map f : E ! Fbetween locally convex spaces is smooth if it transports smooth curves in Eto smooth curves in F , where the notion of smooth curves is straightforward(via limits of di�erence quotients). The di�eomorphism invariant theoryof Colombeau algebras on open subsets of Rn introduced in [16] is basedon this notion of di�erentiability. Of the two (equivalent) formalisms fordescribing the di�eomorphism invariant local Colombeau theory analyzedin [16], Section 5 only one (the so-called J-formalism which was employedby Jel��nek in [19]) lends itself naturally to an intrinsic generalization onmanifolds, as the embedding of distributions does not involve a translation(see below). We brie y recall the main features of this theory.Let � Rn open; then de�neA0() = f' 2 D()jZ '(�)d� = 1gAq(Rn) = f' 2 A0(Rn)jZ '(�)��d� = 0; 1 � j�j � q; � 2 Nn0g (q 2 N) :The basic space of the di�eomorphism invariant local Colombeau algebra isde�ned to be E() = C1(A0() � ). The algebra itself is constructed asthe quotient of the space of moderate modulo the ideal of negligible elementsR of the basic space where the respective properties are de�ned by pluggingscaled and transformed \test objects" into R and analyzing its asymptoticbehavior on these \paths" as the scaling parameter " tends to 0. Di�eo-morphism invariance of the whole construction is achieved by di�eomor-phism invariance of this process| termed \testing for moderateness resp.negligibility"|in [16], Section 9. We shall discuss this matter in some moredetail at the end of this section but now proceed by de�ning the actual \test4

objects." Set I = (0; 1]. Let C1b (I�;A0(Rn)) be the space of smooth maps� : I � ! A0(Rn) such that for each K �� (i.e., K a compact subsetof ) and any � 2 Nn0 , the set f@�x�("; x) j " 2 (0; 1]; x 2 Kg is bounded inD(Rn). For any m � 1 we setA2m() := f� 2 C1b (I � ;A0(Rn))j supx2K jZ �("; x)(�)��d�j = O("m)(1 � j�j � m) 8K �� gA�m() := f� 2 C1b (I � ;A0(Rn))j supx2K jZ �("; x)(�)��d�j = O("m+1�j�j)(1 � j�j � m) 8K �� gElements of A2m() are said to have asymptotically vanishing moments oforder m (more precisely, in the terminology of [17], elements of A2m() areof type [Ag], the abbreviation standing for asymptotic vanishing of momentsglobally, i.e., on each K �� ). The spaces A2m() and A�m() play a crucialrole in the characterizations of the algebra ([16], Section 10) and in Section 4below (cf. also the discussion of di�eomorphism invariance at the end of thissection). Also for later use we note that A2m() � A�m() � A22m�1(). Forx 2 Rn, " 2 I we de�ne translation resp. scaling operators by Tx : D(Rn)!D(Rn), Tx(') = '(:�x) and S" : D(Rn)! D(Rn), S"' = "�n'( :"). Now thesubspaces of moderate resp. negligible elements of E() are de�ned byEm() = fR 2 E() j 8K �� 8� 2 Nn0 9N 2 N8� 2 C1b (I � ;A0(Rn)) : supx2K j@�(R(TxS"�("; x); x))j = O("�N ) ("! 0)gN () = fR 2 E() j 8K �� 8� 2 Nn0 8r 2 N9m 2 N8� 2 C1b (I � ;Am(Rn)) : supx2K j@�(R(TxS"�("; x); x))j = O("r) ("! 0)gBy [16], Th. 7.9 and [17], Corollaries 16.8 and 17.6, R 2 Em() is negligiblei� 8K �� 8� 2 Nn0 8r 2 N 9m 2 N 8� 2 A2m(); 9C > 0 9� > 0 8" (0 <" < �) 8x 2 K: j@�(R(TxS"�("; x); x))j � C"r :The di�eomorphism invariant Colombeau algebra G() on is the quotientalgebra Em()=N (). Partial derivatives in G() are de�ned as(DiR)('; x) = �((d1R)('; x))(@i') + (@iR)('; x) (3)where d1 and @i denote di�erentiation with respect to ' and xi, respectively.Note that formula (3) is exactly the result of translating the usual partial5

di�erentiation from the C-formalism to the J-formalism (cf. [16], Section 5).With these operations, ! G() becomes a �ne sheaf of di�erential algebrason Rn. For R 2 Em() we denote by cl[R] its equivalence class in G(). Themap � : D0() ! G()u ! cl[('; x)! hu; 'i]provides a linear embedding of D0() into G() whose restriction to C1()coincides with the embedding� : C1() ! G()f ! cl[('; x)! f(x)]so � renders C1() a faithful subalgebra and D0() a linear subspace of G().Moreover, � commutes with partial derivatives due to the speci�c form of (3).The pullback of R 2 E() under a di�eomorphism � : ~! is de�ned as(�R)( ~'; ~x) = R(��( ~'; ~x))where ��( ~'; ~x) = (( ~' � ��1) � jdetD��1j; �(~x)). Pullback under di�eomor-phisms then commutes with the embedding of D0() into G().Finally we return to the issue of di�eomorphism invariance. Since the actionof a di�eomorphism onR 2 E() is de�ned in a functorial way di�eomorphism-invariance of the de�nition of the moderate resp. negligible elements of E()indeed is invariant if the class of scaled and transformed \test objects" is;more precisely if�("; x) = S�1" � T�x � pr1 � ��(T��1x � S" ~�("; ��1x)(�); ��1x)= ~�("; ��1x)���1("� + x)� ��1x" � � jdetD��1("� + x)j (4)is a valid \test object" if ~�("; ~x) was. However, if ~� 2 C1b (I � ;Aq(Rn))in general � will neither be de�ned on all of I � nor will its moments bevanishing up to order q. To remedy these defects a rather delicate analysisof the testing procedure is required: Denote by C1b;w(I�;A0(Rn)) the spaceof all � : D ! A0(Rn) where D is some subset (depending on ') of (0; 1]�and for D;' the following holds:For each L �� there exists "0 and a subset U of D which is open in(0; 1]� such that(0; "0]� L � U(� D) and � is smooth on U (5)f@��("; x) j 0 < " � "0; x 2 Lg is bounded in D(Rn) 8� 2 Nn0 (6)6

(the subscript w signi�es the weaker requirements on the domain of de�nitionof �). Then by [16], Th. 10.5 R 2 E() is moderate i� 8K �� 8� 2Nn0 9N 2 N 8� 2 C1b;w(I � ;A0(Rn)); � : D ! A0(Rn))9C > 0 9� >0 8" (0 < " < �) 8x 2 K: ("; x) 2 D andj@�(R(TxS"�("; x); x))j � C"�NMoreover, by [16], Th. 7.14, (4) is an element of C1b;w(I� ~;A0(Rn)) for every� 2 C1b (I � ;A0(Rn)). The subspace of C1b;w(I � ;A0(Rn)) consistingof those � whose moments up to order m vanish asymptotically on eachcompact subset of will be written as A2m;w() (also, A�m;w() is de�nedanalogously). By the proof of [16], Cor. 10.7 R 2 Em() is negligible i�8K �� 8� 2 Nn0 8r 2 N 9m 2 N 8� 2 A2m;w(); � : D ! A0(Rn))9C >0 9� > 0 8" (0 < " < �) 8x 2 K: ("; x) 2 D andj@�(R(TxS"�("; x); x))j � C"rand if � 2 A2m;w() then (4) de�nes an element of A2[m+12 ];w(~). These factsdirectly imply di�eomorphism invariance of local Colombeau algebras ([16],Thms. 7.15 and 7.16).The di�eomorphism invariance of the scaled, transformed \test objects" demon-strated above suggests to choose an analogue of these as the main \test ob-jects" on the manifold, where no natural scaling resp. translation operatoris available. It is precisely this notion which is captured in the de�nition ofthe smoothing kernels below.3 Smoothing kernels and basic function spacesIn this section we introduce the basic de�nitions and operations needed foran intrinsic de�nition of algebras of generalized functions on manifolds.3.1 De�nition A0(M) := f! 2 nc (M) : Z ! = 1gThe basic space for the forthcoming de�nition of the Colombeau algebra onM is de�ned as follows:3.2 De�nition E(M) = C1(A0(M)�M)7

Then ( �� � �1� )(A0( �(U�))� �(U�)) � A0(M)�Mand locally we have E( �(U�)) �= E( �(U�)), the isomorphism being e�ectedby E( �(U�)) 3 R 7! [('; x) 7! R(' dny; x)]. In what follows, we willtherefore use E( �(U�)) and E( �(U�)) interchangeably. Clearly the map �� � �1� : nc ( �(U�))� �(U�)! nc (U�)� U� ,! nc (M)�Mis smooth. Therefore, for any R 2 E(M) its local representation( �1� )^R := R � ( �� � �1� )is an element of E( �(U�)). More generally, if � : M1 ! M2 is a di�eomor-phism and R 2 E(M2) then its pullback �(R) 2 E(M1) under � is de�nedas R � �� where ��(!; p) = ((��1)�!; �(p)) and clearly (�1 � �2) = �2 � �1 fordi�eomorphisms �1, �2.Let f : M �M ! Vn T �M be smooth such that for each (p; q) 2 M �M ,f(p; q) belongs to the �ber over q or, equivalently, that for each �xed p 2M ,fp := (q 7! f(p; q)) represents a member of n(M). Obviously, p 7! fp 2C1(M;n(M)) in this case. Given a smooth vector �eld X on M , we de�netwo notions of Lie derivatives of f with respect to X which, essentially, ariseas Lie derivatives of q 7! f(p; q) resp. p 7! f(p; q):On the one hand, viewing n(M) together with the topology of pointwise(i.e., �berwise) convergence as a locally convex space we de�ne(L0Xf)(p; q) := LX(p 7! f(p; q)) = ddt����0 (f(FlXt )(p); q); (7)on the other hand, we set(LXf)(p; q) := LX(q 7! f(p; q)) = ddt����0 ((FlXt )�fp)(q) (8)where the latter symbol LX denotes the usual Lie derivative on the bundlen(M).Now we are ready to introduce the space of smoothing kernels which willserve as an analogue for the (unbounded) sequences ('")" used in [7].3.3 De�nition � 2 C1(I�M; A0(M)) � C1(I�M�M;�nT �M) is calleda smoothing kernel if it satis�es the following conditions(i) 8K ��M 9 "0, C > 0 8p 2 K 8" � "0: supp�("; p) � B"C(p)8

(ii) 8K ��M 8k; l 2 N0 8X1; : : : ;Xk; Y1; : : : ; Yl 2 X(M)supp2Kq2M kLY1 : : : LYl(L0X1 + LX1) : : : (L0Xk + LXk)�("; p)(q)k = O("�(n+l))The space of smoothing kernels on M is denoted by ~A0(M) .In (i) the radius of the ball B"C(p) has to be measured with respect to theRiemannian distance induced by a Riemannian metric h on M . By Lemma3.4 below, (i) then holds in fact for the distance induced by any Riemannianmetric h0 with a new set of constants "0(h0) and C(h0). Similarly in (ii), k : kdenotes the norm induced on the �bers of nc (M) by any Riemannian metricon M (i.e. convergence with respect to k : k amounts to convergence of allcomponents in every local chart). Thus both (i) and (ii) are independent ofthe Riemannian metric chosen on M , hence intrinsic.3.4 Lemma Let M be a smooth paracompact manifold and let h1, h2 beRiemannian metrics on M. Then for all K �� M there exist "0(K) andC > 0 such that 8p 2 K 8" � "0:B(2)" (p) � B(1)C" (p)where B(i)" (p) = fq 2M : di(p; q) < "g and di denotes Riemannian distancewith respect to hi.Proof. We �rst show that 8K ��M 9C � 0 such thath1(p)(v; v) � Ch2(p)(v; v) 8p 2 K 8v 2 TpM: (9)Without loss of generality we may assumeK �� U� for some chart ( �; U�).Denoting by h�i (i = 1; 2) the local representations of hi in this chart it followsthat f : (x; v)! h�1 (x)(v; v)h�2 (x)(v; v)is continuous (even smooth) on �(U�)�Rn n f0g. Thussupx2 �(K)v2Rn nf0g f(x; v) = supx2 �(K)v2@Beucl1 (0) f(x; v) <1with Beucl1 (0) the Euclidian ball of radius 1 around 0.Next we choose a geodesically convex (with respect to h2) relatively compactneighborhood Up of p 2 M (cf. e.g. [25], Prop. 5.7). Moreover, let C be the9

square root of the constant in (9) with K = Up. Given any q; q0 2 Up let �be the unique h2-geodesic in Up connecting q and q0. Then d1(q; q0) � L1(�)�CL2(�) = Cd2(q; q0) where Li denotes the length of � with respect to hi.Now for each p 2 K there exist Up and Cp as above and we choose "psuch that B(2)"p (p) � Up. Then there exist some p1; : : : ; pm in K with K �Smi=1B(2)"pi2 (pi) =: U and we set "0 := min(dist2(K;@U); "p12 ; : : : ; "pm2 ) andC := max1�i�m Cpi.Let p 2 K, " � "0, q 2 B(2)" (p). There exists some i with d2(p; pi) � "pi2and by construction d2(p; q) � "pi2 , so p; q 2 B(2)"pi (pi) � Upi. Hence d1(p; q) �Cpid2(p; q) and, �nally, q 2 B(1)C" (p). 2The next step is to introduce the following grading on the space of smoothingkernels.3.5 De�nition For each m 2 N we denote by ~Am(M) the set of all � 2~A0(M) such that 8f 2 C1(M) and 8K ��Msupp2K jf(p) � ZM f(q)�("; p)(q)j = O("m+1)3.6 Remark. 3.5 is modelled with a view to reproducing the main tech-nical ingredient for proving �jC1 = � (i.e. the fact that the embedding ofdistributions into G coincides with the \identical" embedding on C1, cf.e.g., [16], Th. 7.4 (iii)) in the local theory. Essentially, the argument in thelocal case consists in (substitution of y0 = y�x" and) Taylor expansion ofR (f(x)� f(y))"�n'(y�x" )dy (' 2 A0) yielding appropriate powers of " as toestablish this term to be negligible. In fact, this argument is at the veryheart of Colombeau's construction and may be viewed as the main technicalmotivation for the concrete form of the sets Aq and, a fortiori, of the idealN . In 3.5 we turn the tables and de�ne the sets ~Am(M) by the analogousestimate. Moreover, as we shall see shortly (cf. 4.2), locally there is a one-to-one correspondence between elements of ~Am(M) and elements of A�m, so thetwo approaches are in fact equivalent (although only one of them, namelythe one occurring in 3.5 admits an intrinsic formulation on M).We are now in a position to prove the nontriviality of the space of smoothingkernels as well as of the spaces ~Am(M):3.7 Lemma(i) ~A0(M) 6= ;. 10

(ii) ~Am(M) 6= ; (m 2 N).Proof. (i) Let (U�; �)�2A be an oriented atlas of M such that each U� isrelatively compact. Let f��j� 2 Ag be a subordinate partition of unity andpick ' 2 A0(Rn). For each �, choose �1� 2 D(U�) such that �1� � 1 in an openneighborhood W� �� U� of supp ��. Let �� := �� � �1� , �1� := �1� � �1� .Now set �0�("; x)(y) := "�n'(y � x" ) :There exists some "�0 = "�0 (supp��) in (0; 1] such that for each x 2 supp(��)and each 0 < " � "�0 we have supp �0�("; x) �� �(W�). Choose �� : R![0; 1] smooth, �� � 1 on (0; "�03 ] and �� � 0 on ( "�02 ; 1]. Finally, let ! 2 A0(M).Then we de�ne our prospective smoothing kernel by�("; p)(q) :=X� ��(p) ���(") �� ��0�("; �(p))( : )�1�( : )dny�+ (1 � ��("))!�(q)(10)By construction, � is smooth and R �("; p) = 1 for all " 2 (0; 1] and allp 2 M . For any given K �� M , set "K := min "�03 where � ranges overthe (�nitely many) � with K \ U� 6= 0; then the terms in (10) containing !vanish for p 2 K and " � "K.In order to prove 3.3 (i), let K �� M . Since for p 2 K and " � "K,supp�("; p) is contained in a �nite union of supports ofq 7! [ �� ��0�("; �(p))( : )�1�( : )dny�](q)it su�ces to consider only one term of the latter form. Clearly, there existC and (0 <) "0 (� "K) such that supp �0�("; x) � Beucl"C (x) for x 2 supp��and all " � "0. Extending the pullback under � of a suitable cut-o� of theEuclidian metric on �(U�) to a Riemannian metric onM , the result followsfrom 3.4.Concerning 3.3 (ii), since K = Smi=1K�i , K�i �� U�i, it su�ces to estimate� on each K�i � (0; "i] for some "i > 0. Thus we may assume that K � U�0for some �xed �0. Let L be a compact neighborhood of K in U�0 and let" � "L in what follows. Each of the terms in (10) for which supp �� doesnot intersect K vanishes in some open neighborhood of K and can thereforebe neglected. Each of the �nitely many remaining terms vanishes outsidesupp ��. Since K � (K n supp ��) [ (K \W �) it is su�cient to let p rangeover K \W � in the estimate. Now for p 2 L, �1� can be omitted from the11

corresponding term in (10). Thus, using local coordinates of (U�0; �0) wehave to estimate a �nite number of Lie derivatives of terms of the form(��1)��~y 7! "�n'� ~y � ~x" � dn~y�= y 7! "�n'���1(y)� ~x" � � detD��1(y) dny ; (11)each for ~x 2 �(K \W �), respectively, where � = �0 � �1� and �(~z) = z.Note that for ~x in a compact subset of �(U�0 \ U�) and su�ciently small "(say, " � "1(� "K)) the support of~y 7! "�n'� ~y � ~x" � dn~yis contained in �(U�0 \ U�), rendering (11) well-de�ned. Setting�("; x)(y) := '���1(x+ "y)� ��1(x)" � � detD��1(x+ "y) ; (12)the coe�cient of the right hand side of (11) can be written as TxS"�("; x).By [16], Th. 7.14 and the remark following it, for each compact set L thereexists "2 (� "1) such that f@�x�("; x)( : ) j 0 < " � "2; x 2 Lg is (de�ned and)bounded in D for each � 2 Nn0 . Of course (12) corresponds to (2) in [9] resp.to (42) in [19], the only di�erence being that since we use an oriented atlasand our test objects are forms the determinants are automatically positive.Since LX(� dny) = (LX�) dny + �LX(dny), in order to verify 3.3 (ii) we haveto estimate terms of the formLY1 : : :LYl0 (L0X1 + LX1) : : : (L0Xk0 + LXk0 )TxS"�("; x)(y) (13)where Xi = nXri=1 airi@ri Yj = nXsj=1 bjsj@sjare local representations in �i(U�i) of vector �elds on M and 0 � k0 � k,0 � l0 � l. Explicitly, (13) is given byl0Yj=10@ nXsj=1 bjsj(y) @@ysj1A k0Yi=1 nXri=1�airi(x) @@xri + airi(y) @@yri�! TxS"�("; x)(y):12

Note that @@ysj TxS"� = "�1TxS" @@ysj � and�airi(x) @@xri + airi(y) @@yri�TxS"� =TxS"�airi(x) @@xri + 1"(airi(x+ "y)� airi(x)) @@yri�� :Each of the maps("; x; y)! 8<:airi(x+ "y)� airi(x)" for " 6= 0;Dairi(x)y for " = 0 (14)is smooth (hence uniformly bounded) on each relatively compact subset ofits domain of de�nition. Due to the boundedness of f@�x�("; x)(:) j 0 < " �"2; x 2 �0(K \ W �)g only values of y from a bounded region of Rn arerelevant in (14). Thus one further restriction of the range of " establishes theclaim.(ii) Choose the �0� as in (i), yet this time additionally requiring ' 2 Am(Rn).To estimate supp2K j RM �("; p)(q)f(q) � f(p)j for some K �� M and f 2C1(M) again only �nitely many terms of (10) have to be taken into account;also, for small ", the terms involving ! vanish and �1� can be neglected. Thensupp2K\supp�� jZM ��(�0�("; �(p))(:) dny)(q)f(q)� f(p)j == supx2 �(K\supp ��) jZ �(U�) 1"n'(y � x" )f(y) dny � f(x)j = O("m+1) 2Next, we introduce the appropriate notion of Lie derivative for elements ofE(M).3.8 De�nition For any R 2 E(M) and any X 2 X(M) we set(LXR)(!; p) := �d1R(!; p)(LX!) + LX(R(!; : ))jp (15)Here, d1R(!; x) denotes the derivative of ! ! R(!; x) in the sense of [16],section 4. In order to obtain a structural description of this Lie derivative(which, at the same time, entails LXR 2 E(M) for R 2 E(M)), for anyX 2 X(M) de�ne XA 2 X(A0(M)) byXA : A0(M) ! A00(M)XA(!) = �LX!13

(where A00(M) is the linear subspace of nc (M) parallel to A0(M), i.e.A00(M) = f! 2 nc (M)j R ! = 0g). ThenXAR( : ; p)(!) = d1R(!; p)(�LX!),so that (LXR)(!; p) = (L(XA;X)R)(!; p) = (L(�LX : ;X)R)(!; p)i.e. LX is the Lie derivative of R with respect to the smooth vector �eld(XA;X) on A0(M) �M . Thus, indeed LXR 2 E(M).3.9 Remark. (Local description of LX) Let X 2 X(M) and set X� =( �1� )�(XjU�). Since the derivative of the (linear and continuous) map �� :nc ( �(U�)) ! nc (U�) in any point equals the map itself, on U� we obtain(writing R in place of RjU�):(( �1� )^(LXR))('dny; x) = (LXR)( ��('dny); �1� (x))= LX(R( ��('dny); : ))( �1� (x))�(d1R)( ��('dny); �1� (x))(LX( ��('dny))| {z } ��(LX�('dny)) )= [LX�(( �1� )^R))]('dny; x)� [d1(( �1� )^R)('dny; x)](LX�('dny))In particular, if X� = @yi (1 � i � n) then this exactly reproduces the localalgebra derivative with respect to yi given in (3).Finally we are in a position to de�ne the subspaces of moderate and negligibleelements of E(M).3.10 De�nition R 2 E(M) is moderate if 8K �� M 8k 2 N0 9N 2 N8 X1; : : : ;Xk 2 X(M) 8 � 2 ~A0(M)supp2K jLX1 : : : LXk(R(�("; p); p))j = O("�N ) (16)The subset of moderate elements of E(M) is denoted by Em(M) .3.11 De�nition R 2 E(M) is called negligible if it satis�es8K ��M 8k; l 2 N0 9m 2 N 8 X1; : : : ;Xk 2 X(M) 8� 2 ~Am(M)supp2K jLX1 : : : LXk(R(�("; p); p))j = O("l) (17)The set of negligible elements of E(M) will be denoted by N (M) .14

4 Construction of the algebra, localizationThe following result is immediate from the de�nitions:4.1 Theorem(i) Em(M) is a subalgebra of E(M) .(ii) N (M) is an ideal in Em(M) . 2For the further development of the theory it is essential to achieve descrip-tions of both moderateness and negligibility that relate these concepts totheir local analogues [19, 16], thereby making available the host of local re-sults already established for open subsets of Rn also in the global context.As a �rst step, we are going to examine localization properties of smoothingkernels:4.2 Lemma Denote by (U�; �) a chart in M .(A) Transforming smoothing kernels to local test objects.(i) Let � be a smoothing kernel. Then the map � de�ned by�("; x)(y)dny := "n (( �1� )��("; �1� x))("y + x) (x 2 �(U�); y 2 Rn)(18)is an element of C1b;w(I � �(U�);A0(Rn)).(ii) If, in addition, � 2 ~Am(M) for some m 2 N then � 2 A�m;w( �(U�)),i.e., Z �("; x)(y)y�dy = O("m+1�j�j) (1 � j�j � m) (19)uniformly on compact sets. In particular, if � 2 ~A2m�1(M) then � 2A2m;w( �(U�)).(B) Transporting local test objects onto the manifold.(i) Let � 2 C1b (I � ;A0(Rn)) and �1 2 ~A0(M). Let K �� �(U�),�;�1 2 D( �(U�)) with � � 1 on an open neighborhood of K and�1 � 1 on an open neighborhood of supp�.�("; p) := (1 � �(p)�("))�1("; p)+�(p)�(") ��� 1"n�("; �p)(y � �p" )�1(y)dny� (20)is a smoothing kernel (the smooth cut-o� function � is de�ned in theproof). 15

(ii) If, in addition, � 2 A�m( �(U�)) and �1 2 ~Am(M) then � 2 ~Am(M).In particular, if � 2 A2m( �(U�)) and �1 2 ~Am(M) then � 2 ~Am(M).Proof.(A) (i) Let D1 := f("; p) 2 I � U� : supp�("; p) � U�g and set D =int((id� �)(D1)). Then evidently � is smooth on D. Furthermore (5)is obvious from (i) in 3.3. Concerning (6), set�0("; x)dny := ( �1� )�(�("; �1� (x))) (("; x) 2 D1)Then we have@�y @ x� = @�y @ xS�1" T�1x �0 = "j�jS�1" T�1x @�y (@x + @y) �0: (21)Let now K �� �(U�), x 2 K and " � "(K). By De�nition 3.3 (i) andLemma 3.4 there exists a constant C such that diam(supp �0("; x)) �"C for all x 2 K. But then diam(supp �("; x)) � C for all suchx. Hence S Sx2K supp @ x�("; x) is bounded uniformly in ". More-over, observing that L�@y(L@x + L@y) (�0("; x)(y)dny) = (L�@y(L@x +L@y) �0("; x)(y))dny, fromDe�nition 3.3 (ii) we obtain @�y (@x+@y) �0 =O("�(j�j+n)) which together with equation (21) gives the desired bound-edness property.(ii) Now suppose that � 2 ~Am(M) and for each j�j � m choose f� 2D(U�) � D(M) such that f� � �1� (x) = x� in a neighborhood ofK �� �(U�). Then by assumption we havesupx2K jZM f�(q)�("; �1� (x))(q)� f�( �1� (x))j = O("m+1)For " su�ciently small this impliessupx2K jZRn "�n�("; x)(y � x" )y�dy � x�j = O("m+1)or, upon substituting z = "�1(y � x) and expanding:supx2K ����� X0< ���� � "j jx�� ZRn �("; x)(z)z dz����� = O("m+1)From this, we prove (19) by induction with respect to j�j: for � = ek(the k-th unit vector) we obtain j RRn �("; x)(z)zkdzj = O("m). Sup-posing now that (19) has already been proved for j�j � 1, the result for16

j�j follows fromsupx2K j X0< <��� � "j jx�� ZRn �("; x)(z)z dz| {z }O("m+1�j j) +"j�j ZRn �("; x)(z)z�dzj= O("m+1)(B) (i) First note that by the boundedness assumption all supports of �(I�supp(�)) are in a �xed compact subset of Rn. By [16], Lemma 6.2 weconclude that there exists � > 0 such that for all " � � and x 2supp� supp � 1"n�("; x)( :�x" )� � f�1 � 1g. Now we are in the positionto de�ne � as follows. Let � 2 C1(R), 0 � � � 1 and � � 1 on(�1; �=3) and � � 0 on (�=2;1). This actually implies R �("; p) = 18p; ". Smoothness again is evident while condition 3.3 (i) is an easyconsequence of [16], Lemma 6.2. 3.3 (ii) then follows exactly as (13) isproved in Lemma 3.7.(ii) Let f 2 C1(M) and p 2 K �� U�; thenjZM �("; p)(q)f(q)� f (p)j �j1 � �(p)�(")j jZM �1("; p)(q)f(q)� f (p)j+j�(p)�(")j jZM ��("�n�("; �(p))(y � �(p)" )�1(y)dny)(q)f(q)�f(p)jThe �rst summand is of order "m+1 by assumption and the second one(apart from � and �) for su�ciently small " and setting f = f � �1�equalsZRn �("; x)(y)f(x+ "y)dny � f(x) = X0<j�j�m @�f(x)�! "j�j Z �("; x)(y)y�dy| {z }O("m+1�j�j )so the claim follows. 2Restriction of an element R of E(M) to an open subset U of M is de�ned byRjU := RjA0(U)�U.4.3 Theorem (Localization of moderateness) Let R 2 E(M). ThenR 2 Em(M), ( �1� )^(RjU�) 2 Em( �(U�)) 8�:17

Proof. ()) For R 2 Em(M) and for (U�; �) some chart in M let R0 :=( �1� )^(RjU�). Let K �� �(U�) =: V�, � 2 Nn0 and let � 2 C1b (I � �(U�);A0(Rn)). We have to show thatsupx2K j@�(R0(TxS"�("; x); x))j = O("�N ) (22)for someN 2 N. To this end we �x some �1 2 ~A0(M) and de�ne � 2 ~A0(M)by (20). Let @� = @i1 : : : @ij�j and for 1 � ij � n choose Xij 2 X(M) suchthat the local expression of Xij coincides with @ij on a neighborhood of K.Then for " su�ciently small (22) equalssupp2K jLXi1 : : : LXij�jR(�("; p); p)j ;so we are done.(() Let X1; : : : ;Xk 2 X(M) and (without loss of generality) K �� U� forsome chart (U�; �). Let � 2 ~A0(M) and de�ne � by (18). Since � : D !A0(Rn) belongs to C1b;w(I� �(U�);A0(Rn)) it follows from [16], Th. 10.5 thatgiven � 2 Nn0 there exists some N 2 N and some "0 > 0 such that for x 2 Kand " � "0 we have ("; x) 2 D and j@�( �1� )^(RjU�)(TxS"�("; x); x))j =O("�N ). Inserting this into the local representation of (16) immediatelygives the result. 24.4 Theorem (Localization of negligibility) Let R 2 Em(M). ThenR 2 N (M), ( �1� )^(RjU�) 2 N ( �(U�)) 8�:Proof. ()) Let R 2 Em(M), (U�; �) some chart in M and set R0 :=( �1� )^(RjU�). Let K �� �(U�) =: V�, l 2 N and � 2 Nn0 . Set k = j�j andchoose m 2 N0 such thatsupp2K jLX1 : : : LXk(R(�("; x); x))j = O("l) (23)for all X1; : : : ;Xk 2 X(M) and all � 2 ~Am(M). Now let � 2 A2m(V�) andconstruct � from � according to (20). By 4.2, � 2 ~Am(M), so (with Xij asin the proof of 4.3)supx2K j@�(R0(TxS"�("; x); x))j = supp2K jLXi1 : : :LXij�j (R(�("; x); x))j = O("l)Thus the claim follows from the characterization of negligibility following thede�nition of N () (section 2). 18

(() Let k 2 N0, l 2 N, X1; : : : ;Xk 2 X(M) and K �� U�. By the discussionat the end of section 2 there exists m0 such that (with R0 = ( �1� )^RjU�)supx2K j@�(R0(TxS"�("; x); x))j = O("l) (24)for all j�j � k and all � 2 A2m0;w(V�). Now set m = 2m0 � 1 and let� 2 ~Am(M). Then � de�ned by (18) is in A2m0 ;w(V�) by 4.2 (A) (ii). Henceinserting local representations of X1; : : : ;Xk, (24) immediately implies thevalidity of (23), thereby �nishing the proof. 2It was shown in [17], sec. 13 that for all variants of (local) Colombeau algebrasmembership of any element R of Em to the ideal N can be tested on thefunction R itself, without taking into account any derivatives of R. As a �rstimportant consequence of the above localization results we note that thisrather surprising simpli�cation also holds true for the global theory:4.5 Corollary Let R 2 Em(M). ThenR 2 N (M), (17) holds for k = 0:Proof. This follows directly from Th. 4.4 by taking into account [16], Th.7.13 and [17], Th. 13.1. 2Moreover, stability of Em(M) and N (M) under Lie derivatives also followsfrom the local description:4.6 Theorem Let X 2 X(M). Then(i) LX Em(M) � Em(M).(ii) LXN (M) � N (M).Proof. Let R 2 Em(M), X 2 X(M). By Th. 4.3 for any chart (U�; �) wehave ( �1� )^(RjU�) 2 Em( �(U�)). Thus by [16], Th. 7.10 also LX�( �1� )^(RjU�) = ( �1� )^(LXRjU�) 2 Em( �(U�)) (where X� denotes the local rep-resentation of X), which, again by Th. 4.3 gives the result. The claim forN (M) follows analogously from [16], Th. 7.11. 2Finally we are in a position to de�ne our main object of interest:4.7 De�nition G(M) := Em(M)=N (M)is called the Colombeau algebra on M .19

By construction, every LX induces a Lie derivative (again denoted by LX )on G(M) , so G(M) becomes a di�erential algebra. If R 2 E(M), its class inG(M) will be denoted by cl[R].4.8 Theorem G(M) is a �ne sheaf of di�erential algebras on M .Proof. This is a straightforward consequence of [16], Th. 8.1. 25 Embedding of distributions and smoothfunctionsIn this section we show that in the global context G(M) displays the sameset of (optimal) embedding properties as the local versions do on open sets ofRn. Most importantly, we shall see that taking Lie derivatives with respectto arbitrary smooth vector �elds commutes with the embedding.To begin with, let u 2 D0(M). The natural candidate for the image of u inG(M) is Ru(!; x) = hu; !i. We �rst show that Ru 2 Em(M) using Th. 4.3.Let ! 2 D( �(U�)); then(( �1� )^(RujU�))(!; x) = (RujU�)( ��(! dny); �1� (x)) =hu; ��(! dny)i = h( �1� )�(ujU�); !iSince ( �1� )�(ujU�) 2 D0( �(U�)) it follows from the local theory that indeed( �1� )^(RujU�) 2 Em( �(U�)). Suppose now that Ru 2 N (M). By the samereasoning as above (!; x)! h( �1� )�(ujU�); !i 2 N ( �(U�)) for each �. Thusagain by the respective local result ( �1� )�(ujU�) = 0 for each �, i.e. u = 0.Therefore � : D0(M)! G(M)�(u) = cl[(!; x)! hu; !i]is a linear embedding. What is more, as a direct consequence of (15) (notingthat distributions are linear and continuous, hence equal to their di�erentialin any point) we obtain�(LXu)(!; p) = �((!; p)! �hu;LX!i) = �d1Ru(!; p)(LX!)+LX(Ru(!; : ))jp| {z }=0 = (LXRu)(!; p) = LX(�(u))(!; p)i.e., � commutes with arbitrary Lie derivatives.20

The natural operation for embedding smooth functions into G(M) is givenby � : C1(M)! G(M)�(f) = cl[(!; x)! f(x)]Obviously, � is an injective algebra homomorphism that commutes with Liederivatives by (15). Moreover, � coincides with � on C1(M). Making useof Th. 4.4 this again follows directly from the local result. Summing up, wehave5.1 Theorem � : D0(M) ! G(M), is a linear embedding that commuteswith Lie derivatives and coincides with � : C1(M) ! G(M) on C1(M).Thus � renders D0(M) a linear subspace and C1(M) a faithful subalgebra ofG(M) .The following commutative diagram illustrates the compatibility propertiesof Lie derivatives with respect to embeddings established in this section:-? ?-JJJJJ �-� JJJJJC1 C1LXLXLXG G� �� �D0 D0

6 AssociationThe concept of association or coupled calculus is one of the distinguishingfeatures of local Colombeau algebras. It introduces a (linear) equivalencerelation on the algebra identifying those elements which are \equal in thesense of distributions", thereby allowing to identify \distributional shadows"of certain elements of the algebra. This construction amounts to determiningthe macroscopic aspect of the regularization procedure encoded in them.Phrased more technically, a linear quotient of Em resp. G is formed containing21

D0 as a subspace. Especially in physical modelling this notion provides auseful tool for analyzing nonlinear problems involving singularities (cf. e.g.[5], [21], [27], [28]). In what follows we extend the notion of association toG(M).6.1 De�nition An element [R] of G(M) is called associated to 0 ([R] � 0)if for some (hence every) representative R of [R] we have: 8! 2 nc (M)9m > 0 with lim"!0ZM R(�("; p); p)!(p) = 0 8� 2 ~Am(M) (25)Two elements [R], [S] of G(M) are called associated ([R] � [S]) if [R�S] � 0.We say that [R] 2 G(M) admits u 2 D0(M) as an associated distribution if[R] � �(u), i.e. if 8! 2 nc (M) 9m > 0 withlim"!0ZM R(�("; p); p)!(p) = hu; !i 8� 2 ~Am(M) (26)Finally, by the same methods as in the local theory we obtain consistency inthe sense of association of classical multiplication operations with multipli-cation in the algebra:6.2 Proposition(i) If f 2 C1(M) and u 2 D0(M) then�(f)�(u) � �(fu) (27)(ii) If f; g 2 C(M) then �(f)�(g) � �(fg) (28)References[1] J. Aragona, H. A. Biagioni, Intrinsic de�nition of the Colombeau algebraof generalized functions, Analysis Mathematica 17 (1991), 75 - 132.[2] H. Balasin, Geodesics for impulsive gravitational waves and the multi-plication of distributions, Class. Quantum Grav. 14 (1997), 455{462.[3] H. A. Biagioni, M. Oberguggenberger, Generalized solutions to theKorteweg-de Vries and the regularized long-wave equations, SIAM J.Math. Anal. 23 (1992), 923{940.22

[4] H. A. Biagioni, M. Oberguggenberger, Generalized solutions to Burgers'equation, J. Di�erential Equations 97 (1992), 263{287.[5] C. J. S. Clarke, J. A. Vickers, J. Wilson, Generalised functions anddistributional curvature of cosmic strings, Class. Quantum Grav. 13(1996), 2485{2498.[6] J. F. Colombeau, New Generalized Functions and Multiplication of Dis-tributions, North Holland, Amsterdam 1984.[7] J. F. Colombeau, Elementary Introduction to New Generalized Func-tions, North Holland, Amsterdam 1985.[8] J. F. Colombeau, Multiplication of Distributions, Bull. Am. Math. Soc.,New Ser. 23, No. 2 (1990), 251-268.[9] J. F. Colombeau, A. Meril, Generalized Functions and Multiplicationof Distributions on C1 Manifolds, J. Math. Analysis Appl. 186 (1994),357{364.[10] J. F. Colombeau, A. Heibig, M. Oberguggenberger, Le probleme deCauchy dans un espace de fonctions generalisees I, C. R. Acad. Sci.Paris S�er. I Math. 317 (1993), 851{855.[11] J. F. Colombeau, A. Heibig, M. Oberguggenberger, Le probleme deCauchy dans un espace de fonctions generalisees II, C. R. Acad. Sci.Paris S�er. I Math. 319 (1994), 1179{1183.[12] J. F. Colombeau, M. Oberguggenberger, On a hyperbolic system witha compatible quadratic term: Generalized solutions, delta waves, andmultiplication of distributions, Comm. Partial Di�erential Equations 15(1990), 905{938.[13] M. Damsma, J. W. de Roever, Colombeau algebras on a C1-manifold,Indag. Mathem., N.S. (3) (1991), 341-358.[14] G. de Rham, Di�erentiable Manifolds, Springer, Berlin 1984.[15] J. Dieudonn�e, �El�ements d'Analyse, Vol 3, Gauthier-Villars, Paris, 1974.[16] E. Farkas, M. Grosser, M. Kunzinger, R. Steinbauer,On the Foundationsof Nonlinear Generalized Functions I, Preprint, 1999.[17] M. Grosser, On the Foundations of Nonlinear Generalized Functions II,Preprint, 1999. 23

[18] L. H�ormander, The Analysis of Linear Partial Di�erential Operators I,Grundlehren der mathematischenWissenschaften 256, Springer, Berlin,1990.[19] J. Jel��nek, An intrinsic de�nition of the Colombeau generalized func-tions, Comment. Math. Univ. Carolinae 40, 1 (1999), 71{95.[20] A. Kriegl, P. W. Michor, The Convenient Setting of Global Analysis,AMS Math. Surv. and Monographs 53, 1997.[21] M. Kunzinger, R. Steinbauer, A rigorous solution concept for geodesicequations in impulsive gravitational waves, J. Math. Phys. Vol. 40, No.3 (1999), 1479{1489.[22] J. E. Marsden, Generalized Hamiltonian Mechanics, Arch. Rat. Mech.Anal. 28 (1968), 323{361.[23] M. Oberguggenberger, Multiplication of Distributions and Applicationsto Partial Di�erential Equations, Pitman Research Notes in Mathemat-ics 259, Longman 1992.[24] M. Oberguggenberger, Nonlinear theories of generalized functions, in: S.Albeverio, W.A.J.Luxemburg, M.P.H.Wol� (Eds.), Advances in Analy-sis, Probability, and Mathematical Physics-Contributions from Nonstan-dard Analysis, Kluwer, Dordrecht 1994.[25] B. O'Neill, Semi-Riemannian Manifolds, Academic Press, 1983.[26] L. Schwartz, Sur L'impossibilite de la Multiplication des Distributions,C.R. Acad.Sci. Paris, 239 (1954), 847-848.[27] J. A. Vickers, J. Wilson, Invariance of the distributional curvature of thecone under smooth di�eomorphisms, Class. Quantum Grav. 16 (1999),579-588.[28] J. A. Vickers, Nonlinear generalised functions in general relativity,in: M. Grosser, G. H�ormann, M. Kunzinger, M. Oberguggenberger(Eds.), Nonlinear Theory of Generalized Functions, 275-290, Chapman& Hall/CRC, Boca Raton 1999.Electronic Mail:M.G.: michael@mat.univie.ac.atM.K.: Michael.Kunzinger@univie.ac.atR.S.: Roland.Steinbauer@univie.ac.atJ.V.: jav@maths.soton.ac.uk 24