Comment.Math.Univ.Carolin. 38,4 (1997)627–634 627 Results on Colombeau product of distributions Blagovest Damyanov Abstract. The differential C-algebra G(Rm) of generalized functions of J.-F. Colombeau contains the space D′(Rm) of Schwartz distributions as a C-vector subspace and has a notion of ‘association’ that is a faithful generalization of the weak equality in D′(Rm). This is particularly useful for evaluation of certain products of distributions, as they are embedded in G(Rm), in terms of distributions again. In this paper we propose some a (p) results of that kind for the products of the widely used distributions x± and δ (x), with x in Rm, that have coinciding singular supports. These results, when restricted to dimension one, are also easily transformed into the setting of regularized model products in the classical distribution theory. Keywords: multiplication of Schwartz distributions, Colombeau generalized functions Classification: 46F10 1. Notation and definitions We will recall first the basic definitions of Colombeau algebra G(Rm), following their recent presentation in [7, Chapter 3]. Notation 1. If N0 stands for the nonnegative integers and p = (p1,p2,...,pm) Nm m is a multiindex in 0 , we let |p| = i=1 pi and p! = p1! ...pm!. Then, if Rm p p1 p2 pm p x = (x1,...,xm) is in , we denote by x = (x1 , x2 ,...,xm ) and ∂ x = |p| p1 pm P ∂ /∂x1 ...∂xm . Also, by x < 0 is meant: x1 ≤ 0,...,xm ≤ 0 and x 6= 0. j Now for any q in N0, denote by Aq(R)= {ϕ(x) ∈ D(R) : R x ϕ(x) dx = δ0j for m 0 ≤ j ≤ q, where δ00 = 1, δ0j = 0 for j > 0}. This also extends to R as an Rm Rm R m m-fold tensor product: Aq( )= {ϕ(x) ∈ D( ) : ϕ(x1,...,xm)= i=1 χ(xi) −m −1 for some χ in Aq(R)}. Finally, we will denote by ϕε = ε ϕ(ε x), for any ϕ m Q in Aq(R ) and ε> 0. m m Definition 1. Let E [R ] stand for the set of functions f(ϕ, x) : A0(R ) × Rm→ C that are C∞-differentiable with respect to x by a fixed ‘parameter’ ϕ, which, with the point-wise function operations, is clearly a C-algebra. Then each generalized function of Colombeau is an element of the quotient algebra m m m m m G(R ) = EM [R ] / I [R ]. Here the subalgebra EM [R ] of E [R ] is the set of ‘moderate’ functions f(ϕ, x) in E [Rm] such that for each compact subset K of Rm Nm N Rm and any p in 0 there is a q in so that : for each ϕ in Aq( ) there are p −q c > 0, η > 0 satisfying supx∈K |∂ f(ϕε, x) | ≤ cε for 0 <ε<η. In turn, the m m ideal I [R ] of EM [R ] is the set of functions f(ϕ, x) such that for each compact 628 B. Damyanov Rm Nm N subset K of and any p in 0 there is q in so that: for every r ≥ q and Rm p r−q each ϕ in Ar( ) there are c> 0, η> 0 satisfying supx∈K |∂ f(ϕε, x) |≤ cε , for 0 <ε<η. The Colombeau algebra G(Rm) contains all distributions (and C∞-differenti- able functions) on Rm, canonically embedded as a C-vector subspace (respectively, a subalgebra) by the map i : D′(Rm)→ G(Rm) : u 7→ u˜ = [˜u(ϕ, x)]. The repre- sentatives here are given byu ˜(ϕ, x) = (u ∗ ϕˇ)(x), withϕ ˇ(x) = ϕ(−x) and ϕ m in Aq(R ). Equivalently, one writesu ˜(ϕ, x) = huy, ϕ(y − x)i. A basic example is the embedding δ˜ in G(Rm) of the Dirac δ-function given by a representative m ˜δ(ϕ, x)= hδy, ϕ(y − x)i = ϕ(−x), for any ϕ in Aq(R ). Definition 2. A generalized function f in G(Rm) is said to admit some u in D′(Rm) as associated distribution, which is denoted by f ≈ u, if f has a repre- m m sentative f(ϕε, x) in EM [R ] such that for any test-function ψ(x) in D(R ) there m exists q in N0 so that, for all ϕ(x) in Aq(R ), lim f(ϕε, x)ψ(x)dx = hu, ψi. ε→0 Rm Z This definition is independent of the representative chosen and the distribution associated is unique if it exists; the image in G(Rm) of every distribution is asso- ciated with that distribution ([1]). The concept of association is thus a faithful generalization of the equality of distributions in D′(Rm). Now by ‘Colombeau product of distributions’ is denoted the product of some distributions as they are embedded in Colombeau algebra G(Rm) whenever the result admits an associated distribution in D′(Rm) (see [5] for a comparison with other distribution products). This notion helps to bring the results ‘down to the level’ of distributions, connecting thus Colombeau theory with the classical distri- bution theory. Below we give some results on products of distribution with coin- ciding singularities in Colombeau algebra G(Rm), or else — on their Colombeau product. 2. Preliminary results The technical lemmas below will be needed later to prove our main results. Lemma 1. For an arbitrary ϕ in A0(R), i.e. ϕ in D(R) with R ϕ(t) dt = 1, suppose that supp ϕ ⊆ [a,b], for some a,b in R. Then, for any p in N , it holds R 0 b t (−1)p p! (1) ϕ(t) (y − t)pϕ(p)(y) dy dt = . 2 Za Za Results on Colombeau product of distributions 629 Proof: On expanding the term (y − t)p on the l.h.s. of (1) and then on multiple integrating by parts, we get : p b t j p j p−j (p) Ip ≡ (−1) t ϕ(t) y ϕ (y) dy dt j a a jX=0 Z Z p p−j p b (p − j)! = (−1)j tj ϕ(t) (−1)k tp−j−kϕ(p−k−1)(t) dt j a (p − j − k)! jX=0 Z kX=0 p p−j p! b = (−1)j+k tp−kϕ(t)ϕ(p−k−1)(t) dt j!(p − j − k)! a jX=0 kX=0 Z p p−k p! p = (−1)k J (−1)j . (p − k)! p−k j kX=0 jX=0 b p−k (p−k−1) Here we have denoted by Jp−k = a t ϕ(t)ϕ (t) dt, where, if k = p, (−1) t ϕ (t) stands for a ϕ(y) dy. For anyR q = p − k > 0, however, it holds (see [6, §21.5–1(b)]) : q (−1)j q = 0. Whence I = (−1)p p! J . As for the R j=0 j p 0 remaining term with p − k = j = 0, we get, by our assumption, P b t b t t J0 = ϕ(t) ϕ(y) dy dt = ϕ(y) dy d ϕ(y) dy Za Za Za Za Za b 1 t 2 1 = ϕ(y) dy = . 2 a 2 Z a This proves equation (1). ′ Rm m i Lemma 2. Let u and v be distributions in D ( ) such that u(x)= i=1 u (xi), v(x) = m vi(x ) with each ui and vi in D′(R), and suppose that their em- i=1 i Q beddings in G(R) satisfy u˜i.v˜i ≈ wi, for i = 1,... ,m. Then u.˜ v˜ ≈ w, where mQ i w = i=1 w (xi). Proof: Q Suppose we have confined ourselves to the subspace of test-functions m R ψ(x) = i=1 ψi(xi), with each ψi in D( ). In view of the tensor-product struc- ture of the distributions u, v in D′(Rm) as well as that of the elements ϕ of m Q A0(R ), by applying a Fubini-type theorem for tensor-product distributions (see [4, §4.3]), we get : m i i hu˜(ϕε, x)˜v(ϕε, x), ψ(x)i = hu˜ (χε, xi)˜v (χε, xi), ψi(xi)i i=1 Ym i i = hw (xi), ψi(xi)i + f (ε) . iY=1 630 B. Damyanov Here, by assumption, one has the asymptotic evaluation f i(ε) = o(1) (ε→0) for each i =1,... ,m. Thus m i i lim hu˜(ϕε, x)˜v(ϕε, x), ψ(x)i = hw , ψ i = hw, ψi, ε→0 iY=1 m i ′ Rm where w = i=1 w (xi) is a uniquely determined distribution in D ( ). More- m Rm over, since ψ(x)= i=1 ψi(xi) is running a dense subset of D( ) ([4, §4.3]), it follows, byQ Definition 2, that the productu. ˜ v˜ in G(Rm) admits w as associated distribution. Q 3. Main results Nm ˜(p) p Proposition 1. For an arbitrary p in 0 , let δ (x) and x˜+ be the embeddings Rm (p) p p in G( ) of the distributions δ (x) and x+ = {x for x ≥ 0, =0 elsewhere} in D′(Rm). Then (−1)|p| p! (2)x ˜p . ˜δ(p)(x) ≈ δ(x). + 2m Proof: R N p In the one-variable case (x ∈ ,p ∈ 0),x ˜+ is represented by ∞ ∞ p −1 p p x˜+(ϕε, x)= ε y ϕ((y − x)/ε) dy = (x + εt) ϕ(t) dt, Z0 Z−x/ε where the substitution (y − x)/ε = t is made. Also, on differentiation in D′(R), we have (p) p −p−1 (p) p −p−1 (p) ˜δ (ϕε, x) = (−1) ε hδy, ϕ ((y − x)/ε)i = (−1) ε ϕ (−x/ε).