aa r X i v : . [ m a t h . F A ] A p r Colombeau algebras without asymptotics
Eduard A. Nigsch ∗ November 5, 2018
Abstract
We present a construction of algebras of generalized functions of Colombeau-type which, instead of asymptotic estimates with respect to a regularizationparameter, employs only topological estimates on certain spaces of kernelsfor its definition.
MSC2010 Classification:
Keywords:
Nonlinear generalized functions, Colombeau algebras, asymptotic es-timates, elementary Colombeau algebra, diffeomorphism invariance
Colombeau algebras, as introduced by J. F. Colombeau [Col84; Col85], today rep-resent the most widely studied approach to embedding the space of Schwartz dis-tributions into an algebra of generalized functions such that the product of smoothfunctions as well as partial derivatives of distributions are preserved. These al-gebras have found numerous applications in situations involving singular objects,differentiation and nonlinear operations (see, e.g., [Obe92; Gro+01; NP06]).All constructions of Colombeau algebras so far incorporate certain asymptotic es-timates for the definition of the spaces of moderate and negligible functions, thequotient of which constitutes the algebra. There is a certain degree of freedom in theasymptotic scale employed for these estimates; while commonly a polynomial scaleis used, generalizations in several directions are possible. For an overview we referto works on asymptotic scales [DS98; DS00], p C , E , P q -algebras [Del09], sequencesspaces with exponent weights [Del+07] and asymptotic gauges [GL16].In this article we will present an algebra of generalized functions which instead ofasymptotic estimates employs only topological estimates on certain spaces of kernels ∗ Wolfgang Pauli Institute, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.email:[email protected] N and N denote the sets of positive and non-negative integers, respectively, and R ` the set of nonnegative real numbers. Concerning distribution theory we use thenotation and terminology of L. Schwartz [Sch66].Given any subsets K, L Ď R n (with n P N ) the relation K Ť L means that K iscompact and contained in the interior L ˝ of L .Let Ω Ď R n be open. C p Ω q is the space of complex-valued smooth functions onΩ. For any K, L Ť Ω, m, l P N and any bounded subset B Ď C p Ω q we set k f k K,m : “ sup x P K, | α |ď m |B α f p x q| p f P C p Ω qq , k ~ϕ k K,m ; L,l : “ sup x P K, | α |ď my P L, | β |ď l ˇˇ B αx B βy ~ϕ p x qp y q ˇˇ p ~ϕ P C p Ω , D p Ω qqq , k ~ϕ k K,m ; B : “ sup x P K, | α |ď mf P B |x f p y q , B αx ~ϕ p x qp y qy| p ~ϕ P C p Ω , E p Ω qqq . Note that k ¨ k K,m , k ¨ k K,m ; L,l and k ¨ k K,m ; B are continuous seminorms on the respectivespaces.We define ~δ P C p Ω , E p Ω qq by ~δ p x q : “ δ x for x P Ω, where δ x is the delta distributionat x . D L p Ω q is the space of test functions on Ω with support in L . For two locally convexspaces E and F , L p E, F q denotes the space of linear continuous mappings from E to F , endowed with the topology of bounded convergence. By U x p Ω q we denote thefilter base of open neighborhoods of a point x in Ω, and by U K p Ω q the filter base ofopen neighborhoods of K . By csn p E q we denote the set of continuous seminormsof a locally convex space E . B r p x q : “ t y P R n : k y ´ x k ă r u is the open Euclideanball of radius r ą x P R n .Our notion of smooth functions between arbitrary locally convex spaces is that ofconvenient calculus [KM97]. In particular, d k f denotes the k -th differential of asmooth mapping f . 2 Construction of the algebra
Throughout this section let Ω Ď R n be a fixed open set. Let C be the category oflocally convex spaces with smooth mappings in the sense of convenient calculus asmorphisms. Definition 1.
Consider C p´ , D p Ω qq and C p´q as sheaves with values in C . Wedefine the basic space of nonlinear generalized functions on Ω to be the set of sheafhomomorphisms B p Ω q : “ Hom p C p´ , D p Ω qq , C p´qq . Hence, an element of B p Ω q is given by a family p R U q U of mappings R U P C p C p U, D p Ω qq , C p U qq p U Ď Ω open q satisfying R U p ~ϕ q| V “ R V p ~ϕ | V q for all open subsets V Ď U and ~ϕ P C p U, D p Ω qq .We will casually write R in place of R U . Remark 2.
The basic space B p Ω q can be identified with the set of all mappings R P C p C p Ω , D p Ω qq , C p Ω qq such that for any open subset U Ď Ω and ~ϕ, ~ψ P C p Ω , D p Ω qq the equality ~ϕ | U “ ~ψ | U implies R p ~ϕ q| U “ R p ~ψ q| U (cf. [GN17]). B p Ω q is a C p Ω q -module with multiplication p f ¨ R q U p ~ϕ q “ f | U ¨ R U p ~ϕ q for R P B p Ω q , f P C p Ω q , U Ď Ω open and ~ϕ P C p U, D p Ω qq . Moreover, it is anassociative commutative algebra with product p R ¨ S q U p ~ϕ q : “ R U p ~ϕ q ¨ S U p ~ϕ q .A distribution u P D p Ω q defines a sheaf morphism from C p´ , D p Ω qq to C p´q .In fact, for U Ď Ω open and ~ϕ P C p U, D p Ω qq the function x ÞÑ x u, ϕ p x qy is anelement of C p U q (see [Sch66, Chap. IV, §
1, Th. II, p. 105] or [Tre76, Theorem40.2, p. 416]). More abstractly, this can be seen using the theory of topologicaltensor products [Sch55; Sch57; Tre76] as follows: C p U, D p Ω qq – C p U q p b D p Ω q – L p D p Ω q , C p U qq , where C p U q p b D p Ω q denotes the completed projective tensor product of C p U q and D p Ω q . The assignment ~ϕ ÞÑ x u, ~ϕ y is smooth, being linear and continuous [KM97,1.3, p. 9]. Hence, we have the following embeddings of distributions and smoothfunctions into B p Ω q : Definition 3.
We define ι : D p Ω q Ñ B p Ω q and σ : C p Ω q Ñ B p Ω q by p ιu qp ~ϕ qp x q : “ x u, ~ϕ p x qy p u P D p Ω qqp σf qp ~ϕ qp x q : “ f p x q p f P C p Ω qq for ~ϕ P C p U, D p Ω qq with U Ď Ω open and x P U . ι is linear and σ is an algebra homomorphism. Directional derivatives on B p Ω q then are defined as follows: Definition 4.
Let X P C p Ω , R n q be a smooth vector field and R P B p Ω q . Wedefine derivatives r D X : B p Ω q Ñ B p Ω q and p D X : B p Ω q Ñ B p Ω q by p r D X R qp ~ϕ q : “ D X p R U p ~ϕ qqp p D X R qp ~ϕ q : “ ´ d R U p ~ϕ qp D SK X ~ϕ q ` D X p R U p ~ϕ qq for ~ϕ P C p U, D p Ω qq with U Ď Ω open, where we set D SK X ~ϕ : “ D X ~ϕ ` D wX ˝ ~ϕ. Here, p D X ~ϕ qp x q is the directional derivative of ~ϕ at x in direction X p x q and p D ωX ˝ ~ϕ qp x q is the Lie derivative of ~ϕ p x q considered as a differential form, given byD ωX p ~ϕ p x qq “ D X p ~ϕ p x qq ` p Div X qp x q ¨ ~ϕ p x q .Note that both r D X and p D X satisfy the Leibniz rule. We have r D x ˝ σ “ σ ˝ r D X , p D X ˝ σ “ σ ˝ p D X , p D X ˝ ι “ ι ˝ p D X . While r D X is C p Ω q -linear in X , p D X is only C -linear in X . We refer to [Nig15;Nig16] for a discussion of the role of these derivatives in differential geometry. Definition 5.
For k P N we set P k : “ R ` r y , . . . , y k s , I k : “ t λ P R ` r y , . . . , y k , z , . . . , z k s | λ p y , . . . , y k , , . . . , q “ u . More explicitly, P k is the commutative semiring of polynomials in the k ` y , . . . , y k with coefficients in R ` . Similarly, I k is the commutativesemiring in the 2 p k ` q commuting variables y , . . . , y k , z , . . . , z k with coefficientsin R ` and such that, if λ P I k is given by the finite sum λ “ ÿ α,β P N k ` λ αβ y α z β , then λ α “ α . Note that P k is a subsemiring of P k ` and I k a subsemiringof I k ` . Furthermore, I k is an ideal in P k if P k is considered as a subsemiringof R ` r y , . . . , y k , z , . . . , z k s . Given λ P P k and y i ď y i for i “ . . . k we have λ p y q ď λ p y q . For λ, µ P P k we write λ ď µ if λ p y q ď µ p y q for all y P p R ` q k ` , andsimilarly for λ, µ P I k .We can now formulate the following definitions of moderateness and negligibility,not involving any asymptotic estimates: Definition 6.
An element R P B p Ω q is called moderate if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U q p@ m, k P N qpD c, l P N q pD λ P P k q p@ ~ϕ , . . . , ~ϕ k P C p U, D L p U qqq : k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q . The subset of all moderate elements of B p Ω q is denoted by M p Ω q . efinition 7. An element R P B p Ω q is called negligible if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U q p@ m, k P N q pD c, l P N qpD λ P I k q pD B Ď C p Ω q bounded q p@ ~ϕ , . . . , ~ϕ k P C p U, D L p U qqq : k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ϕ k K,c ; B , . . . , k ~ϕ k k K,c ; B q . The subset of all negligible elements of B p Ω q is denoted by N p Ω q . It is worthwile to discuss possible simplifications of these definitions, which atthis stage should be considered more as a proof of concept than as the definiteform they should have. First, we note that we cannot replace p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U q by p@ K, L Ť Ω q . In fact, in the second case K and L can be distant from each other, while in the first case it suffices to control thesituation where K and L are close to each other. However, the following re-sult shows that we can always assume K Ť L and that the ~ϕ , . . . , ~ϕ k are givenmerely on an arbitrary open neighborhood of K , i.e., as elements of the direct limit C p K, D L p Ω qq : “ lim ÝÑ V P U K p Ω q C p V, D L p Ω qq : Proposition 8.
Let R P B p Ω q . Then R is moderate if and only if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U : K Ť L q p@ m, k P N qpD c, l P N q pD λ P P k q p@ ~ϕ , . . . , ~ϕ k P C p K, D L p U qqq : k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q . Similarly, R is negligible if and only if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U : K Ť L q p@ m, k P N q pD c, l P N qpD λ P I k q pD B Ď C p U q bounded q p@ ~ϕ , . . . , ~ϕ k P C p K, D L p U qqq : k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ϕ k K,c ; B , . . . , k ~ϕ k k K,c ; B q . Proof.
Obviously each of these conditions is weaker than the corresponding one ofDefinition 6 or Definition 7.Suppose we are given R P B p Ω q such that the condition stated for moderatenessholds. Given x P Ω there hence exists some U P U x p Ω q . Now given arbitrary K, L Ť U we choose a set L Ť U such that K Y L Ť L . Fixing m, k P N for themoderateness test, for p K, L q we hence obtain c, l P N and λ P P k . Now fix some ~ϕ , . . . , ~ϕ k P C p U, D L p U qq ; each of those represents an element of C p K, D L p U qq ,whence we have the estimate k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ p k ~ϕ k K,c ; L ,l , . . . , k ~ϕ k k K,c ; L ,l q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q . where the last equality follows because the ~ϕ , . . . , ~ϕ k take values in D L p U q . Thisshows that R is moderate. 5or the case of negligibility we proceed similarly until we obtain c, l P N , λ P I k and B Ď C p U q . Let χ P D p U q be such that χ ” L andset B : “ t χf | f P B u Ď C p Ω q , which is bounded. For any ~ϕ , . . . , ~ϕ k we thenobtain k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ďď λ p k ~ϕ k K,c ; L ,l , . . . , k ~ϕ k k K,c ; L ,l , k ~ϕ ´ ~δ k K,c ; B , k ~ϕ k K,c ; B , . . . , k ~ϕ k k K,c ; B q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ϕ k K,c ; B , . . . , k ~ϕ k k K,c ; B q which proves negligibility of R .If the test of Definition 6, Definition 7 or Proposition 8 holds on some U thenclearly it also holds on any open subset of U . The following characterization ofmoderateness and negligiblity is obtained by applying polarization identities to thedifferentials of R : Lemma 9.
Let R P B p Ω q .(i) R is moderate if and only if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U q p@ m, k P N qpD c, l P N q pD λ P P min p ,k q q p@ ~ϕ, ~ψ P C p U, D L p U qqq : k d k R p ~ϕ qp ~ψ, . . . , ~ψ q k K,m ď λ p k ~ϕ k K,c ; L,l q if k “ ,λ p k ~ϕ k K,c ; L,l , k ~ψ k K,c ; L,l q if k ě . (ii) R is negligible if and only if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U q p@ m, k P N q pD c, l P N qpD λ P I min p ,k q q pD B Ď C p Ω q bounded q p@ ~ϕ, ~ψ P C p U, D L p U qqq : k d k R p ~ϕ qp ~ψ, . . . , ~ψ q k K,m ď λ p k ~ϕ k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B q if k “ ,λ p k ~ϕ k K,c ; L,l , k ~ψ k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ψ k K,c ; B q if k ě . Proof.
We assume k ě
1, as for k “ ñ ”: One obtains λ P P k such that k d k R p ~ϕ qp ~ψ, . . . , ~ψ q k K,m ď λ p k ~ϕ k K,c ; L,l , k ~ψ k K,c ; L,l , . . . , k ~ψ k K,c ; L,l q“ λ p k ~ϕ k K,c ; L,l , k ~ψ k K,c ; L,l q with λ P P given by λ p y , y q “ λ p y , y , . . . , y q .“ ð ”: One obtains λ P P . We then use the polarization identity [Tho14, eq. (7),p. 471] d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q “ n ! k ÿ a “ p´ q k ´ a ÿ J Ďt ...k u| J |“ a ∆ ˚ p d k R p ~ϕ qqp S J q S J : “ ř i P J ~ϕ i and we have set ∆ ˚ p d k R p ~ϕ qqp ~ψ q “ d k R p ~ϕ qp ~ψ, . . . , ~ψ q . Hence, k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď n ! k ÿ a “ ÿ | J |“ a k ∆ ˚ p d k R p ~ϕ qqp S J q k K,m ď n ! k ÿ a “ ÿ | J |“ a λ p k ~ϕ k K,c ; L,l , k S J k K,c ; L,l qď n ! k ÿ a “ ÿ | J |“ a λ p k ~ϕ k K,c ; L,l , ÿ i P J k ~ϕ i k K,c ; L,l q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q with λ P P k given by λ p y , . . . , y k q “ n ! k ÿ a “ ÿ | J |“ a λ p y , ÿ i P J y i q . (ii) “ ñ ”: We have λ P I k such that k d k R p ~ϕ qp ~ψ, . . . , ~ψ q k K,m ď λ p k ~ϕ k K,c ; L,l , k ~ψ k K,c ; L,l , . . . , k ~ψ k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ψ k K,c ; B , . . . , k ~ψ k K,c ; B q“ λ p k ~ϕ k K,c ; L ; l , k ~ψ k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ψ k K,c ; B q with λ P I k given by λ p y , y , z , z q “ λ p y , y , . . . , y , z , z , . . . , z q . “ ð ”: We obtain λ P I such that, as above, k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď n ! k ÿ a “ ÿ | J |“ a λ p k ~ϕ k K,c ; L,l , k S J k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k S J k K,c ; B qď n ! k ÿ a “ ÿ | J |“ a λ p k ~ϕ k K,c ; L,l , ÿ i P J k ~ϕ i k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , ÿ i P J k ~ϕ i k K,c ; B q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , k ~ϕ k K,c ; B , . . . , k ~ϕ k k K,c ; B q with λ P I k given by λ p y , . . . , y k , z , . . . , z k q “ n ! k ÿ a “ ÿ | J |“ a λ p y , ÿ i P J y i , z , ÿ i P J z i q . Note that the polarization identities could be applied also in the formulation ofProposition 8. 7 roposition 10. N p Ω q Ď M p Ω q .Proof. Let R P N p Ω q and fix x P Ω for the moderateness test. By negligibility of R there exists U P U x p Ω q as in Definition 7. Let K, L Ť U and m, k P N be arbitrary.Then there exist c, l, λ and B such that the estimate of Definition 7 holds. We knowthat λ P I k is given by a finite sum λ p y , . . . , y k , z , . . . , z k q “ ÿ α,β λ αβ y α z β . It suffices to show that there are λ , λ P P such that for any ~ϕ P C p U, D L p U qq we have the estimates k ~ϕ ´ ~δ k K,c ; B ď λ p k ~ϕ k K,c ; L,l q , (1) k ~ϕ k K,c ; B ď λ p k ~ϕ k K,c ; L,l q . (2)In fact, these inequalities imply k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď ÿ α,β λ αβ k ~ϕ k α K,c ; L,l ¨ . . . ¨ k ~ϕ k k α k K,c ; L,l ¨ k ~ϕ ´ ~δ k β K,c ; B ¨ k ~ϕ k β K,c ; B ¨ . . . ¨ k ~ϕ k k β k K,c ; B ď ÿ α,β λ αβ k ~ϕ k α K,c ; L,l ¨ . . . ¨ k ~ϕ k k α k K,c ; L,l ¨ λ p k ~ϕ k K,c ; L,l q β ¨ λ p k ~ϕ k K,c ; L,l q β ¨ ¨ ¨ λ p k ~ϕ k k K,c ; L,l q β k “ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q with λ P P k given by λ p y , . . . , y k q “ ÿ λ αβ y α λ p y q β λ p y q β ¨ ¨ ¨ λ p y k q β k . Inequality (1) is seen as follows: k ~ϕ ´ ~δ k K,c ; B “ sup x P K, | α |ď cf P B ˇˇˇˇż L f p y qB αx ~ϕ p x qp y q d y ´ B α f p x q ˇˇˇˇ ď | L | ¨ sup f P B k f k L, ¨ k ~ϕ k K,c ; L,l ` sup f P B k f k K,c “ λ p k ~ϕ k K,c ; L,l q with λ p y q “ | L | ¨ sup f P B k f k L, ¨ y ` sup f P B k f k K,c , where | L | denotes the Lebesguemeasure of L . Similarly, inequality (2) results from k ~ϕ k K,c ; B “ sup x P K, | α |ď cf P B ˇˇˇˇż L f p y qB αx ~ϕ p x qp y q d y ˇˇˇˇ ď | L | ¨ sup f P B k f k L, ¨ k ~ϕ k K,c ; L,l “ λ p k ~ϕ k K,c ; L,l q with λ p y q “ | L | ¨ sup f P B k f k L, ¨ y . 8 roposition 11. M p Ω q is a subalgebra of B p Ω q and N p Ω q is an ideal in M p Ω q .Proof. This is evident from the definitions.
Theorem 12.
Let u P D p Ω q and f P C p Ω q . Then(i) ιu is moderate,(ii) σf is moderate,(iii) ιf ´ σf is negligible, and(iv) if ιu is negligible then u “ .Proof. (i): Fix x for the moderateness test and let U P U x p Ω q be arbitrary. Fix any K, L Ť U and m P N . Then there are constants C “ C p L q P R ` and l “ l p L q P N such that |x u, ϕ y| ď C k ϕ k L,l for all ϕ P D L p Ω q . Hence, we see that k p ιu qp ~ϕ q k K,m “ k x u, ~ϕ y k K,m “ sup x P K, | α |ď m |x u, B αx ~ϕ p x qy|ď C ¨ sup x P K, | α |ď my P L, | β |ď l ˇˇ B αx B βy ~ϕ p x qp y q ˇˇ “ C k ~ϕ k K,m ; L,l “ λ p k ~ϕ k K,m ; L,l q . with λ p y q “ Cy . Moreover, we have k d p ιu qp ~ϕ qp ~ϕ q k K,m ď C k ~ϕ k K,m ; L,l “ λ p k ~ϕ k K,m ; L,l , k ~ϕ k K,m ; L,l q with λ p y , y q “ Cy . Higher differentials of ιu vanish and the moderateness test issatisfied with λ “ k ě x and let U P U x p Ω q be arbitrary. For any K, L Ť U and m P N we have k p σf qp ~ϕ q k K,m “ k f k K,m “ λ p k ~ϕ k K, L, q with λ p y q “ k f k K,m . Differentials of σf vanish, i.e., λ “ k ě x and let U P U x p Ω q be arbitrary. For any K, L Ť U and m, k P N wehave p ιf ´ σf qp ~ϕ q “ x f, ~ϕ ´ ~δ y , d p ιf ´ σf qp ~ϕ qp ~ϕ q “ x f, ~ϕ y , d k p ιf ´ σf qp ~ϕ qp ~ϕ , . . . , ~ϕ k q “ k ě . Hence, with c “ m , l “ B “ t f u the negligibility test is satisfied with λ p y , z q “ z for k “ λ p y , y , z , z q “ z for k “ λ “ k ě x P Ω has an open neighborhood V such that u | V “
0, which implies u “ x P Ω, let U P U x p Ω q be as in the characterization of negligibility inProposition 8. Choose an open neighborhood V of x such that K : “ V Ť U r ą L : “ B r p K q Ť U . With k “ m “
0, Proposition 8 gives c, l P N , λ P I and B Ď C p U q , where λ has the form λ p y, z q “ ÿ α P N n ,β P N λ αβ y α z β . Choose ϕ P D p R n q with supp ϕ Ď B p q , ş ϕ p x q d x “ ş x γ ϕ p x q d x “ γ P N n with 0 ă | γ | ď q , where q is chosen such that β p q ` q ą α p n ` c ` l q for all α, β with λ αβ ‰ q “ p n ` c ` l q deg y λ , where deg y λ is thedegree of λ with respect to y ). For ε ą ϕ ε p y q “ ε ´ n ϕ p y { ε q . Then for ε ă r , ~ϕ ε p x qp y q : “ ϕ ε p y ´ x q defines an element ~ϕ ε P C p K, D L p Ω qq becausesupp ϕ ε p . ´ x q “ x ` supp ϕ ε Ď B ε p x q Ď B r p K q Ď L for x P B r ´ ε p K q . Consequently,we have k p ιu qp ~ϕ ε q k K, ď λ p k ~ϕ ε k K,c ; L,l , k ~ϕ ε ´ ~δ k K,c ; B q . Because of the estimates k ~ϕ ε k K,c ; L,l “ O p ε ´p n ` l ` c q q k ~ϕ ε ´ ~δ k K,c ; B “ O p ε q ` q , which may be verified by a direct calculation, we have k p ιu qp ~ϕ ε q k K, ď ÿ α,β λ α,β ¨ O p ε ´ α p n ` c ` l q q ¨ O p ε β p q ` q q Ñ q , which means that p ιu qp ~ϕ ε q| V Ñ C p V q and hence also in D p V q . On the other hand, we have x u, ~ϕ ε y| V Ñ u | V in D p V q , as is easily verified. This completes the proof. Theorem 13.
For X P C p Ω , R n q we have(i) r D X p M p Ω qq Ď M p Ω q and p D X p M p Ω qq Ď M p Ω q ,(ii) r D X p N p Ω qq Ď N p Ω q and p D X p N p Ω qq Ď N p Ω q .Proof. The claims for r D X are clear because k d k p r D X R qp ~ϕ qp ~ψ, . . . , ~ψ q k K,m “ k D X p d k R p ~ϕ qp ~ψ, . . . , ~ψ qq k K,m ď C k d k R p ~ϕ qp ~ψ, . . . , ~ψ q k K,m ` for some constant C depending on X . As to p D X , we have to deal with terms of theform d k ` R p ~ϕ qp D SK X ~ϕ, ~ψ, . . . , ~ψ q and d k R p ~ϕ qp D SK X ~ψ, ~ψ, . . . , ~ψ q for which we use the estimate k D SK X ~ϕ k K,c ; L,l ď C k ~ϕ k K,c, ` L,l ` for some constant C depending on X . 10e now come to the quotient algebra. Definition 14.
We define the Colombeau algebra of generalized functions on Ω by G p Ω q : “ M p Ω q{ N p Ω q . G p Ω q is a C p Ω q -module and an associative commutative algebra with unit σ p q . ι is a linear embedding of D p Ω q and σ an algebra embedding of C p Ω q into G p Ω q such that ιf “ σf in G p Ω q for all smooth functions f P C p Ω q . Furthermore, thederivatives p D X and r D X are well-defined on G p Ω q .Finally, we establish sheaf properties of G . Note that for Ω Ť Ω open, the restriction R | Ω p ~ϕ q : “ R p ~ϕ q is well-defined because for U Ď Ω open we have C p U, D p Ω qq Ď C p U, D p Ω qq . Proposition 15.
Let R P B p Ω q and Ω Ď Ω be open. If R is moderate then R | Ω ismoderate; if R is negligible then R | Ω is negligible.Proof. Suppose that R P M p Ω q . Fix x P Ω , which gives U P U x p Ω q . Set U : “ U X Ω P U x p Ω q and let K, L Ť U and m, k P N be arbitrary. Then there are c, l, λ as in Definition 6. Let now ~ϕ , . . . , ~ϕ k P C p U , D L p U qq be given. Choose ρ P D p U q such that ρ ” K . Then ρ ¨ ~ϕ i P C p U, D L p U qq ( i “ . . . k ) and k d k R | Ω p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m “ k d k R | Ω p ρ~ϕ qp ρ~ϕ , . . . , ρ~ϕ k q k K,m “ k d k R p ρ~ϕ qp ρ~ϕ , . . . , ρ~ϕ k q k K,m ď λ p k ρ~ϕ k K,c ; L,l , . . . , k ρ~ϕ k k K,c ; L,l q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q . Hence, the moderateness test is satisfied for R | Ω .Now suppose that R P N p Ω q . For the negligibility test fix x P Ω , which gives U P U x p Ω q . Set U : “ U X Ω and let K, L Ť U and m, k P N be arbitrary. Then D c, l, B, λ as in Definition 7. Let now ~ϕ , . . . , ~ϕ k P C p U , D L p U qq be given. Choose ρ P D p U q such that ρ ” K . Then ρ ¨ ~ϕ i P C p U, D L p U qq ( i “ . . . k ) and k d k R | Ω p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m “ k d k R | Ω p ρ~ϕ qp ρ~ϕ , . . . , ρ~ϕ k q k K,m “ k d k R p ρ~ϕ qp ρ~ϕ , . . . , ρ~ϕ k q k K,m ď λ p k ρ~ϕ k K,c ; L,l , . . . , k ρ~ϕ k k K,c ; L,l , k ρ~ϕ ´ ~δ k K,c ; B , . . . , k ρ~ϕ k k K,c ; B q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l , k ~ϕ ´ ~δ k K,c ; B , . . . , k ~ϕ k k K,c ; B q which shows negligibility of R | Ω . Proposition 16. G p´q is a sheaf of algebras on Ω .Proof. Let X Ď Ω be open and p X i q i be a family of open subsets of Ω such that Ť i X i “ X . 11e first remark that if R P B p X q satisfies R | X i P N p X i q for all i then R P N p X q ,as is evident from the definition of negligibility.Suppose now that we are given R i P M p X i q such that R i | X i X X j ´ R j | X i X X j P N p X i X X j q for all i, j with X i X X j ‰ H . Let p χ i q i be a partition of unitysubordinate to p X i q i , i.e., a family of mappings χ i P C p X q such that 0 ď χ i ď p supp χ i q i is locally finite, ř i χ i p x q “ x P X and supp χ i Ď X i . Choosefunctions ρ i P C p X i , D p X i qq which are equal to 1 on an open neighborhood of thediagonal in X i ˆ X i for each i . For V Ď X open and ~ϕ P C p V, D p X qq we define R V p ~ϕ q P C p V q by R V p ~ϕ q : “ ÿ i χ i | V ¨ p R i q V X X i p ρ i | V X X i ¨ ~ϕ | V X X i q . (3)For showing smoothness of R V consider a curve c P C p R , C p V, D p X qqq . We haveto show that t ÞÑ R V p c p t qq is an element of C p R , C p V qq . By [KM97, 3.8, p. 28]it suffices to show that for each open subset W Ď V which is relatively compact in V the curve t ÞÑ R V p c p t qq| W “ R W p c p t q| W q is smooth, but this holds because thesum in (3) then is finite. Hence, p R V q V P B p Ω q .Fix x P X for the moderateness test. There is a finite index set F and an openneighborhood W P U x p X q such that W X supp χ i ‰ H implies i P F . We can alsoassume that x P Ş i P F X i . Let Y be a neighborhood of x such that ρ i ” Y ˆ Y for all i P F . For each i P F let U i P U x p X i q be obtained from moderateness of R i as in Definition 6. Set U : “ Ş i P F U i X W X Y P U x p X q , and let K, L Ť U aswell as m, k P N be arbitrary. For each i P F there are c i , l i , λ i such that for any ~ϕ , . . . , ~ϕ k P C p U, D L p U qq we have k d k R i p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ i p k ~ϕ k K,c i ; L,l i , . . . , k ~ϕ k k K,c i ; L,l i q . Now we have, for ~ϕ P C p U, D L p U qq , R p ~ϕ q| W “ ÿ i P F χ i | W ¨ p R i q W X X i p ρ i ~ϕ | W X X i q and hence, for ~ϕ , . . . , ~ϕ k P C p U, D L p U qq ,d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q| W “ ÿ i P F χ i | W ¨ d k pp R i q W X X i qp ρ i ~ϕ | W X X i qp ρ i ~ϕ | W X X i , . . . , ρ i ~ϕ k | W X X i q . We see that k d k R p ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď ÿ i P F C p m q ¨ k χ i k K,m ¨ λ i p k ~ϕ k K,c i ; L,l i , . . . , k ~ϕ k k K,c i ; L,l i q“ λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ k k K,c ; L,l q c “ max j P F c j , l “ max j P F l j , some constant C p m q coming from the Leibnizrule, and λ P P k given by λ “ ÿ i P F C p m q k χ i k K,m ¨ λ i . This shows that R is moderate. Finally, we claim that R | X j ´ R j P N p X j q for all j . For this we first note that p R | X j ´ R j qp ~ϕ q “ ÿ i χ i | X j ¨ p R i p ρ i ~ϕ | X i X X j q ´ R j p ~ϕ qq for ~ϕ P C p X j , D p X j qq . Again, for x P X j there is a finite index set F and anopen neighborhood W P U x p X q such that W X supp χ i ‰ H implies i P F , and wecan assume that x P Ş i P F X i . Let Y be a neighborhood of x such that ρ i ” Y ˆ Y for all i P F and let U i P U x p X i X X j q be given by the negligibility test of R i | X i X X j ´ R j | X i X X j according to Definition 7. Set U : “ Ş i P F U i X W X Y . Fixany K, L Ť U and m, k P N . For each i P F there are c i , l i , λ i , B i such that for ~ϕ , . . . , ~ϕ k P C p U, D L p U qq we have k d k p R i | X i X X j ´ R j | X i X X j qp ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď λ i p k ~ϕ k K,c i ; L,l i , . . . , k ~ϕ ´ ~δ k K,c i ; B i , k ~ϕ k K,c i ; B i , . . . , k ~ϕ k k K,c i ; B i q . As above, we then have k d k p R | X j ´ R j qp ~ϕ qp ~ϕ , . . . , ~ϕ k q k K,m ď ÿ i P F C p m q ¨ k χ i k K,m ¨ λ i p k ~ϕ k K,c i ; L,l i , . . . , k ~ϕ ´ ~δ k K,c i ; B i , k ~ϕ k K,c i ; B i , . . . qď λ p k ~ϕ k K,c ; L,l , . . . , k ~ϕ ´ ~δ k K,c ; B , k ~ϕ k K,c ; B , . . . q with c “ max i P F c i , l “ max i P F l i , B “ Ť i P F B i , and λ P I k given by λ “ ÿ i P F C p m q k χ k K,m ¨ λ i . This completes the proof.
We will now give a variant of the construction of Section 3 similar in spirit toColombeau’s elementary algebra [Col85]: if we only consider derivatives along thecoordinate lines of R n we can replace the smoothing kernels ~ϕ P C p U, D L p Ω qq byconvolutions. This way, one can use a simpler basic space which does not involvecalculus on infinite dimensional locally convex spaces anymore: Definition 17.
Let Ω Ď R n be open. We set U p Ω q : “ tp ϕ, x q P D p R n q ˆ Ω | supp ϕ ` x Ď Ω u . and define B c p Ω q to be the set of all mappings R : U p Ω q Ñ C such that R p ϕ, ¨q issmooth for fixed ϕ . D p R n q in place of thespace of test functions whose integral equals one. We now introduce a notation forthe convolution kernel determined by a test function. Definition 18.
For ϕ P D p R n q we define ‹ ϕ P C p R n , D p R n qq by ‹ ϕ p x qp y q : “ ϕ p y ´ x q . In fact, with this definition we have x u, ‹ ϕ y “ u ˚ ˇ ϕ , where as usually we set ˇ ϕ p y q : “ ϕ p´ y q . Furthermore, for c P N we write k ϕ k c : “ sup x P R n , | α |ď c |B α ϕ p x q| p ϕ P D p R n qq . The direct adaptation of Definitions 6 and 7 then looks as follows:
Definition 19.
Let R P B c p Ω q . Then R is called moderate if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U : K Ť L q p@ m P N qpD c P N q pD λ P P q p@ ϕ P D p R n q : K ` supp ϕ Ď L q : k R p ϕ, . q k K,m ď λ p k ϕ k c q . The subset of all moderate elements of B c p Ω q is denoted by M c p Ω q .Similarly, R is called negligible if p@ x P Ω q pD U P U x p Ω qq p@ K, L Ť U : K Ť L q p@ m P N q pD c P N qpD λ P I q pD B Ď C p U q bounded q p@ ϕ P D p R n q : K ` supp ϕ Ď L q : k R p ϕ, . q k K,m ď λ p k ϕ k c , k ‹ ϕ ´ ~δ k K,c ; B q . The subset of all negligible elements of B c p Ω q is denoted by N c p Ω q . It is convenient to work with the following simplification of these definitions.
Proposition 20. R P B c p Ω q is moderate if and only if p@ K Ť Ω q pD r ą B r p K q Ť Ω q p@ m P N q pD c P N qpD λ P P q p@ ϕ P D p R n q : supp ϕ Ď B r p qq : k R p ϕ, . q k K,m ď λ p k ϕ k c q . Similarly, R P B c p Ω q is negligible if and only if p@ K Ť Ω q pD r ą B r p K q Ť Ω q p@ m P N q pD c P N qpD λ P I q pD B Ď C p Ω q bounded q p@ ϕ P D p R n q : supp ϕ Ď B r p qq : k R p ϕ, . q k K,m ď λ p k ϕ k c , k ‹ ϕ ´ ~δ k K,c ; B q . roof. Suppose R is moderate and fix K Ť Ω. We can cover K by finitely manyopen sets U i obtained from Definition 19 and write K “ Ť i K i with K i Ť U i .Choose r ą L i : “ B r p K i q Ť U i for all i . Fixing m , by moderatenessthere exist c i and λ i for each i . Set c “ max i c i and choose λ with λ ě λ i for all i .Now given ϕ P D p R n q with supp ϕ Ď B r p q we also have K i ` supp ϕ Ď L i and wecan estimate k R p ϕ, . q k K,m ď sup i k R p ϕ, . q k K i ,m ď sup i λ i p k ϕ k c i q ď λ p k ϕ k c q . Conversely, suppose the condition holds and fix x P Ω for the moderateness test.Choose a ą B a p x q Ť Ω. By assumption there is r ą B r ` a p x q Ť Ω.Set U : “ B r { p x q . Then, fix K Ť L Ť U and m for the moderateness test. Thereare c and λ by assumption. Now given ϕ with K ` supp ϕ Ď L , we see that for y P supp ϕ and an arbitrary point z P K we have | y | ď | y ` z ´ x | ` | z ´ x | ă r ,which means that supp ϕ Ď B r p q . But then k R p ϕ, . q k K,m ď λ p k ϕ k c q as desired.If R is negligible we proceed similarly until the choice of K i Ť L i Ť U i and m gives c i , λ i and B i . Choose χ i P D p U i q with χ i ” L i , and define B : “ Ť i t χ i f | f P B i u , which is bounded in C p Ω q . Then with c “ max i c i and λ ě λ i for all i we have k R p ϕ, . q k K,m ď sup i λ i p k ϕ k c i , k ‹ ϕ ´ ~δ k K i ,c i ; B i q ď λ p k ϕ k c , k ‹ ϕ ´ ~δ k K,c ; B q . The converse is seen as for moderateness by restricting the elements of B Ď C p Ω q to U .The embeddings now take the following form. Definition 21.
We define ι c : D p Ω q Ñ B c p Ω q and σ c : C p Ω q Ñ B c p Ω q by p ι c u qp ϕ, x q : “ x u, ϕ p . ´ x qy p u P D p Ω qqp σ c f qp ϕ, x q : “ f p x q p f P C p Ω qq . Partial derivatives on B c p Ω q then can be defined via differentiation in the secondvariable: Definition 22.
Let R P B c p Ω q . We define derivatives D i : B c p Ω q Ñ B c p Ω q ( i “ , . . . , n ) by p D i R qp ϕ, x q : “ BB x i p x ÞÑ R p ϕ, x qq . Theorem 23.
We have D i p M c p Ω qq Ď M c p Ω q and D i p N c p Ω qq Ď N c p Ω q ,Proof. This is evident from the definitions.
Proposition 24.
We have D i ˝ ι “ ι ˝ B i and D i ˝ σ “ σ ˝ B i . roof. D i p ιu qp ϕ, x q “ BB x i x u p y q , ϕ p y ´ x qy “ x u p y q , ´pB i ϕ qp y ´ x qy “ xB i u p y q , ϕ p y ´ x qy “ ι pB i u qp ϕ, x q . The second claim is clear. Proposition 25. N c p Ω q Ď M c p Ω q .Proof. The result follows from k ‹ ϕ ´ ~δ k K,c ; B ď λ p k ϕ k c q for suitable λ and c , which is seen as in the proof of Proposition 10.Similarly to Proposition 11 we have: Proposition 26. M c p Ω q is a subalgebra of B c p Ω q and N c p Ω q is an ideal in M c p Ω q . Theorem 27.
Let u P D p Ω q and f P C p Ω q . Then(i) ι c u is moderate,(ii) σ c f is moderate,(iii) ι c f ´ σ c f is negligible, and(iv) if ι c u is negligible then u “ . The proof is almost identical to that of Theorem 12 and hence omitted.
Definition 28.
We define the elementary Colombeau algebra of generalized func-tions on Ω by G c p Ω q : “ M c p Ω q{ N c p Ω q . As before, one may show that G c is a sheaf. In this section we show that the algebra G constructed above is near to beinguniversal in the sense that there exist canonical mappings from it into most of theclassical Colombeau algebras which are compactible with the embeddings.We begin by constructing a mapping G p Ω q Ñ G c p Ω q . Definition 29.
Given R P B p Ω q we define r R P B c p Ω q by r R p ϕ, x q : “ R p ~ϕ qp x q pp ϕ, x q P U p Ω qq where ~ϕ P C p Ω , D p Ω qq is chosen such that ~ϕ “ ‹ ϕ in a neighborhood of x . p ϕ, x q in U p Ω q we have supp ϕ p . ´ x q Ď Ω for x in a neighborhood V of x . Choosing ρ P D p Ω q with supp ρ Ď V and ρ ” x , we can take ~ϕ p x q : “ ρ ‹ ϕ . Obviously, r R p ϕ, x q does not dependon the choice of ~ϕ p x q and r R p ϕ, . q is smooth, so indeed we have r R P B c p Ω q . Proposition 30.
Let R P B p Ω q . Then the following holds:(i) Ă ιu “ ι c u for u P D p Ω q .(ii) Ă σf “ σ c f for f P C p Ω q .(iii) r R P M c p Ω q for R P M p Ω q .(iv) r R P N c p Ω q for R P N p Ω q .Proof. (i): For u P D p Ω q we have Ă ιu p ϕ, x q “ p ιu qp ~ϕ qp x q “ x u, ~ϕ p x qy “ x u, ‹ ϕ p x qy “ x u p y q , ϕ p y ´ x qy “ p ι c u qp ϕ, x q . (ii) is clear.(iii): Suppose that R P M p Ω q . Fixing x P Ω, we obtain U as in Proposition 8. Let K Ť L Ť U and m be given, set k “
0, and choose L such that L Ť L Ť U . ThenProposition 8 gives c, l, λ such that for ~ϕ P C p K, D L p U qq , k R p ~ϕ q k K,m ď λ p k ~ϕ k K,c ; L ,l q . Now for ϕ P D p R n q with K ` supp ϕ Ď L we have ‹ ϕ P C p K, D L p U qq , which gives k r R p ϕ, . q k K,m “ k R p ‹ ϕ q k K,m ď λ p k ‹ ϕ k K,c ; L ,l q ď λ p k ϕ k c ` l q which proves that r R P M c p Ω q .(iv): Similarly, if R P N p Ω q then for x P Ω we have U as in Proposition 8. For K Ť L Ť U , m given, k “
0, and L such that L Ť L Ť U , we obtain c, l, λ, B asin Proposition 8 such that k R p ~ϕ q k K,m ď λ p k ~ϕ k K,c ; L ,l , k ~ϕ ´ ~δ k K,c ; B q and hence k r R p ϕ, . q k K,m “ k R p ‹ ϕ q k K,m ď λ p k ‹ ϕ k K,c ; L ,l , k ‹ ϕ ´ ~δ k K,c ; B qď λ p k ϕ k c ` l , k ‹ ϕ ´ ~δ k K,c ; B q . which gives negligibility of r R . 17 .1 The special algebra We define the special Colombeau algebra G s with the embedding as in [Del05]: fixa mollifier ρ P S p R n q with ż ρ p x q d x “ , ż x α ρ p x q d x “ @ α P N n zt u . Choosing χ P D p R n q with 0 ď χ ď χ ” B p q and supp χ Ď B p q we set ρ ε p y q : “ ε ´ n ρ p y { ε q , θ ε p y q : “ ρ ε p y q χ p y | ln ε |q p ε ą q . Moreover, with K ε “ t x P Ω | d p x, R n z Ω q ě ε u X B { ε p q Ť Ω p ε ą q we choose functions κ ε P D p Ω q such that 0 ď κ ε ď κ ε ” K ε . Then thespecial algebra G s p Ω q is given by E s p Ω q : “ C p Ω q I with I : “ p , s , E sM p Ω q : “ tp u ε q ε P E s p Ω q | @ K Ť Ω @ m P N D N P N : k u ε k K,m “ O p ε ´ N qu , N s p Ω q : “ tp u ε q ε P E s p Ω q | @ K Ť Ω @ m P N @ N P N : k u ε k K,m “ O p ε N qu , G s p Ω q : “ E sM p Ω q{ N s p Ω q , p ι s u q ε : “ x u, ~ψ ε y p u P D p Ω qq , p σ s f q ε : “ f p f P C p Ω qq ,~ψ ε p x qp y q : “ θ ε p x ´ y q κ ε p y q . Definition 31.
For R P B p Ω q we define R s “ p R sε q ε P E s p Ω q by R sε p x q : “ R p ~ψ ε qp x q . Proposition 32. (i) p ιu q s “ ι s u for u P D p Ω q .(ii) p σf q s “ σ s f for f P C p Ω q .(iii) R s P E sM p Ω q for R P M p Ω q .(iv) R s P N s p Ω q for R P N p Ω q .Proof. (i) and (ii) are clear.For (iii) it suffices to show the needed estimate locally. Fix x P Ω, which gives U P U x p Ω q as in Proposition 8. Choose any K, L such that x P K Ť L Ť U , fix m , and set k “
0. Then there are c, l, λ as in Proposition 8. Because supp ~ψ ε p x q Ď B | ln ε | ´ p x q we have ~ψ ε P C p K, D L p U qq for ε small enough, which gives k R sε k K,m ď λ p k ~ψ ε k K,c ; L,l q . p R sε q ε P E sM p Ω q follows from k ~ψ ε k K,c ; L,l “ sup x,α,y,β ˇˇ B αx B βy ` ρ ε p x ´ y q χ pp x ´ y q | ln ε |q κ ε p y q ˘ˇˇ “ O p ε ´ n ´ c ´ l q . For negligibility we proceed similarly; the claim then follows by using that for abounded subset B Ď C p U q we have k ~ψ ε ´ ~δ k K,c ; B “ O p ε N q for all N P N , whichis seen as in [Del05, Prop. 12, p. 38] and actually merely a restatement of the factthat ι s f ´ σ s f “ O p ε N q for all N uniformly for f P B . There are several variants of the diffeomorphism invariant algebra G d ; we will employthe following formulation [Nig15; Nig16; GN17]: E d p Ω q : “ C p D p Ω q , C p Ω qq E dM p Ω q : “ t R P C p D p Ω qq | @ K Ť Ω @ k, m P N @p ~ϕ ε q ε P S p Ω q @p ~ψ ,ε q ε , . . . , p ~ψ k,ε q ε P S p Ω q D N P N : k d k R p ~ϕ ε qp ~ψ ,ε , . . . , ~ψ k,ε q k K,m “ O p ε ´ N qu , N d p Ω q : “ t R P C p D p Ω qq | @ K Ť Ω @ k, m P N @p ~ϕ ε q ε P S p Ω q @p ~ψ ,ε q ε , . . . , p ~ψ k,ε q ε P S p Ω q @ N P N : k d k R p ~ϕ ε qp ~ψ ,ε , . . . , ~ψ k,ε q k K,m “ O p ε N qu , G d p Ω q : “ E dM p Ω q{ N d p Ω q , p ι d u qp ϕ qp x q : “ x u, ϕ y , p σ d f qp ϕ qp x q : “ f p x q . The spaces S p Ω q and S p Ω q employed in this definition are given as follows: Definition 33.
Let a net of smoothing kernels p ~ϕ ε q ε P C p Ω , D p Ω qq I be given anddenote the corresponding net of smoothing operators by p Φ ε q ε P L p D p Ω q , C p Ω qq I .Then p ϕ ε q ε is called a test object on Ω if(i) Φ ε Ñ id in L p D p Ω q , D p Ω qq ,(ii) @ p P csn p L p D p Ω q , C p Ω qqq D N P N : p p Φ ε q “ O p ε ´ N q ,(iii) @ p P csn p L p C p Ω q , C p Ω qqq @ m P N : p p Φ ε | C p Ω q ´ id q “ O p ε m q ,(iv) @ x P Ω D V P U x p Ω q @ r ą D ε ą @ y P V @ ε ă ε : supp ϕ ε p y q Ď B r p y q .We denote the set of test objects on Ω by S p Ω q . Similarly, p ~ϕ ε q ε is called a -test object if it satisfies these conditions with (i) and (iii) replaced by the followingconditions:(i’) Φ ε Ñ in L p D p Ω q , D p Ω qq ,(iii’) @ p P csn p L p C p Ω q , C p Ω qqq @ m P N : p p Φ ε | C p Ω q q “ O p ε m q . he set of all -test objects on Ω is denoted by S p Ω q . Definition 34.
For R P B p Ω q we define R d P E d p Ω q by R d p ϕ qp x q : “ R pr x ÞÑ ϕ sqp x q . Proposition 35. (i) p ιu q d “ ι d u for u P D p Ω q .(ii) p σf q d “ σ d u for f P C p Ω q .(iii) R d P E dM p Ω q for R P M p Ω q .(iv) R d P N d p Ω q for R P N p Ω q .Proof. (i) and (ii) are clear from the definition. (iii) and (iv) follow directly fromthe estimates k ~ϕ ε k K,c ; L,l “ O p ε ´ N q for some N, k ~ϕ ε ´ ~δ k K,c ; B “ O p ε N q for all N, which hold by definition of the spaces S p Ω q and S p Ω q . For Colombeau’s elementary algebra we employ the formulation of [Gro+01, Section1.4], Section 1.4. For k P N we let A k p R n q be the set of all ϕ P D p R n q with integralone such that, if k ě
1, all moments of ϕ order up to k vanish. U e p Ω q : “ tp ϕ, x q P A p R n q ˆ Ω | x ` supp ϕ Ď Ω u E e p Ω q : “ t R : U e p Ω q Ñ C | @ ϕ P A p R n q : R p ϕ, . q is smooth u E eM p Ω q : “ t R P E e p Ω q | @ K Ť Ω @ m P N D N P N @ ϕ P A N p R n q : k R p S ε ϕ, . q k K,m “ O p ε ´ N qu N e p Ω q : “ t R P E e p Ω q | @ K Ť Ω @ m P N @ N P N D q P N @ ϕ P A q p R n q : k R p S ε ϕ, . q k K,m “ O p ε N qu G e p Ω q : “ E eM p Ω q{ N e p Ω qp ι e u qp ϕ, x q : “ x u, ϕ p . ´ x qyp σ e f qp ϕ, x q : “ f p x q Definition 36.
For R P B c p Ω q we define R e P E e p Ω q by R e p ϕ, x q : “ R p ϕ, x q . Proposition 37. (i) p ι c u q e “ ι e u for u P D p Ω q .(ii) p σ c f q e “ σ e u for f P C p Ω q .(iii) R e P E eM p Ω q for R P M c p Ω q .(iv) R e P N e p Ω q for R P N c p Ω q . roof. Again, (i) and (ii) are clear from the definition. For (iii), fix K Ť Ω and m P N . From Proposition 20 we obtain r , c and λ such that for supp ϕ Ď B r p q , k R p ϕ, . q k K,m ď λ p k ϕ k c q . For ϕ P A p R n q and ε small enough, supp S ε ϕ Ď B r p q ,so we only have to take into account that k S ε ϕ k c “ O p ε ´ N q for some N P N .Similarly, (iv) is obtained from the fact that given any N , for q large enough wehave k p S ε ϕ q ˚ ´ ~δ k K,c ; B “ O p ε N q for all ϕ P A q p R n q . Acknowledgments.
This research was supported by project P26859-N25 of theAustrian Science Fund (FWF).
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