Topological properties of regular generalized function algebras
aa r X i v : . [ m a t h . F A ] N ov Topological properties of regular generalizedfunction algebras
H. Vernaeve ∗ Abstract
We investigate density of various subalgebras of regular generalized functionsin the special Colombeau algebra G (Ω) of generalized functions. M. 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 G L a (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). Let Ω ⊆ R d be open. By K ⊂⊂ Ω, we denote a compact subset of Ω.For u ∈ C ∞ (Ω), K ⊂⊂ Ω and α ∈ N d , let p α,K ( u ) := sup x ∈ K | ∂ α u ( x ) | . For k ∈ N , let p k,K ( u ) := max | α | = k p α,K ( u ).The special algebra of Colombeau generalized functions (see e.g. [8]) is G (Ω) := E M (Ω) / N (Ω), where E M (Ω) = (cid:8) ( u ε ) ε ∈ C ∞ (Ω) (0 , : ( ∀ K ⊂⊂ Ω)( ∀ α ∈ N d )( ∃ N ∈ N )( p α,K ( u ε ) ≤ ε − N , for small ε ) (cid:9) N (Ω) = (cid:8) ( u ε ) ε ∈ C ∞ (Ω) (0 , : ( ∀ K ⊂⊂ Ω)( ∀ α ∈ N d )( ∀ m ∈ N )( p α,K ( u ε ) ≤ ε m , for small ε ) (cid:9) . ∗ Dept. Of Mathematics, Ghent University. E-mail: [email protected]
1y [( u ε ) ε ], we denote the generalized function with representative ( u ε ) ε ∈ E M (Ω).The subalgebra G c (Ω) of compactly supported generalized functions consists of those u ∈ 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 ( u ε ) ε ,( ∃ N ∈ N )( ∀ α ∈ N d ) (cid:0) p α,K ( u ε ) ≤ ε − N , for small ε (cid:1) . For ( z ε ) ε ∈ C (0 , , the valuation v( z ε ) := sup { b ∈ R : | z ε | ≤ ε b , for small ε } and theso-called sharp norm | z ε | e := e − v( z ε ) . For u ∈ G (Ω), P α,K ( u ) := | p α,K ( u ε ) | e ( α ∈ N d )and P k,K ( u ) := | p k,K ( u ε ) | e ( k ∈ N ), independent of the representative ( u ε ) ε of u . Theultra-pseudo-seminorms P α,K ( α ∈ N d , K ⊂⊂ Ω) determine a topology on G (Ω) calledsharp topology [2, 6, 16]. Then u ∈ G ∞ ( K ) iff sup k ∈ N P k,K ( u ) < + ∞ . Further, thealgebra G ∞ (Ω) := T K ⊂⊂ Ω G ∞ ( K ) [13].For K ⊂⊂ Ω, the algebra G L a ( K ) of generalized functions of sublinear growth withslope smaller than a > a ∈ R ) on K consists of those u ∈ G (Ω) such that for one(and hence for each) representative ( u ε ) ε ,( ∃ a ′ < a )( ∃ b ∈ R )( ∀ α ∈ N d )( p α,K ( u ε ) ≤ ε − a ′ | α |− b , for small ε )or, equivalently, ( ∃ a ′ < a )( ∃ c ∈ R )( P α,K ( u ) ≤ ce a ′ | α | , ∀ α ∈ N d ) , which can still be expressed concisely by lim sup k →∞ ln P k,K ( u ) k < a . Since P α,K ( uv ) ≤ max β ≤ α ( P β,K ( u ) P α − β,K ( v )) by Leibniz’s rule, G L a ( K ) are subalgebras of G (Ω).For a = 0, G L ( K ) := T a> G L a ( K ). Clearly, G ∞ ( K ) ⊆ G L ( K ).Again, the algebras G L a (Ω) := T K ⊂⊂ Ω G L a ( K ) ( a ≥
0) [3, 4]. Clearly, G ∞ (Ω) ⊆ G L (Ω).By definition, u = [( u ε ) ε ] ∈ G (Ω) is sublinear [1, 15] iff for each K ⊂⊂ Ω and each( x ε ) ε ∈ K (0 , , there exists k ∈ R and ( p n ) n ∈ N ∈ R N such that lim n →∞ p n + kn = ∞ and for each α ∈ N d , | ∂ α u ε ( x ε ) | ≤ ε p | α | , for small ε . It can be shown [1, Thm. 5.7],[15, Thm. 10] that the algebra of sublinear generalized functions exactly contains those u ∈ 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 G L a ( K ): Lemma 2.1.
Let u ∈ G (Ω) . Then u is sublinear iff for each K ⊂⊂ Ω , there exists a > ( a ∈ R ) such that u ∈ G L a ( K ) .Proof. ⇒ : let u be sublinear and suppose that there exists K ⊂⊂ Ω such that u / ∈ G L a ( K ), for each a >
0. Then we find α n ∈ N (for each n ∈ N ), ε n,m ∈ (0 , /m ) (for each n, m ∈ N ) (by enumerating the countable family ( ε n,m ) n,m , we cansuccessively choose the ε n,m such that they are all different) and x ε n,m ∈ K suchthat (cid:12)(cid:12) ∂ α n u ε n,m ( x ε n,m ) (cid:12)(cid:12) > ε − n | α n |− nn,m , for each n, m ∈ N . Let x ε ∈ K arbitrary if ε ∈ (0 , \ { ε n,m : n, m ∈ N } . By assumption, there exist k ∈ R , ( p n ) n ∈ N ∈ R N and N ∈ N such that for each α ∈ N d with | α | ≥ N , | ∂ α u ε ( x ε ) | ≤ ε p | α | ≤ ε − k | α | , forsmall ε . Since u ∈ G (Ω), it follows that there exists b ∈ R such that for each α ∈ N d , | ∂ α u ε ( x ε ) | ≤ ε − k | α |− b , for small ε . This contradicts the fact that for n ∈ N with n ≥ k and n ≥ b , lim m ε n,m = 0 and | ∂ α n u ε n,m ( x ε n,m ) | > ε − n | α n |− nn,m , ∀ m ∈ N .2 : let K ⊂⊂ Ω and ( x ε ) ε ∈ K (0 , . By assumption, there exist a, b ∈ R such that foreach α ∈ N d , p α,K ( u ε ) ≤ ε − a | α |− b , for small ε . Then, for k := a + 1 and p n := − an − b ,lim n p n + kn = ∞ and for each α ∈ N d , | ∂ α u ε ( x ε ) | ≤ p α,K ( u ε ) ≤ ε p | α | , for small ε . G ∞ (Ω) and G L (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 ⊆ R d . 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 \ { } , P k,K ( u ) >P k − ,K ( u ) , then P k,K ( u ) ≤ P k − ,K ( u ) P k +1 ,K ( u ) .Proof. Let first k = 1. Let x ∈ K . Let e , . . . , e d ∈ R d be linearly independent unitvectors such that the line segments [ x, x + r e j ] ⊆ K . Denote the directional derivativein the direction e j by ∂ e j . Let a ∈ R , a >
0. For ε ∈ (0 , θ ε ∈ [0 ,
1] such that ∂ e j u ε ( x ) = ε − a u ε ( x + ε a e j ) − ε − a u ε ( x ) + ε a ∂ e j u ε ( x + ε a θ ε e j ) . Hence for ε ≤ ε (where ε does not depend on x ∈ K ), (cid:12)(cid:12) ∂ e j u ε ( x ) (cid:12)(cid:12) ≤ ε − a sup y ∈ K | u ε ( y ) | + ε a sup y ∈ K (cid:12)(cid:12) ∂ e j u ε ( y ) (cid:12)(cid:12) ≤ ε − a p ,K ( u ε ) + ε a p ,K ( u ε ) . Since e , . . . , e d are linearly independent, we can write ∂ , . . . , ∂ d as a linear com-bination (with coefficients independent of ε and x ) of ∂ e , . . . , ∂ e d . Thus thereexists C ∈ R such that p ,K ( u ε ) ≤ Cε − a p ,K ( u ε ) + Cε a p ,K ( u ε ), and P ,K ( u ) ≤ max( e a P ,K ( u ) , e − a P ,K ( u )). Should P ,K ( u ) ≤ P ,K ( u ), then letting a → P ,K ( u ) ≤ P ,K ( u ), contradicting the hypotheses. Hence P ,K ( u ) > P ,K ( u ), andwe can choose a > e a = P ,K ( u ) /P ,K ( u ) (since the case P ,K ( u ) = 0 istrivial).If k ∈ N \ { } arbitrary, the same reasoning can be applied to all ∂ α u with | α | = k − u . Corollary 3.2. (cf. [8, Thm. 1.2.3]) Let K ⊂⊂ Ω ⊆ R d . Suppose that there exists r ∈ R + such that for each x ∈ K , there exist d line segments of length r containing x in linearly independent directions that are contained in K . Let u ∈ G (Ω) . If for some k ∈ N , P k,K ( u ) = 0 , then P l,K ( u ) = 0 , ∀ l ≥ k .Proof. If P k +1 ,K ( u ) = 0, then P k +1 ,K ( u ) ≤ P k,K ( u ) P k +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 ∈G L ( K ) . Then P k,K ( u ) are decreasing in k , and G ∞ ( K ) = G L ( K ) = { u ∈ G (Ω) : P k,K ( u ) ≤ P ,K ( u ) , ∀ k ∈ N } . In particular, G ∞ ( K ) is closed in G (Ω) . roof. Let u ∈ G (Ω). If P k,K ( u ) are not decreasing in k , then there exists k ∈ N \ { } such that P k,K ( u ) > P k − ,K ( u ) > r := P k,K ( u ) /P k − ,K ( u ) >
1. By proposition 3.1, P k +1 ,K ( u ) ≥ rP k,K ( u ) (in particular, P k +1 ,K ( u ) > P k,K ( u )).Inductively, P k + n,K ( u ) ≥ r n P k,K ( u ), for each n ∈ N . Thus lim sup n →∞ ln P n + k,K ( u ) n + k ≥ lim sup n →∞ ln( r n P k,K ( u )) n + k = ln r >
0, and u / ∈ G L ( K ). In particular, u / ∈ G ∞ ( K ).The fact that G ∞ ( K ) is closed follows by continuity of P k,K . Theorem 3.4. G ∞ (Ω) = G L (Ω) is closed in G (Ω) . In particular, G ∞ (Ω) is not densein G (Ω) .Proof. G ∞ (Ω) = T 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 G L (Ω). The conclusions followfrom proposition 3.3. G L a (Ω) , a > Proposition 4.1.
Let K ⊂⊂ Ω satisfy the hypothesis of proposition 3.1. Let a ∈ R , a ≥ . Then { u ∈ G (Ω) : ( ∃ c ∈ R )( P k,K ( u ) ≤ ca k , ∀ k ∈ N ) } = { u ∈ G (Ω) : P k +1 ,K ( u ) ≤ aP k,K ( u ) , ∀ k ∈ N } . In particular, this describes a closed subset of G (Ω) .Proof. Let u ∈ G (Ω). If P k +1 ,K ( u ) > aP k,K ( u ), for some k ∈ N , then P k,K ( u ) > r := P k +1 ,K ( u ) /P k,K ( u ) > a . By proposition 3.1, P k + n,K ( u ) ≥ r n P k,K ( u ), for each n ∈ N . Thus lim sup n ∈ N P n,K ( u ) /a n ≥ lim sup n ∈ N r n − k P k,K ( u ) a n =+ ∞ .The other inclusion is clear. Theorem 4.2.
Let a ∈ R , a > . Then G L a (Ω) is not dense in G (Ω) .Proof. G L a (Ω) ⊆ T K { u ∈ G (Ω) : ( ∃ c ∈ R )( P k,K ( u ) ≤ ce ak , ∀ k ∈ N ) } =: A , where K 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). In 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 ∞ ( R d ) with ψ ( x ) = 0 if | x | ≥ and R R d ψ = 1 . Denoteby ψ ε ( x ) := ε − d ψ ( x/ε ) , for each ε ∈ (0 , . If u = [( u ε ) ε ] ∈ G (Ω) , then lim n →∞ [( u ε ⋆ψ ε n ) ε ] = u . roof. Let n ∈ N and K ⊂⊂ Ω. Then u ε ⋆ ψ ε n ( x ) = R | t |≤ ε n u ε ( x − t ) ψ ε n ( t ) dt iswell-defined as soon as d ( x, R d \ Ω) > ε n . For small ε , d ( K, R \ Ω) > ε n and thus( u ε ⋆ ψ ε n ) | K can be extended to a C ∞ (Ω)-function. Independent of the extension, bythe mean value theorem, p k,K ( u ε ⋆ ψ ε n − u ε ) = sup x ∈ K, | α | = k | ( ∂ α u ε ) ⋆ ψ ε n ( x ) − ∂ α u ε ( x ) | = sup x ∈ K, | α | = k (cid:12)(cid:12)(cid:12)(cid:12)Z | t |≤ ε n ( ∂ α u ε ( x − t ) − ∂ α u ε ( x )) ψ ε n ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε n p k +1 ,K + r ( u ε ) Z R d | ψ | for small ε , where r > r ∈ R ) such that K + r = { x ∈ R d : d ( x, K ) ≤ r } ⊂⊂ Ω. Proposition 5.2.
Let A be the set of all u = [( u ε ) ε ] ∈ G c (Ω) for which ( ∃ N ∈ N )( ∀ K ⊂⊂ Ω)( ∀ k ∈ N )( p k,K ( u ε ) ≤ ε − Nk − N , for small ε ) . Then A is dense in G (Ω) .Proof. Let u ∈ G c (Ω). Then there exists a representative ( u ε ) ε of u and L ⊂⊂ Ω suchthat supp u ε ⊆ L , for each ε . For each K ⊂⊂ Ω and k ∈ N , p k,K ( u ε ⋆ ψ ε n ) = sup x ∈ K, | α | = k | u ε ⋆ ∂ α ( ψ ε n ) |≤ ε − nk sup x ∈ L | u ε ( x ) | max | α | = k Z R d | ∂ α ψ | ≤ ε − nk − sup x ∈ L | u ε ( x ) | , for small ε . Thus [( u ε ⋆ ψ ε n ) ε ] ∈ A . By proposition 5.1, A is dense in G c (Ω). Further, G c (Ω) is dense in G (Ω) (for u ∈ G (Ω), u = lim n →∞ uχ n , where χ n ∈ D (Ω) with χ n ( x ) = 1, ∀ x ∈ K n , where ( K n ) 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. Scarpal´ezos for very useful discussions.
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