We investigate density of various subalgebras of regular generalized
functions in the special Colombeau algebra of generalized functions.Comment: 6 page

A Delcroix

C Garetto

G Hörmann

H Vernaeve

H. Vernaeve

J Aragona

N Dapić

S Pilipović

English

arXiv

Crossref

Monatshefte für Mathematik

Topological properties of regular generalized function algebras

We investigate the topological density of various subalgebras of regular generalized functions in the Colombeau algebra G(Ω) of generalized functions with its natural (so-called sharp) topology

Vernaeve, Hans

Ghent University Academic Bibliography

arXiv:1011.2190v1  [math.FA]  9 Nov 2010Topological properties of regular generalizedfunction algebrasH. Vernaeve∗2000 Mathematics subject classification: 46F30.AbstractWe investigate density of various subalgebras of regular generalized functionsin the special Colombeau algebra G(Ω) of generalized functions.1 IntroductionM. Oberguggenberger introduced the algebra G∞(Ω) of regular generalized functionsin order to develop a hypoelliptic regularity theory and hyperbolic propagation of sin-gularities in the algebra G(Ω) of Colombeau generalized functions [13], where it takesover the role of the subalgebra of C∞-regular functions in the space D′(Ω) of distri-butions. It thus became the starting point of investigations of microlocal regularityin generalized function algebras (see [5, 7, 9, 10, 12, 16] and the references therein).More recently, various other subalgebras of regular generalized functions have beenconsidered, from the point of view of generalized analytic functions [1], kernel theo-rems [3], propagation of singularities [14] and microlocal analysis [4]. We show that,in contrast with the situation of C∞(Ω) as a subalgebra of D′(Ω) (and therefore maybesurprisingly), the subalgebra G∞(Ω) and the subalgebras GLa(Ω) considered in [3, 4]are not dense in the algebra G(Ω). On the other hand, the subalgebra of sublinear orS-analytic generalized functions is dense in G(Ω).2 NotationsLet Ω ⊆ Rd be open. By K ⊂⊂ Ω, we denote a compact subset of Ω.For u ∈ C∞(Ω), K ⊂⊂ Ω and α ∈ Nd, let pα,K(u) := supx∈K |∂αu(x)|. For k ∈ N, letpk,K(u) := max|α|=k pα,K(u).The special algebra of Colombeau generalized functions (see e.g. [8]) is G(Ω) :=EM(Ω)/N (Ω), whereEM(Ω) ={(uε)ε ∈ C∞(Ω)(0,1) : (∀K ⊂⊂ Ω)(∀α ∈ Nd)(∃N ∈ N)(pα,K(uε) ≤ ε−N , for small ε)}N (Ω) ={(uε)ε ∈ C∞(Ω)(0,1) : (∀K ⊂⊂ Ω)(∀α ∈ Nd)(∀m ∈ N)(pα,K(uε) ≤ εm, for small ε)}.∗Dept. Of Mathematics, Ghent University. E-mail: hvernaev@cage.ugent.be1By [(uε)ε], we denote the generalized function with representative (uε)ε ∈ EM(Ω).The subalgebra Gc(Ω) of compactly supported generalized functions consists of thoseu ∈ G(Ω) such that for some K ⊂⊂ Ω, the restriction of u to Ω \K equals 0 (as anelement of G(Ω \K)).For K ⊂⊂ Ω, the algebra G∞(K) consists of those u ∈ G(Ω) such that for one (andhence for each) representative (uε)ε,(∃N ∈ N)(∀α ∈ Nd)(pα,K(uε) ≤ ε−N , for small ε).For (zε)ε ∈ C(0,1), the valuation v(zε) := sup{b ∈ R : |zε| ≤ εb, for small ε} and theso-called sharp norm |zε|e := e− v(zε). For u ∈ G(Ω), Pα,K(u) := |pα,K(uε)|e (α ∈ Nd)and Pk,K(u) := |pk,K(uε)|e (k ∈ N), independent of the representative (uε)ε of u. Theultra-pseudo-seminorms Pα,K (α ∈ Nd, K ⊂⊂ Ω) determine a topology on G(Ω) calledsharp topology [2, 6, 16]. Then u ∈ G∞(K) iff supk∈N Pk,K(u) < +∞. Further, thealgebra G∞(Ω) :=⋂K⊂⊂Ω G∞(K) [13].For K ⊂⊂ Ω, the algebra GLa(K) of generalized functions of sublinear growth withslope smaller than a > 0 (a ∈ R) on K consists of those u ∈ G(Ω) such that for one(and hence for each) representative (uε)ε,(∃a′ < a)(∃b ∈ R)(∀α ∈ Nd)(pα,K(uε) ≤ ε−a′|α|−b, for small ε)or, equivalently,(∃a′ < a)(∃c ∈ R)(Pα,K(u) ≤ cea′|α|, ∀α ∈ Nd),which can still be expressed concisely by lim supk→∞lnPk,K(u)k< a. Since Pα,K(uv) ≤maxβ≤α(Pβ,K(u)Pα−β,K(v)) by Leibniz’s rule, GLa(K) are subalgebras of G(Ω).For a = 0, GL0(K) :=⋂a>0 GLa(K). Clearly, G∞(K) ⊆ GL0(K).Again, the algebras GLa(Ω) :=⋂K⊂⊂Ω GLa(K) (a ≥ 0) [3, 4]. Clearly, G∞(Ω) ⊆ GL0(Ω).By definition, u = [(uε)ε] ∈ G(Ω) is sublinear [1, 15] iff for each K ⊂⊂ Ω and each(xε)ε ∈ K(0,1), there exists k ∈ R and (pn)n∈N ∈ RN such that limn→∞ pn + kn = ∞and for each α ∈ Nd, |∂αuε(xε)| ≤ εp|α| , for small ε. It can be shown [1, Thm. 5.7],[15, Thm. 10] that the algebra of sublinear generalized functions exactly contains thoseu ∈ G(Ω) satisfying a natural condition of analyticity (called S-real analyticity in [15]).Sublinearity can still be characterized as follows by means of the algebras GLa(K):Lemma 2.1. Let u ∈ G(Ω). Then u is sublinear iff for each K ⊂⊂ Ω, there existsa > 0 (a ∈ R) such that u ∈ GLa(K).Proof. ⇒: let u be sublinear and suppose that there exists K ⊂⊂ Ω such thatu /∈ GLa(K), for each a > 0. Then we find αn ∈ N (for each n ∈ N), εn,m ∈(0, 1/m) (for each n,m ∈ N) (by enumerating the countable family (εn,m)n,m, we cansuccessively choose the εn,m such that they are all different) and xεn,m ∈ K suchthat∣∣∂αnuεn,m(xεn,m)∣∣ > ε−n|αn|−nn,m , for each n,m ∈ N. Let xε ∈ K arbitrary ifε ∈ (0, 1) \ {εn,m : n,m ∈ N}. By assumption, there exist k ∈ R, (pn)n∈N ∈ RNand N ∈ N such that for each α ∈ Nd with |α| ≥ N , |∂αuε(xε)| ≤ εp|α| ≤ ε−k|α|, forsmall ε. Since u ∈ G(Ω), it follows that there exists b ∈ R such that for each α ∈ Nd,|∂αuε(xε)| ≤ ε−k|α|−b, for small ε. This contradicts the fact that for n ∈ N with n ≥ kand n ≥ b, limm εn,m = 0 and |∂αnuεn,m(xεn,m)| > ε−n|αn|−nn,m , ∀m ∈ N.2⇐: let K ⊂⊂ Ω and (xε)ε ∈ K(0,1). By assumption, there exist a, b ∈ R such that foreach α ∈ Nd, pα,K(uε) ≤ ε−a|α|−b, for small ε. Then, for k := a+1 and pn := −an− b,limn pn + kn =∞ and for each α ∈ Nd, |∂αuε(xε)| ≤ pα,K(uε) ≤ εp|α| , for small ε.3 G∞(Ω) and GL0(Ω)Our method is based upon a quantitative version of an argument used in [8, Thm. 1.2.3](cf. also [10, Prop. 1.6] and [17]), which can in fact be traced back to [11].Proposition 3.1. Let K ⊂⊂ Ω ⊆ Rd. Suppose that there exists r ∈ R+ such that foreach x ∈ K, there exist d line segments of length r containing x in linearly independentdirections that are contained in K. Let u ∈ G(Ω). If for some k ∈ N \ {0}, Pk,K(u) >Pk−1,K(u), then P2k,K(u) ≤ Pk−1,K(u)Pk+1,K(u).Proof. Let first k = 1. Let x ∈ K. Let e1, . . . , ed ∈ Rd be linearly independent unitvectors such that the line segments [x, x+ r2ej ] ⊆ K. Denote the directional derivativein the direction ej by ∂ej . Let a ∈ R, a > 0. For ε ∈ (0, 1), by Taylor’s formula thereexist θε ∈ [0, 1] such that∂ejuε(x) = ε−auε(x+ εaej)− ε−auε(x) +εa2∂2ejuε(x+ εaθεej).Hence for ε ≤ ε0 (where ε0 does not depend on x ∈ K),∣∣∂ejuε(x)∣∣ ≤ 2ε−a supy∈K|uε(y)|+ εa supy∈K∣∣∂2ejuε(y)∣∣ ≤ 2ε−ap0,K(uε) + εap2,K(uε).Since e1, . . . , ed are linearly independent, we can write ∂1, . . . , ∂d as a linear com-bination (with coefficients independent of ε and x) of ∂e1 , . . . , ∂ed. Thus thereexists C ∈ R such that p1,K(uε) ≤ Cε−ap0,K(uε) + Cεap2,K(uε), and P1,K(u) ≤max(eaP0,K(u), e−aP2,K(u)). Should P2,K(u) ≤ P0,K(u), then letting a → 0 wouldyield P1,K(u) ≤ P0,K(u), contradicting the hypotheses. Hence P2,K(u) > P0,K(u), andwe can choose a > 0 such that e2a = P2,K(u)/P0,K(u) (since the case P0,K(u) = 0 istrivial).If k ∈ N \ {0} arbitrary, the same reasoning can be applied to all ∂αu with |α| = k− 1instead of u.Corollary 3.2. (cf. [8, Thm. 1.2.3]) Let K ⊂⊂ Ω ⊆ Rd. Suppose that there existsr ∈ R+ such that for each x ∈ K, there exist d line segments of length r containing xin linearly independent directions that are contained in K. Let u ∈ G(Ω). If for somek ∈ N, Pk,K(u) = 0, then Pl,K(u) = 0, ∀l ≥ k.Proof. If Pk+1,K(u) 6= 0, then Pk+1,K(u)2 ≤ Pk,K(u)Pk+2,K(u) = 0 by proposition 3.1,a contradiction. The result follows inductively.Proposition 3.3. Let K ⊂⊂ Ω satisfy the hypothesis of proposition 3.1. Let u ∈GL0(K). Then Pk,K(u) are decreasing in k, andG∞(K) = GL0(K) = {u ∈ G(Ω) : Pk,K(u) ≤ P0,K(u), ∀k ∈ N}.In particular, G∞(K) is closed in G(Ω).3Proof. Let u ∈ G(Ω). If Pk,K(u) are not decreasing in k, then there exists k ∈ N \ {0}such that Pk,K(u) > Pk−1,K(u) > 0 by corollary 3.2. Let r := Pk,K(u)/Pk−1,K(u) >1. By proposition 3.1, Pk+1,K(u) ≥ rPk,K(u) (in particular, Pk+1,K(u) > Pk,K(u)).Inductively, Pk+n,K(u) ≥ rnPk,K(u), for each n ∈ N. Thus lim supn→∞lnPn+k,K(u)n+k≥lim supn→∞ln(rnPk,K(u))n+k= ln r > 0, and u /∈ GL0(K). In particular, u /∈ G∞(K).The fact that G∞(K) is closed follows by continuity of Pk,K .Theorem 3.4. G∞(Ω) = GL0(Ω) is closed in G(Ω). In particular, G∞(Ω) is not densein G(Ω).Proof. G∞(Ω) =⋂K G∞(K), where K runs over all compact subsets of Ω that are afinite union of d-dimensional cubes parallel with the coordinate axes (hence satisfyingthe hypothesis of proposition 3.1), and similarly for GL0(Ω). The conclusions followfrom proposition 3.3.4 GLa(Ω), a > 0Proposition 4.1. Let K ⊂⊂ Ω satisfy the hypothesis of proposition 3.1. Let a ∈ R,a ≥ 1. Then{u ∈ G(Ω) : (∃c ∈ R)(Pk,K(u) ≤ cak, ∀k ∈ N)}= {u ∈ G(Ω) : Pk+1,K(u) ≤ aPk,K(u), ∀k ∈ N}.In particular, this describes a closed subset of G(Ω).Proof. Let u ∈ G(Ω). If Pk+1,K(u) > aPk,K(u), for some k ∈ N, then Pk,K(u) > 0by corollary 3.2. Let r := Pk+1,K(u)/Pk,K(u) > a. By proposition 3.1, Pk+n,K(u) ≥rnPk,K(u), for each n ∈ N. Thus lim supn∈N Pn,K(u)/an ≥ lim supn∈Nrn−kPk,K(u)an=+∞.The other inclusion is clear.Theorem 4.2. Let a ∈ R, a > 0. Then GLa(Ω) is not dense in G(Ω).Proof. GLa(Ω) ⊆⋂K{u ∈ G(Ω) : (∃c ∈ R)(Pk,K(u) ≤ ceak, ∀k ∈ N)} =: A, whereK runs over all compact subsets of Ω that are a finite union of d-dimensional cubesparallel with the coordinate axes. The set A is closed by proposition 4.1 and is a strictsubset of G(Ω).5 Sublinear generalized functionsIn order to investigate the density of the algebra of sublinear generalized functions, westart with the following proposition (see also [16, Prop. 4.3.1]):Proposition 5.1. Let ψ ∈ C∞(Rd) with ψ(x) = 0 if |x| ≥ 1 and∫Rdψ = 1. Denoteby ψε(x) := ε−dψ(x/ε), for each ε ∈ (0, 1). If u = [(uε)ε] ∈ G(Ω), then limn→∞[(uε ⋆ψεn)ε] = u.4Proof. Let n ∈ N and K ⊂⊂ Ω. Then uε ⋆ ψεn(x) =∫|t|≤εnuε(x − t)ψεn(t) dt iswell-defined as soon as d(x,Rd \ Ω) > εn. For small ε, d(K,R \ Ω) > εn and thus(uε ⋆ ψεn)|K can be extended to a C∞(Ω)-function. Independent of the extension, bythe mean value theorem,pk,K(uε ⋆ ψεn − uε) = supx∈K,|α|=k|(∂αuε) ⋆ ψεn(x)− ∂αuε(x)|= supx∈K,|α|=k∣∣∣∣∫|t|≤εn(∂αuε(x− t)− ∂αuε(x))ψεn(t) dt∣∣∣∣ ≤ εnpk+1,K+r(uε)∫Rd|ψ|for small ε, where r > 0 (r ∈ R) such that K + r = {x ∈ Rd : d(x,K) ≤ r} ⊂⊂ Ω.Proposition 5.2. Let A be the set of all u = [(uε)ε] ∈ Gc(Ω) for which(∃N ∈ N)(∀K ⊂⊂ Ω)(∀k ∈ N)(pk,K(uε) ≤ ε−Nk−N , for small ε).Then A is dense in G(Ω).Proof. Let u ∈ Gc(Ω). Then there exists a representative (uε)ε of u and L ⊂⊂ Ω suchthat supp uε ⊆ L, for each ε. For each K ⊂⊂ Ω and k ∈ N,pk,K(uε ⋆ ψεn) = supx∈K,|α|=k|uε ⋆ ∂α(ψεn)|≤ ε−nk supx∈L|uε(x)|max|α|=k∫Rd|∂αψ| ≤ ε−nk−1 supx∈L|uε(x)| ,for small ε. Thus [(uε ⋆ ψεn)ε] ∈ A. By proposition 5.1, A is dense in Gc(Ω). Further,Gc(Ω) is dense in G(Ω) (for u ∈ G(Ω), u = limn→∞ uχn, where χn ∈ D(Ω) withχn(x) = 1, ∀x ∈ Kn, where (Kn)n∈N is a compact exhaustion of Ω).Theorem 5.3. The subalgebra of sublinear generalized functions is dense in G(Ω).Proof. With the notations of proposition 5.2, A ⊆ {u ∈ G(Ω) : u is sublinear}.Acknowledgment. We are grateful to D. Scarpale´zos for very useful discussions.References[1] J. Aragona, R. Fernandez, S. O. Juriaans, A discontinuous Colombeau differentialcalculus, Monatsh. Math. 144, 1329 (2005).[2] H. Biagioni, A Nonlinear Theory of Generalized Functions, Lec. Notes Math.1421, Springer, 1990.[3] A. Delcroix, Generalized integral operators and Schwartz kernel type theorems, J.Math. Anal. Appl. 306(2), 481–501 (2005).[4] A. Delcroix, Regular rapidly decreasing nonlinear generalized functions. Applica-tion to microlocal regularity, J. Math. Anal. Appl. 327(1), 564–584 (2007).5[5] N. Dapic´, S. Pilipovic´ and D. Scarpale´zos, Microlocal analysis of Colombeau’sgeneralized functions: propagation of singularities, J. Analyse Math. 75, 51–66(1998).[6] C. Garetto, Topological structures in Colombeau algebras: topological C˜-modulesand duality theory, Acta Appl. Math. 88(1), 81–123 (2005).[7] C. 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Topological properties of regular generalized function algebras

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