The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria On the Foundations of Nonlinear Generalized Functions I E. Farkas M. Grosser M. Kunzinger R. Steinbauer Vienna, Preprint ESI 811 (1999) December 27, 1999 Supported by Federal Ministry of Science and Transport, Austria Available via

2 On the foundations of nonlinear generalized functions I E. Farkas, M. Grosser, M. Kunzinger and R. Steinbauer Universitat Wien Institut fur Mathematik Abstract. We construct a dieomorphism invariant (Colombeau-type) dierential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing dierential calculus in innite dimensional (convenient) vector spaces, previous attempts in this direction are unied and completed. Several classication results are achieved and applications to nonlinear dierential equations involving singularities are given Mathematics Subject Classication. Primary 46F30; Secondary 26E15, 46E50, 35D05. Key words and phrases. Algebras of generalized functions, Colombeau algebras, calculus on innite dimensional spaces, convenient vector spaces, dieomorphism invariance. 1 Introduction In his celebrated impossibility result ([40]), L. Schwartz demonstrated that the space D 0 () of distributions over some open subset of R n cannot be embedded into an associative commutative algebra (A(); +; ) satisfying (i) D 0 () is linearly embedded into A() and f(x) 1 is the unity in A(). (ii) There exist derivation i linear and satisfy the Leibnitz rule. : A()! A() (i = 1; : : : ; n) that are i j D 0 () is the usual partial derivative (i = 1; : : : ; n). (iv) j C()C() coincides with the pointwise product of functions. Since this result remains valid upon replacing C() by C k () for any nite k, the best possible result would consist in constructing an embedding of D 0 () as above with (iv) replaced by (iv') j C 1 ()C 1 () coincides with the pointwise product of functions. The actual construction of dierential algebras satisfying these optimal properties is due to J. F. Colombeau ([9], [10], [11], [12]). The need for algebras of this type 1

3 arises, for example, from the necessity of considering non-linear PDEs where either the respective coecients, the data or the prospective solutions are non-smooth. Classical linear distribution theory clearly does not permit the treatment of such problems. Colombeau algebras, on the other hand, have proven to be a useful tool for analyzing such questions (for applications in nonlinear PDEs, cf. e.g., [4], [5], [16], [14], [15], [29], [35], for applications to numerics, see e.g., [3], [6], [7], for applications in mathematical physics, e.g., [44], [32], [24] as well as the literature cited in these works). For alternative approaches to algebras of generalized functions, cf. [37], [38]. Since Colombeau's monograph [9], there have been introduced a considerable number of variants of Colombeau algebras, many of them adapted to special purposes. From the beginning, however, the question of the functor property of the construction was at hand as a crucial one: If : ~! denotes a dieomorphism between open subsets ~ ; of R s, is it possible to extend the operation : f 7! f on smooth distributions on to an operation ^ on the Colombeau algebra such that ( )^ = ^ ^ and (id)^ = id are satised? To phrase it dierently, is it possible to achieve a dieomorphism invariant construction of Colombeau algebras? As long as this question could not be answered in the positive, there remained the serious objection that there is no way of dening such algebras on manifolds, based on intrinsic terms, exclusively. (This discussion does not take into account the so called \special or \simplied variant of Colombeau's algebra whose elements are classes of nets of smooth functions indexed by > 0 (cf. [35], p. 109). Although dieomorphism-invariant, these algebras lack a canonical embedding of distributions ([17], [41]), so we do not consider them here.) The rst variants of Colombeau algebras, though serving as a valuable tool in the treatment of non-linear problems, indeed did not have the property of dieomorphism invariance: Some of the key ingredients used in dening them (in particular, the \test objects (see section 3) being employed as well as the denition of the subsets A q (R s ) of the set of all test objects) turned out not to be invariant under the natural action of a dieomorphism. Colombeau and Meril in their paper [13] made the rst decisive steps to remove this aw by proposing a construction of Colombeau algebras which they claimed to be dieomorphism invariant. As an essential tool, they had to use calculus on locally convex spaces. However, they did not give the details of the application of that calculus; moreover, their denition of the objects constituting the Colombeau algebra was not unambiguous and, which amounts to the most serious objection, their notion of test objects still was not preserved under the action of the dieomorphism. Nevertheless, despite these defects (which, apparently, went unnoticed by nearly all workers in the eld) their construction was quoted and used many times (see, e.g., 2

4 [31], [18], [1], [45], [42], [34], [43], [19], [44], [2], [32]). It was only in 1998 that J. Jelnek in [28] pointed out the error in [13] by giving a (rather simple) counterexample. In the same paper, he presented another version of the theory avoiding the shortcomings of [13] and forming the basis for the approach taken here. The present article is the rst in a series of two papers. It is organized as follows: After xing notation and terminology in section 2, a general scheme of construction for dieomorphism-invariant Colombeau-type algebras of generalized functions is introduced in section 3. Section 4 gives a quick overview of calculus in convenient vector spaces providing the necessary results for the development of the theory. Especially with a view to applications (in particular: partial dierential equations) we feel that this approach has several advantages over the concept of Silva-dierentiability employed so far. Section 5 introduces a translation formalism that allows to freely switch between what we call the C- and J- (Colombeauand Jelinek-) formalism of the dieomorphism invariant theory to be constructed in section 7. In the actual construction of this algebra, smooth functions dened on sets denoted by U () play a central r^ole. Dierentials of such functions are of utmost importance in the development of the theory. However, U () is not a linear space. Sections 6 thus provides the framework necessary for doing calculus on U (). A complete presentation of the resulting dieomorphism invariant algebra, based on the general construction scheme of section 3, is the focus of section 7. The sheaf-theoretic properties of this algebra are discussed in section 8. This is followed by a short section on the separation of testing procedures and denition of objects in algebras of generalized functions. Section 10 provides several new characterizations of the fundamental building blocks E M and N of the algebra. In particular, these characterizations will constitute the key ingredient in obtaining an intrinsic description of the theory on manifolds ([26]). Finally, we present some applications to partial dierential equations in section 11. The second paper of this series gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the dieomorphism-invariant quotient algebra G d = E M N introduced in section 7 and (the smooth version of) Colombeau's original algebra G e introduced in [10] (which, to be sure, is the standard version among those being independent of the choice of a particular approximation of the delta distribution). Three main results are established: First, a simple criterion describing membership in N (applicable to all types of Colombeau algebras) is given (section 13). Second, two counterexamples demonstrate that G d is not injectively included in G e (section 15); their construction is based on a completeness theorem for spaces of smooth functions in the sense of sections 4 and 6 (section 14). Finally, it is shown that in the range \between G d and G e only one more construction leads to a dieomorphism invariant algebra. In analyzing the latter, several classication results essential for obtaining an intrinsic description of G d on 3

5 manifolds are derived (sections 16, 17). The concluding section 18 points out that also weaker invariance properties than with respect to all dieomorphisms should be envisaged for Colombeau algebras, in particular regarding applications. 2 Notation and Terminology Throughout this paper,, ~ will denote non-empty open subsets of R s. For any A R s, A denotes its interior. C 1 () is the space of smooth, complex valued functions on. If f 2 C 1 () then Df denotes its (total) derivative. Also, we set f(x) = f(?x). On any cartesian product, pr i denotes the projection onto the i-th factor. For r 2 R, [r] is the largest integer r. We set I = (0; 1]. Concerning locally convex spaces our basic reference is [39]. In particular, by a locally convex space we mean a vector space endowed with a locally convex Hausdor topology. The space of test functions (i.e., compactly supported smooth functions) on is denoted by D() and is equipped with its natural (LF)-topology; its dual, the space of distributions on is termed D 0 (). The action of any u 2 D 0 () on a test function ' will be written as hu; 'i. denotes the Dirac delta distribution. K A (A R s ) means that K is a compact subset of A. For K, D K () is the space of smooth functions on supported in K. We set A 0 () = f' 2 D()j A q () = f' 2 A 0 ()j Z Z '() d = 1g '() d = 0; 1 jj q; 2 N s 0g (q 2 N) A q0 () is the linear subspace of D() parallel to the ane space A q () (q 2 N 0 ). For any maps f; g; h such that g f and f h are dened we set g (f) := g f and h (f) := f always stand i ( 2 N 0 i For any locally convex space F the space C 1 (; F ) of smooth functions from into F will always carry the topology of uniform convergence in all derivatives on compact subsets of. In particular, a subset B of this space will be said to be bounded if, for any K and any 2 N s, the set 0 f@ (x) j 2 B; x 2 Kg is bounded in F. Observe that in case that the image of a map 2 C 1 (; F ) is contained in some ane subspace F 0 of F then the derivatives of take their values in the linear subspace parallel to F 0. For locally convex spaces E, F the space C 1 (E; F ) (resp. C 1 (E) for F = C ) is introduced in section 4. In what follows, A 0 (R s ) may be replaced by any closed ane subspace of D(R s ). By C [1;] b (I ; A 0 (R s )) we denote the space of all maps : I! A 0 (R s ) which are smooth with respect to the second argument and bounded in the sense that the 4

6 corresponding map ^ : I! C 1 (; A 0 (R s )) has a bounded image as dened above, i.e., for every K and any 2 N s 0, the set f@ (^())(x) j 2 I; x 2 Kg is bounded in A 0 (R s ) resp. D(R s ), which, in turn, is equivalent to saying that C 1 b 1. for every K as above and any 2 N s 0, the supports of x (; x) ( 2 I, x 2 K) are contained in some xed bounded set (depending only on K) and 2. supfj@ (@ x (^())(x))()j 2 I; x 2 K; 2 R s g (or, expressed in terms of itself) x ((; x))()j 2 I; x 2 K; 2 R s g is nite. (I ; A 0(R s )) is the subspace of C [1;] b (I ; A 0 (R s )) whose elements are smooth in both arguments. Finally, for any K and any q 1 an element of C 1(I ; b D(Rs )) is said to have asymptotically vanishing moments of order q on K if Z sup j x2k (; x)() dj = O( q ) (1 jj q) : For this notion to make sense it is obviously sucient for to be dened on (0; 0 ]K for some 0 > 0. 3 Scheme of construction As was already pointed out in section 1, due to the lack of a canonical embedding of the space of distributions into \special variants of Colombeau algebras we shall not consider these. Instead, we focus on \full algebras (in the sense of [31], p. 31), distinguished by the fact that such a canonical embedding is always available. Elements of full Colombeau algebras are equivalence classes of functions R taking as arguments certain pairs ('; x) consisting of a suitable test function ' 2 D(R s ) and a point x of. Every (full) Colombeau algebra is constructed according to the following blueprint (where (Di), (Tj) stand for Denition i and Theorem j, respectively). (D5) and (T6){(T8) are only relevant if a dieomorphism invariant type of algebra is to be obtained. (D1) (D2) E() (the \basic space, see the remarks below); maps : C 1 ()! E(), : D 0 ()! E(). Derivations D i on E() (i = 1; : : : ; s) extending i of partial dierentiation on D 0 () resp. on C 1 (), i.e., D i and D i. 5

7 (D3) (D4) E M () ( E(); the subspace of \moderate functions). N () ( E(); the subspace of \negligible functions). (T1) (D 0 ()) E M (); (C 1 ()) E M (); (? )(C 1 ()) N (); (T2) (T3) (D 0 ()) \ N () = f0g. E M () is a subalgebra of E(). N () is an ideal in E M (). (T4) E M () is invariant under each D i. (T5) N () is invariant under each D i. (D5) (T6) For each dieomorphism : ~!, a map : D ~! D 1 is dened in a functorial way such that its \transpose ^ : E()! E(~), ^(R) := R, extends the usual eect has on distributions, i.e., ^ = where for u 2 D 0 (), u is dened by h u; 'i := hu; ('?1 ) j det D?1 ji. Similarly, we require ^ = on C 1 (). The class of \scaled test objects (see below) is invariant under the action induced by. (T7) E M is invariant under ^, i.e., ^ maps E M () into E M ( ~ ). (T8) (D6) N is invariant under ^, i.e., ^ maps N () into N (~). G() := E M () N (). For R 2 E M (), the class R + N () of R in G() will be denoted by [R]. The following comments are intended to motivate and clarify the preceding admittedly very formal denition schemes and theorems. ad (D1): Here, E() denotes some algebra of complex-valued functions having appropriate smoothness properties on a suitable domain D D(R s ). has to be an injective algebra homomorphism, whereas just has to be linear and injective. ad (D3), (D4): Membership of R 2 E() in N () respectively E M () depends on the \asymptotic behaviour of R on certain paths in D(R s ), where the second component is constant whereas the rst component, depending on as parameter, tends to the delta distribution weakly as! 0. Essentially, these paths are obtained by applying the scaling operator S : ' 7! 1 s '( : ) (thereby introducing the parameter ) to so-called test objects. Typically, a test object is some xed element ' 2 D(R s ) satisfying R ' = 1 or a suitable bounded family (; x) 2 D(), parametrized by 2 I, x 2, where again R (; x)() d 1. Roughly speaking, R is dened to be 1 Concerning D ~ ; D, see the remark on (D1) below. 6

8 negligible if the values R attains on those \scaled test objects tend to zero faster than any positive power of, while it is called moderate if these values are bounded by some xed (negative) power of. In both cases, convergence in each derivative, uniformly on compact subsets of, is required. We will refer to those dening procedures as testing for negligibility resp. moderateness (see also section 9). ad (T1), (T3): N () has to be large enough to contain all (f)? (f) (f 2 C 1 ()) (this renders C 1 () an algebra homomorphism by passing to a quotient by N ()), however small enough to intersect D 0 () just in f0g (this guarantees D 0 () to be contained injectively in the quotient by N ()). E M (), on the other hand, clearly has to be large enough to contain C 1 () and D 0 () (via resp. ), yet small enough such that N () is an ideal in it: This will allow us to form the quotient E M () N (). ad (D5): as dened above extends : f 7! f where the latter is viewed as the action induced by on the smooth distribution f 2 C 1 (). Hence we regard distributions (and, in the sequel, non-linear generalized functions) as generalizations of functions on the respective open set, acting as functionals on (smooth, compactly supported) densities. This is in agreement with, for example, [27], however has to be distinguished clearly from constructing distributions as distributional densities, acting on (smooth, compactly supported) functions, as it is done, e.g., in [20]. ad (T7), (T8): Because of the forms of (D3) and (D4) as tests to be performed on the elements R of E(), with the appropriate type of (scaled) test objects being inserted, (T7) as well as (T8) follow immediately from (T6), taking into account (D5). ad (D6): By this denition, G() is a dierential algebra containing D 0 () via followed by the canonical quotient map ((T1){(T5)); by abuse of notation, we will denote this embedding also by. Each dieomorphism : ~! induces a map ^ : G()! G( ~ ) extending the usual action of on distributions such that composition and identities are preserved by 7! ^ ((T7), (T8)). Without the requirement of dieomorphism invariance (as, for example, in [9]), the smoothness property of R mentioned above only needs to refer to the variable x in the pair ('; x), thus involving only classical calculus. However, as mentioned already in the introduction, to obtain a dieomorphism invariant algebra we also have to consider smoothness with respect to the test function '. Therefore, in the following section, we are going to outline the elements of calculus on locally convex spaces which are required for the subsequent constructions. The path we will pursue in this respect is dierent from the approaches taken so far and, in our view, has some decisive advantages over these. 7

9 4 Calculus In the rst versions of Colombeau algebras (on R n or open subsets thereof), the main ingredient was the algebra of smooth functions ' 7! R(') on the ((LF)-)space D of test functions (see [9]). Thus, from the very beginning, there had to be a theory of dierentiation on (certain non-banach) locally convex spaces at the basis of the construction of these algebras. Colombeau's approach in [9] employs the notion of Silva-dierentiability ([46], [8]) where a map f : E U! F from an open subset U of a locally convex space E into another locally convex space F is called Silva-dierentiable in x 2 U if there exists a bounded linear map (called f 0 (x)) E! F such that the restriction of the corresponding remainder function to suciently small homothetic images of bounded subsets may be viewed as a map between suitable normed spaces and satises a condition thereon which is completely analogous to the classical remainder condition for Frechet-dierentiable maps. In later versions, Colombeau managed to circumvent this necessity by introducing an additional variable x 2 R n into R which could carry the burden of smoothness: For the construction of the algebra G() of [10] he now used functions R('; x) which, for each xed ' from (a certain ane subspace of) D, are smooth in x (in the usual elementary sense hence the title of [10]) whereas the dependence on ' is completely arbitrary; ' just plays the r^ole of a parameter in this setting. Apart from simplifying the general setup of the theory the introduction of x as a separate variable was also crucial for solving dierential equations in G(). However, when Colombeau and Meril in [13] began to develop a dieomorphism invariant version of the algebra G() of [10], they had to reintroduce the smooth dependence of R on ': Under the action of a dieomorphism, ' changes to some ~' x depending on x. For the smoothness of the -transform of R (which, according to (D5), is of the form (^R)('; x) = R(('; x)) = R( ~' x ; x)) with respect to x, obviously the smooth dependence of R also on its rst argument ' is needed ([13], p. 263). Concerning calculus on locally convex spaces, the authors as the rst of them did already in [9] refer to [8]. Omitting any details in this respect, they rather invite the reader to admit the respective smoothness properties (p. 263). Jelnek in [28] includes a section on calculus (items 9{16): In addition to [8], he quotes [46] as reference for some results needed. The relevant statements are formulated in terms of higher Frechet dierentials. Contrary to the above, we prefer to base our presentation on the notion of smoothness as it is outlined in [30]. This approach seems to us to have a number of striking advantages: On the one hand, the basic denition is very simple and easy to work 8

10 with, a smooth map between locally convex spaces E; F being one that takes smooth curves R! E to smooth curves R! F (by composition); the notion of a smooth curve into a locally convex space obviously is without problems. We will denote by C 1 (E; F ) the space of smooth maps between E and F. For C 1 (E;C ), we will simply write C 1 (E). On the other hand, all the basic theorems of dierential calculus can be reconstructed in this setting (see, e.g., the version of the mean value theorem given in 4.5) and more than that (see, e.g., the exponential laws stated in 4.2 and 4.3 below and the dierentiable uniform boundedness principle 4.7). As smooth curves are continuous, the above denition of smoothness carries over to open subsets of locally convex spaces. We will make use of this in the sequel and want to note that in any of the theorems of this section, we may replace the respective locally convex (domain) spaces by open subsets thereof whenever their linear structure is not needed. This notion of smoothness is a weaker one than Silva-dierentiability but turns out to be equivalent for a huge class of spaces, e.g. those which are complete and Montel so that the two notions coincide in particular on the regular 2 strict inductive limit D() of Frechet spaces and each closed subspace thereof. The seeming drawback of this (and any other reasonable such as Colombeau's abovementioned) theory of dierentiation is the fact that smooth maps (resp. their dierentials) need no longer be continuous. The fundamental r^ole played by continuity in the classical context is taken over by the notion of boundedness: Indeed, the difference quotients of smooth curves converge in a stronger sense than the topological one, so that continuity is not a necessary property for a map to be smooth. In order to be able to test smoothness by composition with suitable families of linear functionals (see, e.g., 4.7) one needs, in addition, a completeness property which is weaker than completeness of the locally convex topology. Separated bornological locally convex spaces which have this property are called convenient spaces and are in some sense the most general class of linear spaces in which one can perform differentiation and integration. As for each locally convex space there exists a ner bornological locally convex topology with the same bornology, i.e., the same system of bounded sets, bornologicity of the topology is not essential. It will be enough for our purpose to conne ourselves to the particular case of complete locally convex spaces. In the sequel, we will endow the space C 1 (R; F) of smooth curves into the locally convex space F with the locally convex topology of uniform convergence on compact intervals in each derivative separately. More generally, we may consider on the space 2 A strict inductive limit lim?! E is called regular if each bounded subset is contained in some E. Note that every strict inductive limit of an increasing sequence E n is regular, as is D(M ) for any paracompact (not necessarily separable) smooth manifold M. 9

11 C 1 (E; F ) the initial locally convex topology induced by the pullbacks along smooth curves R! E. It can be shown that the bounded sets associated with this topology are the same as the ones associated with the topology of uniform convergence on compact subsets in each dierential (as dened for such maps in 4.4) separately. Moreover, as mentioned in the introduction of this paper, for complete F, the latter is again complete; see section 14 of the second part of this series ([25]) for details. Testing of smoothness is particularly simple in the case of a linear map: A linear map is smooth if and only if it is bounded. L(E; F ) will stand for the space of bounded (smooth) linear maps between E; F. 4.1 Theorem. A map f from D() into a locally convex space E is smooth if and only if for each K, the restriction of f to D K () is smooth. Proof. For the non-trivial part of the proof, consider a smooth curve c : R! D(). Its restriction to any bounded interval J has a relatively compact, hence bounded image. Therefore, c maps J into some D K () and the same holds for each derivative of c since D K () is a closed subspace of D. By assumption, f c is smooth on J. Since smoothness is a local property, we are done. 2 The obvious generalization of the preceding theorem is true for any strict inductive limit of a sequence of Frechet spaces. Its trivial part has an important consequence: D K () being a Frechet space, the restriction to D K () of any smooth map f from D() to any metrizable locally convex space E is continuous: Both on D K () and E the so-called c 1 -topology (see [30]) coincides with the metric topology ([30], 4.11.(1)); moreover, smooth maps are continuous with respect to the c 1 -topology ([30], p. 8). One of the particular features of the Frolicher-Kriegl-theory which considerably simplify its application is the exponential law (cf. Theorem 3.12 and Corollary 3.13 in [30]): 4.2 Theorem. Let E; F; G be locally convex spaces. Then the two spaces C 1 (E F; G) and C 1 (E; C 1 (F; G)) are isomorphic algebraically and bornologically, i.e., they have the same bounded sets. Replacing C 1 by L in 4.2 yields the exponential law for linear smooth maps. By iteration one obtains (see Proposition 5.2 in [23]): 4.3 Theorem. Let n; k 2 N and E i ; F (i = 1; : : : ; n + k) locally convex spaces. Then there is a bornological isomorphism L(E 1 ; : : : ; E n+k ; F ) = L(E 1 ; : : : ; E n ; L(E n+1 ; : : : ; E n+k ; F )): 10

12 For later use, we present the analoga of items 10{16 in [28] in the setting of [30]: 4.4 Theorem. (Theorem 3.18 and Corollary 5.11 in [30]) Let E; F be locally convex spaces. Then the dierentiation operator d : C 1 (E; F )! C 1 (E; L(E; F )) given by f(x + tv)? f(x) df(x)v := lim t!0 t exists and is linear and bounded (smooth). Hence, for n 2 N one can form the iterated dierentiation operator d n : C 1 (E; F )! C 1 (E; L(E; : : : ; L(E; F ) : : : )) = C 1 (E; L(E; : : : ; E; F )) which is smooth and linear and has values in C 1 (E; L sym (E; : : : ; E; F )), where L sym (E; : : : ; E; F ) stands for the space of smooth n-linear symmetric maps between E E and F. Also, the chain rule holds: d(f g)(x)v = df(g(x))dg(x)v: It is shown in [30], 1.4, that, given a curve which is smooth from (an open neighborhood of) R [a; b] to E, the dierence quotient c(b)?c(a) is an element of b?a convfc 0 (t) : t 2 [a; b]g, where conv denotes the closed convex hull. By virtue of the chain rule given in 4.4, this is equivalent to 4.5 Proposition. (Mean Value Theorem) Let f : E U! F be smooth, where U is an open neighborhood of a segment [x; x + v] E. Then f(x + v)? f(x) 2 convfdf(x + tv)(v) : t 2 [0; 1]g: As a consequence of 4.2, for each smooth map f 2 C 1 (F; G), the maps f : C 1 (E; F )! C 1 (E; G) and f : C 1 (G; E)! C 1 (F; E) are smooth. In particular, for a smooth map f 2 C 1 (E F; G) we may dene smooth linear \operators of partial dierentials d 1 ; d 2 as and d 1 := ( E) d : C 1 (E F; G)! C 1 (E F; L(E; G)) d 2 := ( F ) d : C 1 (E F; G)! C 1 (E F; L(F; G)); where E ; F denote the natural embeddings of E resp. F into E F. Obviously, we have f(x + t E (v))? f(x) d 1 f(x)(v) = df(x)( E (v)) = lim ; t!0 t which yields an alternative denition of d 1, which makes sense also for maps f : E F! G which are not a priori known to be smooth on E F. 11

13 4.6 Proposition. A map on E F is smooth if and only if both partial dierentials d 1 ; d 2 exist and are smooth as maps on E F. In this case the dierential d equals the sum (pr 1) d 1 + (pr 2) d 2 of the partial dierentials; the iterated mixed second derivatives coincide via the isomorphism L(E; L(F; G)) = L(F; L(E; G)) which is a consequence of 4.2. Proof. Necessity follows by what has been remarked above together with the symmetry of iterated derivatives stated in 4.4. For suciency, consider the map ~df 2 C 1 (E F; L(E F; G)) dened by df(x)(v ~ 1 ; v 2 ) := d 1 f(x)(v 1 ) + d 2 f(x)(v 2 ). Then obviously for xed x the map (t; v) 7! df(x ~ + tv)(v) is smooth from [0; 1] E F! G and hence can be viewed as an element of C 1 ([0; 1]; C 1 (E F; G)). By [30], 2.7, a smooth curve is Riemann integrable, the Riemann integral leads again into C 1 (E F; G) and commutes with the application of smooth linear maps. It follows that the map v 7! f(x) + Z 1 0 ~df(x + tv)(v)dt is smooth on E F and it suces to verify that the expression on the right hand side equals f(x+v) in order to obtain smoothness of f on E F. For this, note that for each xed segment [x; x + v], we can recover the claimed identity from the nite dimensional one by composing the restriction of f to the segment with bounded linear functionals. 2 The dierentiable uniform boundedness principle (see in [23]) constitutes an extremely useful tool for testing smoothness of linear maps into spaces of smooth functions: 4.7 Theorem. Let E; F; G be locally convex spaces, E; G complete. A linear map E! C 1 (F; G) is smooth if and and only if its composition with the evaluation ev x for each x 2 F is smooth. If we endow the space C 1 (X;R) (in the present paper, X will be one of the spaces D(); D(); A 0 () or A 0 (R s )) with the topology of uniform convergence on compact subsets in each derivative, i.e., in each iterated dierential separately, then by considering the corresponding seminorms one sees that taking the dierential constitutes a continuous linear operation. To be precise, the space A 0 (R s ) is not a linear space itself but the ane image of the closed linear subspace E := A 00 (R s ) D(R s ) and may be identied with the latter. A map on A 0 (R s ) is then said to be smooth if it is the pullback of a smooth map on E under the ane isomorphism. We say that the smooth structure on A 0 (R s ) is induced by its isomorphism with 12

14 E. This is a simple example of the notion of a smooth space as introduced in [23]. Locally convex spaces may be viewed as smooth spaces with a compatible linear structure. 4.8 Proposition. The following maps (to be dened in section 5) are smooth: The linear maps S : D(R s )! D(R s ), T x : D(R s )! D(R s ) and ('; x) 7! X ('; x), R 7! ^ X R (X 2 fc; Jg), as well as the non-linear maps S, T, x 7! T x, x 7! T x ', ('; x) 7! T x '. Proof. Smoothness of S ; T x ; X ; ^ X follows by our remarks preceding 4.1 and following 4.5, respectively, as each of these maps is essentially a pullback of a smooth map by denition. As the map S : (; ') 7! S ') is linear in ', it follows by the exponential law 4.2 and the uniform boundedness principle 4.7 that S is smooth i it is separately smooth, i.e., if and only if the maps S and ( 7! S ') are smooth. While smoothness of the former is already established, the latter is a curve which is obviously smooth o 0 and we are done. In a similar fashion, we obtain smoothness of T and all the maps associated with it. 2 5 C- and J-formalism Colombeau in [10] and in [13] (together with Meril) on the one hand and Jelnek in [28] on the other hand used dierent, yet equivalent formalisms to describe their respective constructions of Colombeau algebras: For embedding the space D 0 (R s ) of distributions on R s into the space E M (R s ) of representatives of generalized functions, they chose dierent (linear injective) maps which we denote by C ([10], [13]) and J ([28], compare also [9]), respectively. On a distribution given by a smooth function f on R s, C and J are dened by ( C f)('; x) := Z f(y)'(y? x) dy (1) resp. ( J f)('; x) := Z f(y)'(y) dy: (2) Here, ' denotes a test function from the subspace A 0 (R s ) of D(R s ) while x 2 R s. There are good reasons for either of these choices of the embedding; we are going to discuss their respective merits below. In this section we show that both formalisms are actually equivalent and establish a translation formalism allowing to change from one setting to the other at any stage of the presentation. 13

15 5.1 Denition. For 2 I and x 2 R s dene the following operators: T x : D(R s ) 3 ' 7! T x ' := '(:? x) 2 D(R s ) (3) S : D(R s ) 3 ' 7! S ' := 1 s ' : 2 D(R s ) (4) S : (0; 1) D(R s ) 3 (; ') 7! S ' 2 D(R s ) (5) T : D(R s ) R s 3 ('; x) 7! T ('; x) := (T x '; x) 2 D(R s ) R s (6) S () : D(R s ) R s 3 ('; x) 7! S () ('; x) := (S '; x) 2 D(R s ) R s : (7) T x and S are linear. All the operators introduced in the preceding denition are one-one and onto; moreover, they are continuous and smooth with respect to the natural topologies (see section 4). In a next step, we take (1) and (2) as a starting point for the determination of suitable domains for representatives of generalized functions on an open subset of R s : Assuming x 2 in (1) and (2), it is immediate that in (2) ' has to have its support in, whereas for (1) to be well-dened for any smooth function f on, the support of ' must be contained in? x. This motivates the introduction of the following sets: 5.2 Denition. Let 2 I. U() := T?1 (A 0 () ) = f('; x) 2 A 0 (R s ) j supp '? xg U () := (S () )?1 (U()) = = (T S () )?1 (A 0 () ) = f('; x) 2 A 0 (R s ) j supp '?1 (? x)g The notation U() is due to Colombeau ([10], 1.2.1). By denition, the maps T : U()! A 0 () and S () : U ()! U() are algebraic isomorphisms in the sense that they are bijective and linear in the rst argument. The question of topology, however, is somewhat subtle: Let denote the product of the (LF)- topology of D() and the Euclidean topology on ; abbreviate R s as 0. Then on A 0 (), the topology without doubt is the appropriate one to consider, rather than (the restriction of) 0. For U(), on the other hand, the topology 1 induced by 0 and the topology 2 := T?1 both seem to be natural choices. (Note that 1 can be obtained equally as T?1 0, due to T being a homeomorphism with respect to 0.) As the following example (which can easily be generalized to arbitrary non-trivial open subsets of R s ) shows, 1 is strictly coarser than 2 in general. 5.3 Example. Let := fx 2 R j x >?1g. Choose ' 2 D() with supp ' = [0; 1] and R ' = 0. Pick any 2 A 0 () such that supp [1; 2]. Letting n := 14

16 + 1 n?1 '(: + ) 2 A n n 0(), it is easy to check that T?1 ( n ; 0) = ( n ; 0) 2 U() tends to (; 0) 2 U() with respect to 1, yet is not convergent (in fact, not even bounded) with respect to 2. The situation is similar in the case of U (): Apart from the topology 1; induced by the topology 0 of A 0 (R s )R s via inclusion, the natural topology of A 0 () via T S () induces a topology 2; which, in general, is strictly ner than 1;. Now, in order to have the respective formalisms of Colombeau and Jelnek equivalent, we want T : U()! A 0 () and S () : U ()! U() to be also topological isomorphisms (hence dieomorphisms). This amounts to endowing U() and U () with the topologies 2 resp. 2; induced via T resp. T S (). Thus we adopt the following convention: Whenever questions of topology (in particular, boundedness) or smoothness on U() or U () are discussed, we consider their topologies to be 2 resp. 2;, i.e., those induced by the natural topology of A 0 () via T resp. T S (). To phrase it dierently, U() can be viewed as (innite-dimensional) smooth manifold, modelled over A 0 (), having an atlas consisting of a single chart T. A similar statement is valid for U () and T S (). The importance as well as the subtlety of distinguishing between 1 and 2 are highlighted in example 5.9 below. We are now able to introduce the basic spaces of smooth functions on which the construction of dieomorphism invariant Colombeau algebras is built. 5.4 Denition. E J () := C 1 (A 0 () ) (8) E C () := C 1 (U()) (9) By our above choice of topologies, T indeed maps E J () bijectively onto E C (). The next denition shows how the space of distributions on is to be embedded into E J () resp. E C (). 5.5 Denition. For u 2 D 0 (), dene J : D 0 ()! E J () ( J u)('; x) := hu; 'i C : D 0 ()! E C () ( C u)('; x) := hu; '(:? x)i By denition, C = T J. It remains to introduce the respective extensions of partial dierentiation from D 0 () to E C () resp. E J () and the respective actions of a dieomorphism. 15

17 5.6 Denition. For i = 1; : : : ; s, dene D C i : E C ()! E C () D C i i ; D J i : E J ()! E J () D J i := (T i T ; i.e., for R 2 E J (); ('; x) 2 A 0 () we set (D J i R)('; x) :=?((d 1 R)('; x))(@ i ') + (@ i R)('; x): Of course we have to demonstrate that for given R 2 E C () and ('; x) 2 U(), (D C i R)('; x) in fact exists and that ('; x) 7! (D C i R)('; x) is smooth on U() with respect to 2 (and similar for R 2 E J () and D J i ). This being a non-trivial task in particular for the case of the innocent-looking map D C i i [sic!] requiring some technical prerequisites, we have to defer it to the following section. Commutativity of the following diagram is immediate: D 0 ()?? y i???! D 0 ()??y C E C () D C i???! E C () x? x?t??t E J () D J i???! E J () 5.7 Denition. Let : ~! be a dieomorphism. Dene by J : A 0 (~) ~! A 0 () C : U(~)! U() : U ( ) ~! U () J ( ~'; ~x) :=? ( ~'?1 ) j det D?1 j ; ~x ; C ( ~'; ~x) :=? T?1 J T ( ~'; ~x) =? ~'(?1 (: + ~x)? ~x) j det D?1 (: + ~x)j ; ~x :? ( ~'; ~x) := (S () )?1 T?1 J T S () ( ~'; ~x) = ~'?1 (: + ~x)? ~x 16 j det D?1 (: + ~x)j ; ~x :

18 5.8 Denition. Let : ~! be a dieomorphism and 2 I. Dene ^ J : E J ()! E J ( ) ~ ^ C : E C ()! E C ( ) ~ ^ : C 1 (U ())! C 1 (U (~)) by ^ J := ( J ), ^ C := ( C ), ^ := ( ), i.e., (^ J R)( ~'; ~x) := R( J ( ~'; ~x)) (R 2 E J (); ( ~'; ~x) 2 A 0 ( ~ ) ~ ); (^ C R)( ~'; ~x) := R( C ( ~'; ~x)) (R 2 E C (); ( ~'; ~x) 2 U(~)); (^ R)( ~'; ~x) := R( ( ~'; ~x)) (R 2 C 1 (U ()); ( ~'; ~x) 2 U ( ~ )): For X 2 fc; Jg we obtain D 0 ()?? y X???! D 0 ( ~ )??y X E X () ^ X???! E X ( ~ ) where for u 2 D 0 (), u is dened by h u; 'i := hu; ('?1 ) j det D?1 ji (' 2 D()) which extends f 7! f = f for f 2 C 1 (). U (~)???! U ()???ys ys () () C 1 (U (~)) x??(s () ) ^??? C 1 (U ()) x??(s () ) U(~)?? yt C???! U()?? yt E C (~) ^ C??? E C () x? x?t??t A 0 ( ~ ) ~ J???! A 0 () E J ( ~ ) ^ J??? E J () Denitions 5.7 and 5.8 reect the fact that in Denition (D5) of section 3, we chose to regard distributions (and, in the sequel, non-linear generalized functions) as generalizations of functions, acting as functionals on test densities (compare, e.g., [27]). This approach has to be distinguished from constructing distributions as distributional densities, acting on test functions (see, e.g., [20]). In the following table, we compare the C-formalism and the J-formalism regarding simplicity of the respective denitions and, in the last item, the degree of familiarity. 17

19 C-formalism Feature J-formalism? domain of basic space E() +? smoothness structure +? embedding of D 0 () + + formula for dierentiation? + solving dierential equations?? action induced by a dieomorphism + + formula for testing?? generalization to manifolds + + tradition? The distribution of the +'s and?'s in the table should be rather obvious by inspecting the corresponding denitions. Due to the absence of a linear structure on a general smooth manifold, it is clear that the C-formalism does not lend itself to a denition of non-linear generalized functions on manifolds based only on intrinsic terms, whereas the J-formalism in fact does permit such a construction; see [26]. We conclude this section by presenting an example that emphasizes the importance of carefully distinguishing between the topologies 1 and 2 on U(). 5.9 Example. We will specify an open subset of R 2, a line segment of the form (t) := ('; z) + t( ; v) (?1 t 1) in U() and a distribution u on such that ( C u)((t))? ( C u)((0)) lim t!0 t = 1: This seems to suggest that in the point ('; z), the function C u on U() (which ought to be smooth according to our denitions) has no directional derivative with respect to ( ; v); or, to phrase it dierently, that the composition of the functions C u and (both of which have the appearence of being smooth) is not even dierentiable in ('; z). We will leave the solution to this puzzle for the end of the example. First we give the details of the construction. Let := f(x; y) 2 R 2 j x >?y 2? 1g, z := (0; 0), v := (0; 1). Choose R 1 2 D(R) such that supp 1 [0; 3] and 2 1(x) = exp(? 1) on I. Let c := x 1 and choose 2 2 D(R) such that supp 2 [ 3 2 ; 2] and R 2 = 1. Then := 1? c 2 has its support contained in [0; 2], coincides with exp(? 1 x ) on I and satises R = 0. Pick! 2 D(R) with the properties supp! = [?2; +2], 0! 1 and! 1 on [?1; +1]. Now dene 2 D(R 2 ) by (x; y) :=!(y) (x + 1? y 2 ): Finally, in order to obtain as dened above, take any ' 2 A 0 (R 2 ) whose support is located at the right hand side of the line given by x = 6. It is easy to check that 18

20 (t) = (' + t ; z + tv) belongs to U() for all t 2 [?1; +1]. Now there is still u to be dened. To this end, let f(x) := 1 x 2 exp( 1 x + 2 x 2 ) (0 < x < 1) 0 (x 1) : For 2 D() dene the distribution u 2 D 0 () by Z Z 0 1 hu; i := f(x + 1)(x; 0) dx = f(x)(x? 1; 0) dx:?1 0 For 0 < jtj 1, it follows Z 1 t [(C u)(' + t ; z + tv)? ( C u)('; z)] = 1 1 thu; (x; y? t)i = f(x)(x? t 2 ) dx: t 0 We will show that for 0 < jtj p 1 2, the value of the last integral can be estimated from below by exp( 1? 1)? exp(1), thus tending to innity for t! 0. Substituting t 2 x = 1, u t2 = 1, we obtain v Z 1 f(x)(x? t 2 ) dx = Z v Z v?1 v?1 e u+2u2 e? vu v?u du e 2u2 e? u2 v?u du e 2u2 e?u2 du Z 0 1 Z v?1 e u du = e v?1? e = e 1 t 2?1? e: The apparent inconsistencies mentioned at the beginning of the example dissolve by taking into account that, in fact, both 1 and 2 are involved in the argument: The statement that : [?1; +1]! U() is smooth is true only if it refers to 1 (the image of any neighborhood of 0 under is even unbounded with respect to 2 since the supports of T ((t)) are not contained in any compact subset of around t = 0). The statement that C u is smooth is true only if U() is endowed with the topology 2 induced by the natural topology of A 0 () via T. 2 being strictly ner than 1, we cannot infer the dierentiability of ( C u) from the actual smoothness properties of C u resp.. Another way of capturing the problem is by pointing out that ( ; v) is not a member of the tangent space to U() at ('; z) (in the sense of the following section) since supp is not contained in? z =. 19

21 6 Calculus on U () The purpose of this section is to develop an appropriate framework for dening and handling dierentials of any order of a function f : U ()! C which is smooth with respect to 2; (by denition, f is of the form f 0 T S () where f 0 2 C 1 (A 0 ())). By choosing = 1, this includes the case of smooth functions on U(), i.e., of elements of the basic space E C (). As a matter of fact, the author of [28] has payed only minor attention to these questions. However, it should be clear even from a glimpse at sections 7 and 10, in particular, that a sound denition and a proper handling of the dierentials of R := R J T S () = R C S () are crucial for the construction of a dieomorphism invariant Colombeau algebra. To start with, we discuss an important property of the sets U () which will be fundamental in the sequel at many places. Loosely speaking, every subset of A 0 (R s ) which is \not too large nally gets into U () by scaling and does not feel any difference between 1; and 2;. To this end, we introduce the following notation: 6.1 Denition. For every compact subset K of dene By denition, we have A 0;K () := f' 2 A 0 () j supp ' Kg; A 00;K () := f' 2 A 00 () j supp ' Kg; U K () := T?1 (A 0;K () ); U ;K () := (S () )?1 (U K ()): U K () = f('; x) 2 A 0 (R s ) j supp ' K? xg; U ;K () = (S () )?1 T?1 (A 0;K () ) = f('; x) 2 A 0 (R s ) j supp '?1 (K? x)g: Then it is immediate that for K, the topologies on A 0;K () inherited from the natural topologies of A 0 (R s ) and A 0 (), respectively, coincide. Consequently, on U K () the topologies 1 and 2 are equal, as are 1; and 2; on U ;K (). We now are in a position to complement Denition 5.6 by establishing that the derivation operators D C i and D J i are in fact well-dened. From the explicit formulas for D C i resp. D J i one is certainly tempted to view the former as being the simpler one of them since it does not seem to involve innite-dimensional calculus. Yet appearances are deceiving in this case: Since we have to view U() as a manifold modelled over A 0 () the only legitimate way of interpreting (D C i R C )('; x) = (@ i R C )('; x) is to push forward the curve t 7! ('; x + te i ) via T to A 0 () 20

22 and to study the directional derivative of R C T?1 along c : t 7! T ('; x + te i ) = ('( :? (x + te i )); x + te i ) at t = 0. To this end, rst note that for small absolute values of t, c actually takes values in A 0 (). Moreover, t 7! c(t) is a smooth curve in A 0 () with respect to 0, for the time being, according to Proposition 4.8. Since c maps some interval [?; +] into A 0;K () for a suitable K, the restriction of c to (?; +) is smooth even with respect to. Therefore, the directional derivative of R C T?1 along c : t 7! T ('; x + te i ) = ('( :? (x + te i )); x + te i ) at t = 0 exists. Having established existence, we can calculate its value as being given by 1 lim t!0 t [R T 1?1 (c(t))? R T?1 (c(0))] = lim t!0 t [R('; x + te i)? R('; x)]: Thus the usual formula works for R C 2 C 1 (U()) and D C i not a linear space. i, although U() is Finally, to see i R C is again smooth, we have to convince ourselves that (@ i R C ) T?1 = (T?1 ) D C i R C = D J i (T?1 ) R C = D J i (R C T?1 ) is smooth on A 0 (). Since, by denition, R J := R C T?1 is smooth on A 0 (), so are its dierential d 1 (R C T?1 ) and its partial i (R C T?1 ) on their respective domains. By Denition 5.6, D J i (RC T?1 ) = (D C i RC ) T?1 is smooth which is equivalent to the smoothness of D C i R C on U(). The smoothness of D J i R J for given R J 2 E J (), on the other hand, is immediate solely by the last part of the argument given above. Let us return to studying the sets U (). For the purpose of reference, the following observation is formulated as a lemma. 6.2 Lemma. Let K L R s and let B be a subset of D(R s ) such that all ' 2 B have their supports contained in some bounded set. Then there exists > 0 such that supp S (') L? x for all and ' 2 B, x 2 K. Proof. Set h := Then for each x 2 K, L contains the closed ball B h (x) of radius h around x. If, on the other hand, the compact ball D := B r (0) contains the supports of all ' 2 B then putting := h will do: We have supp S r (') + x D + x L for, ' 2 B, x 2 K Proposition. Let K L and let B be a subset of A 0 (R s ) such that all ' 2 B have their support contained in some xed bounded subset of R s. Then there exists > 0 such that B K U ;L () for all. In particular, B K U () and the restrictions of 1; and 2; to B K are equal. Proof. L, K and B satisfying the assumptions of Lemma 6.2, we obtain > 0 such that supp S (') L? x, i.e., ('; x) 2 U ;L () for all and ' 2 B, x 2 K. 2 21

23 The fact that for small the topologies 1; and 2; agree on sets of the form B K as above is crucial to get the smoothness properties right when it comes to testing for moderateness resp. negligibility, as we will see. With these prerequisites at hand, we now are ready to introduce the tangent space of U () and to dene dierentials of all orders of a smooth function dened on U (). >From an abstract point of view, the tangent space of U () with respect to 2; at the point ('; x) 2 U () is isomorphic to A 00 () ; to the tangent vector (; v) 2 A 00 () R s at (; x) 2 A 0 () there corresponds the \tangent vector (S 1 T?x( + d v); v) 2 A 00 (R s ) R s at ('; x) = (T S () )?1 (; x) = (S 1 T?x; x) 2 U () where d v denotes the directional derivative of with respect to v. The preceding formula is obtained by taking the derivative at t = 0 of the smooth curve t 7! (T S () )?1 ( + t; x + tv). In this sense, the tangent space to U () at ('; x) 2 U () can be identied with the set of all ( ; v) 2 A 00 (R s ) R s satisfying supp?x. Note that in this case the kinematic tangent space coincides with the operational one (the space of derivations dened on the smooth functions): In fact, by [30], 28.7., and [22], this is true for the space D() (more generally, for smooth sections with compact support of vector bundles over a manifold) and hence by [22] for its complemented subspace A 0 (). Essentially, Proposition 6.3 also applies to tangent vectors: 6.4 Proposition. Let K L and let B; C be subsets of A 0 (R s ) resp. A 00 (R s ) such that all! 2 B [ C have their supports contained in a xed bounded subset of R s. Then there exists > 0 such that B K U ;L () and C R s is contained in the tangent space to U () at ('; x) for all ('; x) 2 B K. The proof is virtually the same as for Proposition 6.3, with B now replaced by B[C; it even yields supp L?x for all tangent vectors ( ; v) with 2 C. Now let f : U ()! C be a function which is smooth with respect to 2;. Basically, d n f ought to be dened on the n-fold tangent space to U (), that is, on T n U () := G (';x)2u () f('; x)g f( ; v) 2 A 00 (R s ) R s j supp? x f being assumed as smooth with respect to 2; by denition, we cannot use a priori the structure of the surrounding space A 0 (R s ) to dene d n f. Instead, we will decompose U () (which has to be viewed as a manifold modelled over A 0 () ) into a family of subsets which is characteristic for smoothness of a function with respect to 2; in the sense that f is smooth on U () if and only if the restriction of f to any of these subsets is smooth, yet this time due to equality of 1; and 2; on each of these subsets either with respect to 1; or 2;. This allows the calculus of 22 g n :