Functional properties of Generalized Hörmander spaces of distributions II : Multilinear maps and applications to spaces of functionals with wave front set conditions
aa r X i v : . [ m a t h - ph ] D ec FUNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDERSPACES OF DISTRIBUTIONS II :MULTILINEAR MAPS AND APPLICATIONS TO SPACES OFFUNCTIONALS WITH WAVE FRONT SET CONDITIONS
YOANN DABROWSKI
Abstract.
We continue our study and applications of generalized H¨ormander spaces ofdistributions D ′ γ, Λ with C ∞ wavefront set included in a cone Λ and the union of H s -wavefront sets in a second cone γ ⊂ Λ. We give hypocontinuity results and failure of continuityof tensor multiplication maps between these spaces and deduce hypocontinuity results forvarious compositions on spaces of multilinear maps. We apply this study to a generalizationof microcausal functionals from algebraic quantum field theory with derivatives controlledby spaces either of the form D ′ γ, Λ or some ǫ -tensor product of them. We prove nuclearityand completeness results and give general results to build Poisson algebra structures (withat least hypocontinuous bilinear products). We also apply our general framework to buildretarded products with field dependent propagators. Introduction
In 1992, Radzikowski [Ra, Ra2] showed the wave front set of distributions to be a keyconcept to define quantum fields in curved spacetime. This idea was fully developed intoa renormalized scalar field theory in curved spacetimes by Brunetti and Fredenhagen [BF],followed by Hollands and Wald [HW] and later extended to more general fields, for instanceDirac fields [Kr, H01, A, DHP, Sa10, R11], gauge fields [H08, FR, FR2] and even someattempts of quantization of gravitation [BFR]. Moreover, recent simplifications stronglyimproved the mathematical understanding of the theory [KM14].Following those developments, the natural space where quantum field theory seem to takeplace is not the space of distributions D ′ , but the space D ′ Γ of distributions having their wavefront set in a specified closed cone Γ. This space and its simplest properties were describedby H¨ormander in 1971 [H71]. Since recent developments [BDF], most papers in algebraicquantum field theory used microcausal functionals where the natural space to control thewave front set is rather the dual of the previous space E ′ Λ with control of the wave frontset by an open cone Λ = − Γ c , we started recently in [BD] the investigation of functionalanalytic properties of these spaces. The first paper of this series [D] then computed theircompletion and bornologification.We called “dual wave front set” the union of usual H s -wave front sets (see e.g. [DL, p8])of a distribution : DW F ( u ) = S s ∈ IR W F s ( u ) so that W F ( u ) = DW F ( u ) is recovered as its Mathematics Subject Classification.
Primary 46F05; Secondary : 81T20.
Key words and phrases.
Wave front set, Microcausal functionals, Retarded brackets, Infinite dimensionalPoisson algebras. closure (see [D] for more details). With this definition the completion c E ′ Λ is nothing but : c E ′ Λ = { u ∈ E ′ , DW F ( u ) ⊂ Λ } . We thus introduced and studied the main functional analytic properties of the followingspaces, for γ ⊂ Λ ⊂ γ cones, we define : E ′ γ, Λ ( U ) = { u ∈ E ′ ( U ) : DW F ( u ) ⊂ γ, W F ( u ) ⊂ Λ } , D ′ γ, Λ ( U ) = { u ∈ D ′ ( U ) : DW F ( u ) ⊂ γ, W F ( u ) ⊂ Λ } . This class of spaces is stable by topological and bornological duality, completion, bornologi-fication and their main properties established in [D] are summarized in section 1.1 below forthe reader’s convenience.After this general study, our second goal is to provide tools to study spaces of functionalswith wave front set conditions on their derivatives. For this we gather miscellaneous resultson tensor products of our spaces and spaces of multilinear maps in part 1 of this second paperin the series. The third paper of this series will investigate more systematically those tensorproducts, following the general advice of Grothendieck, and mostly because general resultsdon’t give a full understanding of those tensor products. As for the completion above, thereis a need for concrete (microlocal) representation to obtain a fully satisfactory functional an-alytic understanding. In this paper, we will be content to generalize hypocontinuity resultsof [BDH] to our spaces in proposition 9. We will deduce some non-continuity results showingthe use of hypocontinuity is unavoidable in proposition 11. This is to be contrasted withvague statements of “continuity” meaning sequential continuity spread out in the literature,and which become dangerous when mixed with projective tensor products statements whichare related to full continuity. We will finally give general relations of tensor products with ourgeneralized H¨ormander spaces. This will be crucial to use standard information on propaga-tors and to relate our functionals with usual microcausal functionals of algebraic quantumfield theory. Finally, we define in section 4 composition maps on spaces of multilinear maps.These spaces thus form a kind of topological operad.Concerning spaces of functionals, we first solve technical problems in getting completenuclear topologies on multilocal functionals and variants of microcausal functionals in The-orem 20. This problem remained open in [BFRi] when some nuclear topology was found onordinary microcausal functionals.However, we go beyond this functional analytic study of already used spaces to suggest amore algebraic and functional analytic approach on them to build efficiently Poisson algebrasin infinite dimension with at least hypocontinuity results for the binary operations. Our ideaabout controlling, by wave front set conditions on derivatives, spaces of functionals is that itis much easier to control derivatives in spaces of multilinear maps on our spaces rather thanby our spaces themselves. This especially restore some continuity of products (instead ofhypocontinuity, in the full support case, see Theorem 20) and this reduces definition of mapslike retarded or Poisson products (Peierls brackets) even with field dependent propagatorsto applications of composition of multilinear maps, see theorem 21. We emphasize that ourtheorem is a scheme of result to build various maps on spaces of functionals. Especially, asan application, we give a construction of retarded products in the case of field dependentpropagators, fixing an issue in [BFRi]. This is the goal of section 7.
UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 3
Let us now describe in more detail the content of this paper, as a guiding summary of ourtools for the reader.Section 1 gathers preliminary material, mostly coming from [D], notably in subsection1.1. Subsection 1.2 recalls various notation and definitions of tensor products we will useextensively later.Subsection 1.3 is new and completes the proofs of some results already stated in our firstpaper on the series. It proves approximation properties for our spaces, identifies completionswith more concrete quasi-completions and give a crucial property of vector valued distri-butions based on our spaces to be tested scalarly by duality, the so-called property ( ǫ ) ofSchwartz.As explained before, part 1 starts the study of tensor products building on previous resultsin [BDH] and our general functional analytic properties. Section 2 contains continuity,and hypocontinuity results (most are generalizations of those in the quoted paper but one,proposition 10, is strictly stronger even in the closed cone case they consider). Those resultsare based on known stability properties of hypocontinuity. This section also contains theadvertised non-continuity result. The issue comes from zero sections where cones are notthe most natural set of control for classical tensor products. The non-continuity result isthus based on our improved continuity and proves using duality results and nuclearity thatcontinuity would imply an isomorphism of the hypocontinuous tensor product with somespace of distributions. But plenty of distributions in this space, typically an example ofHormander with wave front set contained in one direction, with one side on zero sections,cannot come from the image of the hypocontinuous product. Since the cones considered inthis non-continuity result are of the form appearing in physics for microcausal functionals,this result is of physical significance.Section 3 gives inclusions between ǫ -products and our generalized H¨ormander spaces ofdistributions, in waiting for an exact microlocal characterization of the former in the thirdpaper of the series. Section 4 is concerned with multilinear maps.Part 2 gives the already described applications to functionals. Schwartz’ papers on vectorvalued distributions [S2] play a key role here as well as the notion of convenient smoothness[KM] and our composition of multilinear maps from section 4. Section 5 defines our generalspaces of functionals and gives completeness, nuclearity and results of (hypo)continuity ofproducts. Section 6 contains a general theorem to construct hypocontinuous bilinear mapsand section 7 applies this to retarded products. Acknowledgments.
The author is grateful to Katarzyna Rejzner and Christian Brouderfor helpful discussions on the physical motivation of part 2. He also thanks Christian Brouderfor many comments on previous versions of this paper that helped improving its exposition.Finally, the author acknowledges the support and hospitality of the Erwin Schr¨odingerInstitute during the workshop “Algebraic Quantum Field Theory: Its Status and Its Future”in May 2014. He also acknowledges the organizers for the stimulating program. Again, hethinks (and hopes) the physical relevance of the content of part 2 greatly benefited from theparticipation at this workshop.
YOANN DABROWSKI Preliminaries
We start by recalling the setting of the first paper of the series without giving the definitionof the topologies but we still state the main results we will most often use in this secondpaper.Let U ⊂ M an open set in a smooth connected manifold (implicitly assumed orientable σ -compact without boundary) of dimension d . We assume given on M a complete Riemannianmetric D giving the topology (so that, by Hopf-Rinow theorem, closed balls for D arecompact.)Let E U a smooth real vector bundle (with finite dimensional fiber of real dimension e ).We will use a fixed smooth partition of unity f i , indexed by I , subordinated to a covering U i , with U i ⊂ U compact and ( U i ) i ∈ I locally finite U i smoothly isomorphic via a chart ϕ i : U i → IR d to an open set in IR d , extending homeomorphically to U i and trivializing E U We consider D ′ ( U ; E ) the space of distributional sections of E with support in U and E ′ ( U ; E ) the space of distributional sections of E with compact support in U . Note that D ′ ( U i , E ) ≃ D ′ ( ϕ i ( U i ) , IR e ) ≃ ( D ′ ( ϕ i ( U i ))) e (see e.g. [GKOS, (3.16) p 234]), via the mapwritten above u u ◦ ϕ − i .Likewise we call E ( U ; E ) the space of smooth sections of E on U and D ( U ; E ) the space of smooth sections of E with compact support in U . We write asusual E ′ for the dual bundle, and E ∗ = E ′ ⊗ IR ∗ its version twisted by the density bundleIR ∗ (cf e.g. [GKOS, chapter 3]). We of course don’t write E in any notation when E isthe trivial line bundle over IR. Note that, since we don’t make explicit our trivialization ofvector bundles, we make the choice for those of E, E ∗ so that for g ∈ D ( U i ) , v ∈ D ( U ): h f i u, gv i = h ( f i u ) ◦ ϕ − i , ( gv ) ◦ ϕ − i i . This reduces duality pairings to those on IR n and the emphasis on the difficult analyticpart of pullback rather than on the multiplication by smooth map part involved in changeof bundle trivialization.1.1. Duality and functional analytic results.
Recall that all the definition of topologiesare given in section 3 there but they are also characterized by some of the properties givenin the next result. The notions related to support properties are defined there in section2. The most used support conditions C will be K the family of compact sets, F the familyof all closed sets and on globally hyperbolic manifolds those explained in [D, Ex 16] (cfalso [Sa]) : timelike-compact closed sets K T = K F ∩ K P , future-compact closed sets K F , past-compact closed sets K P , spacelike-compact closed sets SK , future-spacelike-compactclosed sets SK F and past-spacelike-compact closed sets SK P . In these cases the class ( O C ) o in duality formulas is described as follows ( O K ) o = F , ( O F ) o = K , SK F = ( O K P ) o , SK P =( O K F ) o , K P = ( O SK F ) o , K F = ( O SK P ) o , ( O SK ) o = ( O SK F ) o ∩ ( O SK P ) o = K T , ( O K T ) o = SK . They satisfy the assumptions below since they are all enlargeable, by definition polar, andeither C or ( O C ) o is countably generated in the sense of [D, Ex 3, Def 13].Recall the notation for our spaces in the vector bundle case : D ′ γ, Λ ( U, C ; E ) = { u ∈ D ′ ( U ; E ) | W F ( u ) ⊂ Λ , DW F ( u ) ⊂ γ, supp( u ) ∈ C} . The following theorem is our main result from the first part [D, Prop 34].
UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 5
Theorem 1.
Let γ a cone and C = C oo an enlargeable polar family of closed sets in U and let λ = − γ c . The bounded sets on D ′ γ,γ ( U, C ; E ) coincide for I ppp and I iii and thislast inductive limit is regular. Moreover on D ′ λ,λ ( U, ( O C ) o ; E ∗ ) we have I b = I ibi = I pbp . ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I ibi ) is the strong and Mackey dual of ( D ′ γ,γ ( U, C ; E ) , I borniii = I bornppp ) ,the bornologification I bornppp , the completion of ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I b ) and also for any cone λ ⊂ Λ ⊂ λ the completion of ( D ′ λ, Λ ( U, ( O C ) o ; E ∗ ) , I ibi ) , which is also nuclear.Thus, ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I ibi ) is complete ultrabornological nuclear, especially a Montelspace. Likewise, ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I b = I ibi ) is the bornologification of I iii , so that thesituation is summarized in the two following commuting diagrams where i is the canonicalinjection from a space to its completion, b the canonical map with bounded inverse betweena bornologification and the original space : ( D ′ γ,γ ( U, C ; E ) , I ibi )( D ′ γ,γ ( U, C ; E ) , I ppp ) b < ( D ′ γ,γ ( U, C ; E ) , I ibi ) i ∪ ∧ ( D ′ γ,γ ( U, C ; E ) , I ppp ) i ∪ ∧ b < ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I ibi )( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I ppp ) b ∨ ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I ibi ) i < ⊃ ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I ppp ) b ∨ i < ⊃ Spaces symmetric with respect to the middle vertical line are Mackey duals of one another.All the spaces involved are nuclear locally convex spaces, and in each of them, bounded setswhich are closed in the completion are metrisable compact sets and are equicontinuous setsfrom the stated dualities. When λ, γ are ∆ -cones (i.e. both F σ , G δ ) and if we assume either C or ( O C ) o countably generated, all the space involved are moreover quasi-LB spaces of class G . The following extension of [BDH, Prop 6.1] was proven in [D, Prop 36].
Proposition 2.
Let γ ⊂ Λ ⊂ γ be cones on U and f : U → U a smooth map. Definethe cone df ∗ γ = { ( x, df ∗ ( x )( ξ )) : ( f ( x ) , ξ ) ∈ γ } and for an enlargeable polar family ofclosed sets C define f − ( C ) = { f − ( C ) , C ∈ C , } , and its polar enlargeable variant f − e ( C ) = { ( f − ( C )) (1 − /n ) ǫ ( C ) , C ∈ C , ǫ i ( C ) > } oo (depending on any function ǫ : C → ]0 , I )Assume df ∗ γ ⊂ ˙ T ∗ U , and df ∗ Λ ⊂ ˙ T ∗ U . Then we have continuous maps for I either I ppp or I ibi : f ∗ : ( D ′ γ, Λ ( U , C ) , I ) → ( D ′ df ∗ γ,df ∗ Λ ( U , f − e ( C )) , I ) ,f ∗ : ( D ′ γ,γ ( U , C ) , I ) → ( D ′ df ∗ γ,df ∗ γ ( U , f − e ( C )) , I ) . Especially,
DW F ( f ∗ u ) ⊂ df ∗ ( DW F ( u )) . We will also need some identification of topologies obtained by computing equicontinuoussets in respective duals, we obtained them in [D, lemma 28] for λ, γ ⊂ Λ ⊂ γ any cones:(1) ( D ′ γ, Λ ( U, C ; E ) , I H,ppp ) ≃ lim ←− λ ⊃ γ lim ←− λ ⊃ λ ∪ Λ (cid:16) D ′ λ ,λ ∩ λ ( U, C ; E ) , I H,pmp (cid:17) , YOANN DABROWSKI (2) ( D ′ γ,γ ( U, C ; E ) , I H,pip ) ≃ lim ←− λ ⊃ γ (cid:16) D ′ λ ,λ ( U, C ; E ) , I H,pip (cid:17) , (3) ( D ′ λ,λ ( U, ( O C ) o ; E ∗ ) , I iii ) ≃ lim −→ Π ⊂ λ ( D ′ Π , Π ( U, ( O C ) o ; E ∗ ) , I iii ) . Here λ i indexes projective limits over open cones, Π inductive limits over closed cones.We finally gather in the next lemma the descriptions of bounded, absolutely convex com-pact and equicontinuous sets in the spaces above which are scattered in the proofs of [D].These are the most relevant bornologies for the tensor products we will consider. Recall thatwe introduced in [D, section 3] the following technical space D ′ γ ( U, F : δ ; E ) = { u ∈ D ′ ( U ; E ) | ∀ n, DW F n ( u ) ⊂ Γ n } , for an increasing sequence of closed cones Γ n ⊂ γ gathered in δ = (Γ n ). Modulo, equivalenceby inclusion, this is basically a space where we fix the H n wave front set (which is similarto DW F n , but different) in a family of closed cones included in the cone γ. Lemma 3.
With the setting of theorem , on ( D ′ γ,γ ( U, C ; E ) , I ibi ) and ( D ′ γ,γ ( U, C ; E ) , I ppp ) , bounded sets, precompact sets, sets included in an absolutely convex compact set and equicon-tinuous sets induced by their duals coincide. We have for them the same description as forbounded sets for ( D ′ γ,γ ( U, C ; E ) , I b ) and ( D ′ γ,γ ( U, C ; E ) , I iii ) , namely sets bounded in the senseof the bornology induced by the regular inductive limit ( D ′ γ,γ ( U, C ; E ) , I iii ) , i.e. uniformly sup-ported in a C ∈ C and included and bounded for some δ = (Γ n ) ∈ ∆( γ ) in D ′ γ ( U, F : δ ; E ) . On ( D ′ γ,γ ( U, C ; E ) , I b ) and ( D ′ γ,γ ( U, C ; E ) , I iii ) , equicontinuous sets from their dual coin-cides with sets included in absolutely convex compact sets which are those associated to thebornological inductive limit associated to I iii , namely sets of distributions uniformly supportedon some C ∈ C and bounded in D ′ Π , Π ( U, F ; E ) for some closed cone Π ⊂ γ. Finally, the bornologification maps b : D ′ γ,γ ( U, C ; E ) , I ibi → I ppp , D ′ γ,γ ( U, C ; E ) , I ibi → I ppp are proper. Especially, all the spaces above are defined from those bounded in a bornological inductivelimit associated to I iii on some space. Proof.
From the beginning of the proof of [D, Prop 34] a set bounded in ( D ′ γ,γ ( U, C ; E ) , I ppp )is bounded in the regular inductive limit I iii which corresponds to the description above.For general reasons about bounded sets in completion and bornologification, this givesthe description of bounded sets in any of the spaces above. Since from [D, Prop 34],( D ′ γ,γ ( U, C ; E ) , I ibi ) is nuclear, bounded sets are exactly precompact=“relatively compact”sets. Thus the inverse image of a compact set for I ppp is closed since the bornologificationmap is continuous, bounded since both topologies have same bounded sets thus compact bythe above nuclearity. This gives that the bornologification map is proper. For the secondproper map we reason by diagram chasing, a compact set is also compact in the completionthus from the complete case in the bornologification, and since the topology of the bornologi-fication without completion agrees with the one of the completeion of the bornologification(for our spaces), it is compact there.In D ′ γ,γ ( U, C ; E ), for I ibi equicontinuous sets and strongly bounded sets coincides as in anydual of a bornological space (this is explained from Hogbe-Nlend’s results in [D, lemma 27]),and moreover, I ibi is known to be the strong dual topology by [D, Prop 34]. Finally, on a UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 7 dual, the equicontinuous sets induced by a space and its completion are known to be thesame thus the equicontinuous sets are the same for I ppp . For ( D ′ γ,γ ( U, C ; E ) , I iii ) equicontinuous sets are identified in [D, lemma 28], and as abovethey are the same for I ibi whose dual is the completion of the dual. In ( D ′ γ,γ ( U, C ; E ) , I b )and ( D ′ γ,γ ( U, C ; E ) , I iii ) closed equicontinuous sets are compact as in any dual of a nuclearspace [T, p 519], and conversely absolutely convex compact sets (which coincides for bothtopologies from the proper map property shown above) are shown to be equicontinuous atthe end of the proof of [D, Prop 34]. This completes our proof. (cid:3) Reminder on tensor products of locally convex spaces.
We will be mostly in-terested in five tensor products, projective, injective, ǫ , γ and bornological (with variants).If E, F are locally convex spaces, E ⊗ π F (resp. E ⊗ β F , E ⊗ βe F when E = E ′ , F = F ′ are duals, E ⊗ γ F ) is the algebraic tensor product equipped with the finest locally convextopology on it making E × F → E ⊗ π F continuous (resp E × F → E ⊗ β F hypocon-tinuous, resp. E × F → E ⊗ βe F ǫ -hypocontinuous, i.e. hypocontinuous for equicontin-uous parts of the implicitly used dualities E = E ′ , F = F ′ , resp. E × F → E ⊗ γ Fγ -hypocontinuous, i.e. hypocontinuous for absolutely convex compact parts). We write asusual E ˆ ⊗ π F, E ˆ ⊗ β F, E ˆ ⊗ βe F, E ˆ ⊗ γ F the corresponding completions. We will also sometimesuse the terminology σ − σ hypocontinuous (especially with σ i one of the letters above) tosay hypocontinuous for bounded sets in σ i on the corresponding side [S2].One should note that at various places in the (bornology, convenient vector space) litera-ture, see e.g. [KM], there is another (different) definition of the bornological tensor productas the finest l.c.s topology making E × F → E ⊗ ′ β F bounded (on product of bounded sets).From the characterization of neighborhoods of zero (see e.g. [PC, Prop 11.3.4 p 390]), it iseasy to see that E ⊗ ′ β F = E born ⊗ β F born is merely the bornological/hypocontinuous productof bornologifications.If E, F are barrelled (resp. bornological) so are E ⊗ β F, E ⊗ βe F, E ⊗ γ F [PC, Prop 11.3.6p 390] [DG, Th 5] and E ˆ ⊗ β F, E ˆ ⊗ γ F are barrelled in the barelled case [PC, Prop 4.2.1 p103].The ǫ -product has been extensively used and studied by Laurent Schwartz [S, section 1].By definition EǫF = ( E ′ c ⊗ βe F ′ c ) ′ is the set of ǫ -hypocontinuous bilinear forms on the duals E ′ c equipped with the Arens topology of uniform convergence on absolutely convex compactsubsets of E. When the bounded sets on E ′ c , F ′ c coincide with the equicontinuous parts this is of coursethe same as EǫF = ( E ′ c ⊗ β F ′ c ) ′ and when E, F have their γ topology (i.e. ( E ′ c ) ′ c so thatas explained in [S] equicontinuous sets of the dual coincide with absolutely convex compactsets), this is the same as EǫF = ( E ′ c ⊗ γ F ′ c ) ′ . Note that all our spaces above in theorem1 have their γ topology since they have all their Mackey topology. The topology on EǫF is the topology of uniform convergence on products of equicontinuous sets in E ′ , F ′ (as theinitial topology of E = ( E ′ c ) ′ , by Mackey Thm is the topology of uniform convergence onequicontinuous parts in E ′ ). If E, F are quasi-complete spaces (resp. complete spaces , resp.complete spaces with the approximation property) so is
EǫF (see [S, Prop 3 p29, Corol 1 p47]).
YOANN DABROWSKI E ⊗ ǫ F is the topology on E ⊗ F induced by EǫF (see [S, Prop 11 p46] and E ˆ ⊗ ǫ F ≃ EǫF if E, F are complete and E has the approximation property. All the above tensor productsare commutative and the ⊗ π , ⊗ ǫ , ⊗ ′ β products are associative.We end this section by a technical result in our context that will be used at the end inthe application section 7. This gives a glimpse of the general study of tensor products in thethird paper of this series. Proposition 4.
Let γ ⊂ ˙ T ∗ U a ∆ -cone and C an enlargeable polar family of closed setsof U with either C or ( O C ) o countably generated, and similarly Γ ⊂ ˙ T ∗ V a closed coneon a similar open V . Let E → U, F → V vector bundle maps and I ∈ {I ppp , I ibi } then ( D ′ γ,γ ( U, C ; E ) , I ) ǫ ( D ′ Γ , Γ ( U, F ; F ) , I ppp ) is a quasi-LB space thus a strictly webbed space.Proof. From Theorem 1, ( D ′ γ,γ ( U, C ; E ) , I ) is a complete quasi-LB space thus a strictlywebbed space [ ? , Th 1]. As in [BD, Corol 13] (using the (PLN) property showed in [D, Prop8]), it is easy to show that ( D ′ Γ , Γ ( U, F ; F ) , I ppp ) is a (PLS)-space thus, by definition, can bewritten lim ←− n ∈ IN X n with X n the dual of a Fr´echet-Schwartz space. Now by completeness andnuclearity, ( D ′ γ, Λ ( U, C ; E ) , I ) ǫ ( D ′ Γ , Γ ( U, F ; F ) , I ppp ) ≃ lim ←− n ∈ IN( D ′ γ, Λ ( U, C ; E ) , I ) ˆ ⊗ ǫ X n . Thusthis is a countable inductive limit of webbed spaces (using [DW, Prop V.2.4 p 91] for thetensor product and e.g. [K2, § ? , Th2], since our space is complete, it is also quasi-(LB). (cid:3) Consequences for approximation properties and vector valued distributions.
We gather here results that are useful in themselves in functional analysis, but essential whendealing with vector valued distributions [S]. The study of vector valued distributions basedon H¨ormander spaces of distributions is motivated by their use in quantum field theory,notably in [R, FR]. We will also use them in part 2 below.
Proposition 5.
Let γ ⊂ Λ ⊂ γ cones and C = C oo an enlargeable polar family of closedsets in U and assume either C or ( O C ) o countably generated. Then ( D ′ γ,γ ( U, C ; E ) , I ibi ) , ( D ′ γ,γ ( U, C ; E ) , I ppp ) , ( D ′ γ,γ ( U, C ; E ) , I b ) and ( D ′ γ,γ ( U, C ; E ) , I iii ) all have the sequential approx-imation property (in the strongest variant with uniform convergence on compact subsets ofthe completion [J, p399] ).Proof. We use [S, Prop 2 p 7]. Of course from the same proof as in [S, Prop 1 p 6], F = D ( U ; E ) has the wanted sequential approximation property (since it is complete, theuniform convergence on compact sets of the completion coincide with Schwartz’ notion ofconvergence on absolutely convex compact sets and Grothendieck’s notion of convergence onprecompact sets). If C is countably generated, [D, lemma 22] gives us a subsequence L k n of L n such that for any B bounded in G = ( D ′ γ,γ ( U, C ; E ) , I ibi = I borniii ), L k n ( B ) is bounded inthe same sense, and L k n ( u ) → u . Thus since ( D ′ γ,γ ( U, C ; E ) , I ibi ) is barrelled, from [K2, p 141] L k n → Id uniformly on compact sets of G. This gives the expected sequential approximationof Id by L ( G, F ) in L ( G, G ) in this case. Composing with the bornologification map atthe target and knowing the stronger continuity of each map from [D, lemma 22], and sincethe compact sets on which we want to converge uniformly are the same by the proper mapproperty in lemma 3, this gives the same result for G = ( D ′ γ,γ ( U, C ; E ) , I ppp ) . Finally these twospaces are the completion of the other two, thus since the uniform convergence sets agree asthe topology, we also conclude those cases. For the case where ( O C ) o is countably generated, UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 9 we have the corresponding result for the dual of ( D ′ γ,γ ( U, C ; E ) , I ibi ) whose Arens dual is stillthis space, thus the sequential approximation property in Schwartz’ sense, which coincideswith the stated one in this complete case. The consequence without bornologification andwithout completion then follows similarly. (cid:3) We gather here other consequences of [D, lemma 22].
Proposition 6.
Let γ a cone and C = C oo an enlargeable polar family of closed sets in U .For any bounded set B in ( D ′ γ,γ ( U, C ; E ) , I ibi ) , ( D ′ γ,γ ( U, C ; E ) , I ppp ) , ( D ′ γ,γ ( U, C ; E ) , I b ) and ( D ′ γ,γ ( U, C ; E ) , I iii ) , there is a set B ′ ⊂ D ( U ; E ) and bounded in the same space with B ⊂ B ′ (even in the Mackey-sequential completion).As a consequence, ( D ′ γ,γ ( U, C ; E ) , I ibi ) is the quasi-completion of ( D ′ γ,γ ( U, C ; E ) , I b ) and ( D ′ γ,γ ( U, C ; E ) , I ppp ) the quasi-completion of ( D ′ γ,γ ( U, C ; E ) , I iii ) . Proof.
In any case B is always bounded in ( D ′ γ,γ ( U, C ; E ) , I ibi ), and from [D, lemma 22] B ′ = L k ( B ) ( B ) is bounded for I iii thus for I ibi since this is the bornologification of the previous one.Thus B ′ is bounded in any of these spaces and the conclusion follows from the lemma. (cid:3) In [S, p 53], Schwartz says a space of distributions H (in IR n but the manifold case isidentical, i.e. H ֒ → D ′ ( U, E )) has property ( ǫ ) if for any quasi-complete separated locallyconvex vector space F , the ǫ product (cf. the previous subsection for a reminder) H ǫF is the same as the space of distributions T ∈ D ′ ( U, E ) ǫF ≃ L ( F ′ c , D ′ ( U, E )) such that forany f ∈ F ′ , T ( f ) ∈ H . (Recall F ′ c is the Arens dual of F , i.e. F ′ given the topology ofuniform convergence on compact absolutely convex sets in F ). Said otherwise H -valueddistributions defined as H ǫF can be tested scalarly to belong to H once known to be in D ′ ( U, E )-valued distributions.
Proposition 7.
Let γ cones and C = C oo an enlargeable polar family of closed sets in U andassume either C or ( O C ) o countably generated. Then ( D ′ γ,γ ( U, C ; E ) , I ibi ) , and ( D ′ γ,γ ( U, C ; E ) , I ppp ) have property ( ǫ ) .Proof. One uses (an obvious manifold variant of) [S, proposition 15 p54]. Both our spaces arequasi-complete nuclear thus closed bounded sets are compact. We know that them and theirArens=Mackey duals (since they are semi-Montel) are strictly normal (see [D, lemma 22] andTheorem 1 for the Mackey duals.) It remains to check they have a system of neighborhoods ofzero which is closed in them with the topology induced by D ′ ( U, E ), for which, by Schwartz’remark after his proof, it suffices to check there is, for every equicontinuous set B of theirduals, a set B ′ ⊂ D ( U, E ∗ ) and equicontinuous in their duals such that B ⊂ B ′ the weakclosure. For ( D ′ γ,γ ( U, C ; E ) , I ibi ) for which equicontinuous sets are the same as boundedsets, this is our preceding proposition. In ( D ′ γ,γ ( U, C ; E ) , I ppp ) ′ , equicontinuous sets are inthe bornological inductive limit associated to the definition of I b namely from the proof ofTheorem 1 again, the bounded sets in the bornological inductive limit associated to I iii (thisis also explained in detail in lemma 3). Thus from [D, lemma 22], we can take B ′ = L k ( B ) ( B )(note this lemma only uses the inductive limit bornology). (cid:3) Part A first look at tensor products and operators on GeneralizedH¨ormander spaces of distributions
We start here our investigation of tensor products and spaces of operators on our (slightly)Generalized H¨ormander spaces of distributions. Our intent is to prove here only what weneed for our applications to generalizations of spaces of functionals used in algebraic quantumfield theory [BF, BDF, R, BFRi]. A more systematic study will be carried out in the thirdpaper of this series.2.
Preliminaries on continuity of tensor products
The strongest result is obtained when tensor product is not considered with optimal wavefront set condition but we can get a continuous map (and not only a hypocontinuous map).
Proposition 8.
Let γ i ⊂ ˙ T ∗ U i be closed cones, E i U i fiber bundles.Let γ = ˙ T ∗ U − γ c × γ c , U = U × U ,then the bilinear map . ⊗ . : ( D ′ γ ,γ ( U , F ; E ) , I iii ) × ( D ′ γ ,γ ( U , F ; E ) , I iii ) → ( D ′ γ,γ ( U, F ; E ⊗ E ) , I iii ) is continuous.Proof. and [D, lemma 27] , we can be content with the cases α = iii . We only considertrivial bundles on open sets of IR d , the general case being identical.Since the support is F ( U i ), we are reduced to the usual normal topology I H from [BD].From the well-known continuity of tensor product of distributions with their strong topologyand exhaustion lemmas (any closed supp( f ) × V ⊂ γ c × γ c can be covered by finitely manyparts as below), it suffices to consider seminorms of the form P k,f ⊗ f ,V × V with supp( f i ) × V i ⊂ γ ci and of course we have : P k,f ⊗ f ,V × V ( u ⊗ u ) = sup ( ξ,η ) ∈ V × V (1+ | ( ξ, η ) | ) k |F ( f u )( ξ ) ||F ( f u )( η ) | ≤ P k,f ,V ( u ) P k,f ,V ( u ) . (cid:3) We also have a weaker but much more useful hypocontinuity result when we take optimalwave front set conditions, which essentially comes from [BDH, Th. 5.5]. This is especiallyimportant once considered our next result that explains why we don’t have continuity inmost cases different from the previous one. Note that above and below E ⊗ E → U × U always denotes the exterior tensor product of vector bundles. Proposition 9.
Let γ i ⊂ Λ ( i ) ⊂ γ i ⊂ ˙ T ∗ U i cones and C i enlargeable polar families of closedsets of U i . Let E i → U i vector bundle maps. Let π : ˙ T ∗ U i → U i the first projection and thenlet γ = γ × γ ∪ γ × U × { } ∪ U × { } × γ =: γ ˙ × γ , Λ = Λ (1) ˙ × Λ (2) and C = ( C × C ) oo . The bilinear maps . ⊗ . : ( D ′ γ , Λ (1) ( U , C ; E ) , I α ) × D ′ γ , Λ (2) ( U , C ; E ) , I α ) → ( D ′ γ, Λ ( U × U , C ; E ⊗ E ) , I β ) are hypocontinuous (if not otherwise specified) in the following cases : (i) γ i = Λ ( i ) closed cones and α j = β = iii , (ii) Λ ( i ) , γ i any cones as above and α j = β = ibi. (iii) γ i = Λ ( i ) any cone α = b, iii , α = β = iii in which case we only have γ -hypocontinuity, but full hypocontinuity if γ closed for α = b . UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 11 (iv) γ = Λ (1) any cone and α = b and γ , Λ (2) any cones and α = β = ppp in whichcase we only have ( γ − β ) -hypocontinuity, but full hypocontinuity if γ closed. The various cases give (some kind of) hypocontinuity for most pairs of main topologies inTheorem 1, most notably for pairs of dual topologies and for identical topologies, with theexception of I ppp in the complete case (where there is only a result with its dual topology),because the non-complete case is only given with γ -hypocontinuity and thus not enoughbounded sets to go to the completion.Note that case (ii) includes γ i = Λ ( i ) open cones and α j = β ∈ { iii, iip, ipi, ipp } sincethose topologies coincide with I b in this case except on the target space since γ may not beopen, but it is still stronger (see [D, Prop 33]) and for the same reason, case (ii) also includes γ i = Λ ( i ) with γ i open α j = β = ppp in the case since I ppp = I bornppp = I ibi in this case fromTheorem 1. Proof.
Note first that γ c = ( ˙ T ∗ U − γ ) × ˙ T ∗ U ∪ ( ˙ T ∗ U ) × ( ˙ T ∗ U − γ ) ∪ γ c × U × { } ∪ U × { } × γ c so that γ is again a ∆ -cone if γ i ’s are (since this formula and the definingone keep the F σ property).Using [D, Th 23, lemma 27] to identify the topologies with what we called normal topologyin [BD], Case (i) is exactly [BDH, Th. 5.5] at least in the case C i = F ( U i ), and trivial vectorbundles. Note also that γ is indeed a closed cone when γ i ’s are. We don’t explain more themanifold case which is a direct consequence of their result.One can use a well-known result for hypocontinuity and inductive limits (see [M, Th 1 c]),since the inductive limit on support are strict regular (especially quasiregular in the sense ofMelnikov), it suffices to get hypocontinuity on each term of the inductive limit, and since thetopology is induced there from the one without support condition and the support conditionof tensor products is well known, the result is obvious.For case (ii), we start with γ i = Λ ( i ) in which case I ibi = I b and we proved in The-orem 1 that the spaces considered are ultrabornological thus barrelled and we can againbe content to prove separate continuity using [T, Th 41.2 p424]. Thus if we take u ∈D ′ Π , Π ( U , C ) and Π ⊂ γ closed cone we have to check that u u ⊗ u is continu-ous from ( D ′ Π , Π ( U , C ) , I bornH,iii ) → ( D ′ γ,γ =Λ ( U × U , C ) , I b ) using the definition of continuityon inductive limits, but for this it suffices to check it is bounded ( D ′ Π , Π ( U , C ) , I H,iii ) → ( D ′ Π ˙ × Π ( U × U , C ) , I H,iii ) (since then it will be bounded thus continuous between bornologi-fications). The boundedness statement is included in the above hypocontinuity alreadychecked.We continue with the case Λ ( i ) = γ i , note that Λ = γ in this case. Since the sourcespaces are the completions of those of our first subcase in case (ii) by our Theorem 1, weuse [K2, § D ′ γ,γ ( U, F ; E ⊗ E ) . Now the topology induced on our desired target by the completion is of course I ibi by thesame completion property at the target from Theorem 1 and it suffices to check the extendedmap is valued in the stated space. For, thanks to the computation of the completion, weonly have to compute wave front sets (since the DWF condition is not changed at the completion level). But for any element in pair in the source, wave front sets are closed cones,and applying (i) to them and increasingness of ˙ × in both arguments, one concludes to theexpected inclusion W F ( u ⊗ v ) ⊂ Λ . In case (iii), we use the inductive limit description for I b and (3). We checked in the lemma3 that absolutely convex compact sets are equicontinuous and that such sets are only in thebornology of the quoted bornological inductive limits. We can thus use a result in the spirit of[M, Th 1 c] (replacing the strong assumption for bounded sets he uses namely a regularity ofthe inductive limit by the right assumption for his result which is to take an hypocontinuitywith respect to the bornologies of the inductive limits). Then the required hypocontinuitiesfor applying this result all follow from the closed case (i). [Note that the case with α = b also follows directly from the case α = iii by composition with the bornologification map.]Noting the first space with topology I b is ultrabornological thus barrelled, we can use[K2, § γ closed, boundedand equicontinuous sets coincide in the left hand side, one gets full hypocontinuity in thiscase. This gives the special result in the closed case.Finally in case (iv) note first taking an equicontinuous or an absolutely convex compactset in the left hand side, thus a bounded set with wave front set controlled by a closedcone Γ ⊂ γ so that the statement reduces to prove hypocontinuity in the case γ closed inreasoning as before with a variant of [M, Th 1 c]. We thus stick to this case. Starting from(iii) with same γ i gives the case γ = Λ (2) (since I ppp = I iii in this case in the source and I iii is stronger in the targe). Next we deal with the case γ = Λ (2) , and fixing as target spacethe completion of the space we want we have from (iii) hypocontinuity. We can apply [K2, § (cid:3) The advertised non-continuity result will be a consequence of an improvement of theprevious result. We will investigate more its meaning in the third paper of this series.We concentrate on the closed cone case, with U i ⊂ IR d i and without vector bundles untilthe end of this subsection, for which we need to introduce extra seminorms. We leave themanifold slight generalization to the reader. Let γ i ⊂ ˙ T ∗ U i be cones and γ = γ ˙ × γ .Define for D > , k ∈ IN ∗ C D,k, = { ( ξ, η ) ∈ IR d × IR d , D | η | k ≥ | ξ |} C D,k, = { ( ξ, η ) ∈ IR d × IR d , D | ξ | k ≥ | η |} and for a set A , f ∈ D ( U × U ) UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 13 p l,A,f ( u ) = sup ( ξ,η ) ∈ A (1 + | ξ | + | η | ) l |F ( uf )( ξ, η ) | Then we call I β , the topology on ( D ′ γ, Λ ( U × U , F ) , I β ) for γ = γ ˙ × γ generated by theseminorms for I ppp and, for ϕ i ∈ D ( U i ) and cones V i such that (supp( ϕ i ) × V i ) ∩ γ i = ∅ , any D, k, l , the seminorms p l,C D,k, ∩ ( IR d × V ) ,ϕ ⊗ ϕ and p l,C D,k, ∩ ( V × IR d ) ,ϕ ⊗ ϕ . Proposition 10.
Let γ i ⊂ ˙ T ∗ U i be cones and γ = γ ˙ × γ . ( D ′ γ,γ ( U × U , F ) , I β ) is completeand the tensor product bilinear map is γ -hypocontinuous : . ⊗ . : ( D ′ γ ,γ ( U , F ) , I iii ) × ( D ′ γ ,γ ( U , F ) , I iii ) → ( D ′ γ,γ ( U × U , F ) , I β ) . Proof.
For the completeness statement, taking any Cauchy net u n for I β it converges to some u for I ppp from the completeness of this topology and F ( u n f )( ξ, η ) thus converges pointwise,and since moreover, from completeness of our supplementary decay seminorms on spaces ofcontinuous functions, they have limits in those spaces, the limit is necessarily F ( uf )( ξ, η ) sothat we have indeed the full convergence for I β . We could of course formulate this in termsof closed subspace of a product of complete spaces.It remains to check the improved bonds near the cones γ × U × { } and U × { } × γ for hypocontinuity. Note that as in the proof of proposition 9 (iii) it suffices to prove theclosed γ i case and reason by inductive limits.By symmetry we are content to bound (in the setting before the proposition): p l,C D,k, ∩ ( IR d × V ) ,ϕ ⊗ ϕ ( u ⊗ u ) := sup ( ξ,η ) ∈ C D,k, ,η ∈ V (1 + | ξ | + | η | ) l |F ( u ϕ )( ξ ) ||F ( u ϕ )( η ) | in two cases to prove hypocontinuity of the tensor product with those seminorms added.First if we assume u ∈ B some bounded set in D ′ ( U ) thus say with F ( u ϕ ) polynomiallybounded of order m , in this case, one gets (using (1 + | ξ | + | η | ) ≤ (1 + | ξ | )(1 + | η | ) ≤ (1 + D | η | k )(1 + | η | ) ≤ max(1 , | D | )(1 + | η | ) k +1 on C D,k, ):sup u ∈ B p l,C D,k, ∩ ( IR d × V ) ,ϕ ⊗ ϕ ( u ⊗ u ) ≤ max(1 , D ) l + m (cid:18) sup ξ (1 + | ξ | ) − m |F ( u ϕ )( ξ ) | (cid:19) P ( k +1)( m + l ) ,ϕ ,V ( u ) . Second, if we assume u ∈ B some bounded set in D ′ γ ( U ) one introduces B ′ = { ϕ e ξ (1 + | ξ | + | η | ) l |F ( u ϕ )( η ) | : ( ξ, η ) ∈ C D,k, , η ∈ V , u ∈ B } so that obviouslysup u ∈ B p l,C D,k, ∩ ( IR d × V ) ,ϕ ⊗ ϕ ( u ⊗ u ) ≤ P B ′ ( u )and it remains to prove B ′ bounded in D ( U ) . First B ′ is indeed uniformly supported onsupp( ϕ ) . We have to boundsup f ∈ B ′ sup x ∈ K X | α |≤ n | f α ( x ) | ≤ sup ( ξ,η ) ∈ C D,k ,η ∈ V ,u ∈ B C n,ϕ (1 + | ξ | ) n (1 + | ξ | + | η | ) l |F ( u ϕ )( η ) |≤ max(1 , D ) l + n C n,ϕ sup u ∈ B P ( k +1)( l + n ) ,V ,ϕ ( u ) for some compact K say containing a neighborhood of supp( ϕ ) , and some constant C n,ϕ depending on the norms of ϕ . But now the last term is finite by definition of boundednessof B . Those two estimates together prove the stated hypocontinuity. (cid:3) Our next non-continuity result will be based on the argument that if we could have a fullcontinuity, we would have an isomorphism contradicting the improved target topology forhypocontinuity in our previous proposition.
Proposition 11.
Let Γ i ⊂ ˙ T ∗ U i closed cones and Λ i = − Γ ci ⊂ ˙ T ∗ U i open cones (comple-ments being taken in ˙ T ∗ U i , ˙ T ∗ U ), C i polar enlargeable families of closed sets on U i . Let also U = U × U , C = ( O ( O o C × O o C ) oo ) o , Γ = (Γ c ˙ × Γ c ) c ⊂ ˙ T ∗ U , and λ = (Λ c ˙ × Λ c ) c (which is open) Λ = − Γ c = Λ ˙ × Λ ⊂ λ, and γ = − λ c = Γ ˙ × Γ ⊂ Γ . Then the tensor map [( D ′ Γ , Γ ( U , F ) , I ) , ( D ′ Γ , Γ ( U , F ) , I )] → ( D ′ Γ , Γ ( U, F ) , I ) , [( D ′ Λ , Λ ( U , C ) , I ) , ( D ′ Λ , Λ ( U , C ) , I )] → ( D ′ λ,λ ( U, C ) , I ) , are NOT continuous as soon as (Γ , Γ )
6∈ { ( ∅ , ∅ ) , ( ˙ T ∗ U , ˙ T ∗ U ) } , for any topology I among I ppp , I ibi . Of course, from the stated inclusions of cones, our previous proposition proved they are(at least γ )-hypocontinuous. Note that the target cone Γ is only slightly smaller than theone of proposition 8, the only difference being in zero sections, explaining this result is quiteoptimal for full continuity. Proof.
Assume for contradiction that the maps are continuous, so that one gets continuousmaps (the second one by completion)( D ′ Γ , Γ ( U , F ) , I ) ˆ ⊗ π ( D ′ Γ , Γ ( U , F ) , I ) → ( D ′ Γ , Γ ( U, F ) , I ) , (4)(resp. ( D ′ Λ , Λ ( U , C ) , I ) ˆ ⊗ π ( D ′ Λ , Λ ( U , C ) , I ) → ( D ′ λ,λ ( U, C ) , I ) , )(5)and moreover from the hypocontinuity in proposition 9 (ii),(iii) we also have a continuous(note that we know ǫ -hypocontinuity is the same as γ -hypocontinuity in the stated case)( D ′ Λ , Λ ( U , K ) , I ) ⊗ βe ( D ′ Λ , Λ ( U , K ) , I ) → ( D ′ Λ , Λ ( U, K ) , I )(6)(resp. ( D ′ Γ , Γ ( U , ( O C ) o ) , I ibi ) ⊗ β ( D ′ Γ , Γ ( U , ( O C ) o ) , I ibi ) → ( D ′ γ,γ ( U, ( O C ) o ) , I ibi ) . )(7)We used the completion here in order to be able to go to the completed tensor product (noproblem for γ since it is closed).We first show this would imply the topological or algebraic isomorphisms :( D ′ Γ , Γ ( U , F ) , I ) ˆ ⊗ π ( D ′ Γ , Γ ( U , F ) , I ) ≃ alg ( D ′ Γ , Γ ( U, F ) , I ) , ( D ′ Λ , Λ ( U , K ) , I ) ˆ ⊗ βe ( D ′ Λ , Λ ( U , K ) , I ) ≃ top ( D ′ Λ , Λ ( U, K ) , I ) , (resp. ( D ′ Λ , Λ ( U , C ) , I ) ˆ ⊗ π ( D ′ Λ , Λ ( U , C ) , I ) ≃ alg ( D ′ λ,λ ( U, C ) , I ) , UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 15 ( D ′ Γ , Γ ( U , ( O C ) o ) , I ibi ) ˆ ⊗ β ( D ′ Γ , Γ ( U , ( O C ) o ) , I ibi ) ≃ top ( D ′ γ,γ ( U, ( O C ) o ) , I ibi ) . )and we will get a contradiction from these isomorphisms.We start by proving the stated isomorphisms. From (4), (5) we have one map for thealgebraic isomorphisms and we have to build the inverse. Since the spaces for which we takeprojective tensor products are complete nuclear and thus have the approximation propertyof Grothendieck, it suffices to build a continuous map to the Schwartz ǫ product (whichcoincides with the completed projective tensor product, and by density and agreement onsmooth maps, it will canonically be the inverse)( D ′ Γ , Γ ( U, F ) , I ) → ( D ′ Γ , Γ ( U , F ) , I ) ǫ ( D ′ Γ , Γ ( U , F ) , I ) ≃ D ′ Γ , Γ ( U , F ) ˆ ⊗ ǫ D ′ Γ , Γ ( U , F ) . (resp.( D ′ λ,λ ( U, C ) , I ) → ( D ′ Λ , Λ ( U , C ) , I ) ǫ ( D ′ Λ , Λ ( U , C ) , I ) ≃ D ′ Λ , Λ ( U , C ) ˆ ⊗ ǫ D ′ Λ , Λ ( U , C ) ) , But as we reminded, ( D ′ Γ , Γ ( U , F ) , I ) ǫ ( D ′ Γ , Γ ( U , F ) , I ) is the set of ǫ -hypocontinuous bi-linear maps on the product of Arens duals[( D ′ Γ , Γ ( U , F ) , I ) ′ c ⊗ βe ( D ′ Γ , Γ ( U , F ) , I ) ′ c ] ′ ≃ [( D ′ Λ , Λ ( U , K ) , I ) ˆ ⊗ βe ( D ′ Λ , Λ ( U , K ) , I )] ′ , since by propositon 1 the Arens topologies coincide with I on the duals (and we went tothe completions for completed tensor product for a more uniform notation). Respectively,( D ′ Λ , Λ ( U , C ) , I ) ǫ ( D ′ Λ , Λ ( U , C ) , I ) is the set of ǫ -hypocontinuous bilinear maps (whichcoincides here with hypocontinuous by the computation of equicontinuous sets in [D, lemma27] ) on the product of Arens duals[( D ′ Λ , Λ ( U , C )) ′ c ⊗ β ( D ′ Λ , Λ ( U , C )) ′ c ] ′ ≃ [( D ′ Γ , Γ ( U , ( O C ) o ) , I ibi ) ˆ ⊗ β ( D ′ Γ , Γ ( U , ( O C ) o ) , I ibi )] ′ , since by [D, Corol 26] the Arens topology coincides with I ibi .The maps we want are thus obtained by composition with the map (6) (resp. (7)), Wecould check the continuity of the first but we won’t need it.Again for the second isomorphism, both sides are complete and thus we have only toidentify the topologies on the algebraic tensor product which is dense on both sides. By themaps (6), (7) again, the topology induced by the bornological tensor product is stronger.But we just identified their duals (not topologically though) thus both are weaker than theMackey topology of duality by Mackey-Arens theorem. Finally, in [D, Corol 26] we checkedthe Mackey topology on D ′ Λ , Λ ( U, K ) (resp. D ′ γ,γ ( U, ( O C ) o )) is the same as I iii = I ibi = I (resp. I ibi ) thus all topologies coincide as expected. (Note we may also use [K, §
21. 4.(5)] toidentify the Mackey topology of a space and the one of the completion on the tensor productdense subspace of ( D ′ Λ , Λ ( U, K ) , I iii ) etc). We thus deduced the second isomorphism.Let us now get our contradiction in showing the completion of the maps (6), (7) are actuallynot surjective. But from our previous proposition 10, after completing the map obtained oninduced tensor products, we checked any u in the image satisfies the supplementary bound(taking D = 3 k as we can)sup ( ξ,η ) ∈ C k,k, ,η ∈ V (1 + | ξ | + | η | ) l |F ( u ( ϕ ⊗ ϕ ))( ξ, η ) | < ∞ , as soon as supp( ϕ ) × V ∩ Λ = ∅ . It thus suffices to prove this condition is not satisfiedby any u in ( D ′ Λ , Λ ( U, K ) , I iii ) , (respectively in the second case exchanging Λ i ’s and γ i ’s, weonly write the first case and leave the notational changes to the reader in the second). Note that for I ibi this is not written in the previous proposition, but in the closed case, we havefull hypocontinuity which goes to the bornologification and in the open case, we reason asin proposition 9 (ii) where we only have to check separate continuity from barelledness, andthen our proposition 10 is enough.We will use a specific case of H¨ormander’s example 8.2.4 [H97, p. 188] with s = − l, ρ =1 /k. Fix ξ a vector, | ξ | = 1 such that say ( x , ξ ) ∈ Λ × U × { } . This points exists ofcourse if Λ = ∅ , (we can assume that up to exchanging the indexes 1 , x = ( x, y ) ∈ U × U and since we are free about y , we choose y such that there is anon-empty closed cone V with { y } × V ∩ Λ = ∅ and it exists since Λ c = ∅ (without loss ofgenerality, since either Λ c = ∅ , and this is forced by our assumption, or otherwise Λ c = ∅ , and if Λ c = ∅ then Λ = ∅ and we could exchange again indexes 1 , ϕ , ϕ smooth compactly supported on a neighborhood of x, y respectively.Let χ ∈ C ∞ (IR , [0 , −∞ , /
2) and to 0 in (1 , + ∞ ), with 0 ≤ χ ≤ e = ξ , e , ..., e d ) such that ( e , ..., e d ) is an orthonormal basisof IR d × { } , ( e d +1 , ..., e d ) an orthonormal basis of { } × IR d , and write coordinates inthis coordinate system, so that for instance C k ,k, = { ( | ξ d +1 | + ... + | ξ d | ) ≥ ( | ξ | + ... + | ξ d | ) /k } . Note that A = { ( ξ = ... = ξ d = 0 , | ξ | ρ ≥ ( | ξ d +1 | + ... + | ξ d | ) ≥ | ξ | /k , ( ξ d +1 , ..., ξ d ) ∈ V } ⊂ C k ,k ∩ { ( | ξ d +1 | + ... + | ξ d | ) / | ξ | ρ ≤ } has non-emptyintersection with every complement of compact sets (for ρ = 1 /k ).Define u ξ ,s ∈ S ′ (IR n ), for s ∈ IR, by d u ξ ,s ( ξ ) = (1 − χ ( ξ )) ξ − s χ (( ξ + · · · + ξ n ) /ξ ρ ) . Then
W F ( u ξ ,s ) = { (0; ξ ); ξ = · · · = ξ n = 0 , ξ > } = { } × IR ∗ + ξ and u ξ ,s coincides witha function in S (IR n ) outside a neighborhood of the origin [H97, p. 188]. Of course we canconsider the obvious translation u = χ ( T x u ξ ,s ), χ smooth with compact support χ ( x ) = 1with DW F ( u ) = W F ( u ) = { x } × IR ∗ + ξ so that u ∈ D ′ Λ , Λ ( U, K ) . Moreover, since w = (1 − χ ( ϕ ⊗ ϕ ))( T x u ξ ,s ) ∈ S (IR d ) from the results above,(1+ | ξ | + | η | ) l |F ( u ( ϕ ⊗ ϕ ))( ξ, η ) | ≥ (1+ | ξ | + | η | ) l |F ( T x u ξ ,s )( ξ, η ) |− (1+ | ξ | + | η | ) l |F ( w )( ξ, η ) | and ( ξ, η ) (1 + | ξ | + | η | ) l |F ( w )( ξ, η ) | which is rapidly decreasing is not a problem atinfinity. Since we can bound below the sup we are interested in a supremum on A , on which χ (( ξ + · · · + ξ n ) /ξ ρ ) = 1, one deduces :sup ( ξ,η ) ∈ C k,k, ,η ∈ V (1+ | ξ | + | η | ) l |F ( T x u ξ ,s )( ξ, η ) | ≥ sup ( ξ,η ) ∈ A (1+ | ξ | + | η | ) l | (1 − χ ( ξ )) ξ − s | = ∞ , at least for s negative. Thus u is not in the image of the completion of (6) concluding toour statement and to the expected contradiction. (cid:3) Control of tensor products by wave front set conditions
Our final goal in the next paper of this series will be to describe all the tensor products weconsidered here in terms of microlocal conditions, notably in order to obtain extra functionalanalytic properties not given by the general theory. But since the lack of continuity presentedin proposition 11 gave an obstruction to an easy identification of these tensor products (seethe isomorphism from which we obtained a contradiction in the proof), we will still needin our applications a way of checking that a distribution gives an element of a completed
UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 17 tensor product by checking a (dual) wave front set condition. This is the goal of this sectionto give such conditions.For our convenience in order to state results in the context of most topologies of Theorem1, we will write for a cone γ (8) J = J = I ppp , J = J = I ibi , γ (1) = γ (3) = γ, γ (2) = γ (4) = γ, and j = j for j = 1 , j = 5 − j for j = 2 , D ′ γ,γ ( j ) ( U, C ; E ) , J j ) ′ c = ( D ′− γ c , − γ c ( j ) ( U, ( O C ) o ; E ∗ ) , J j ) . Proposition 12.
Let γ i ⊂ ˙ T ∗ U i be cones and C i enlargeable polar families of closed sets of U i , E i → U i vector bundles. Let γ = ( γ c ˙ × γ c ) c , Γ = (( γ ) c × ( γ ) c ) c and C = ( C o × C o ) o . Thereare continuous injections for any j ∈ { , , } : ( D ′ γ,γ ( j ) ( U × U , C ; E ⊗ E ) , J j ) ֒ → ( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ǫ ( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ֒ → ( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ˆ ⊗ π ( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ֒ → ( D ′ Γ , Γ ( U × U , F ; E ⊗ E ) , I ppp ) , and for any j ∈ { , } : ( D ′ γ,γ ( U × U , C ) , J j ) ֒ → ( D ′ γ ,γ ( U , C ) , I ppp ) ǫ ( D ′ γ ,γ ( U , C ) , J j ) . Proof.
We know from [D, Prop 33] that all spaces involved in the first line are nuclear forthe stated topologies, thus completed projective product and injective product coincide, andmoreover, they have the approximation property (see e.g.[ ? , p 110]), thus by [S, prop 11 p46], the ǫ product is a dense subspace of this completed injective product, we only have tobuild a map to/from Schwartz ǫ -product (since the last space ( D ′ Γ , Γ ( U × U , F ; E ⊗ E ) , I ppp )is complete). Using injectivity of the ǫ product of maps [S, p 20] (and the equality I pip = I iii in the closed DWF/WF case and injections in [D, lemma 21]) we have( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ǫ D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ֒ → ( D ′ γ ,γ ( U , C ; E ) , I ppp ) ǫ ( D ′ γ ,γ ( U , C ; E ) , I ppp ) . Using also our proposition 8 and composing with the previous one, the second map is thusknown (and injective via inclusions in distributions).But thanks to the identifications of the Arens duals in the same corollary( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ǫ ( D ′ γ ,γ ( j ) ( U , C ; E ) , J j ) ≃ [( D ′− γ c , − γ c ( j ) ( U , ( O C ) o ; E ∗ ) , J j ) ⊗ βe ( D ′− γ c , − γ c ( j ) ( U , ( O C ) o ; E ∗ ) , J j )] ′ . Our first injective map is thus built in dualizing the dense range tensor multiplication mapof proposition 9 (ii,iii), using also ( O C ) o = ( O ( C o ×C o ) o ) o = ( O o C × O o C ) oo , (as is easily checked,using crucially enlargeability for ⊃ ). To prove continuity, it suffices to prove the tensormultiplication map sends tensor products of equicontinuous sets to equicontinuous sets. For j = 4 this is obvious since equicontinuous sets are the same as bounded sets. For j = 1 , D ′− γ ci , − γ ci ( U i , ( O C i ) o ; E ∗ i ) to bounded sets of( D ′ Π i , Π i ( U i , ( O C i ) o ; E ∗ i ) , I H,iii ) with a closed Π i ⊂ − γ ci . But, since we have hypocontinuity atlevel of each term of the defining inductive limits I iii , I ibi , and since an equicontinuous set B defines an equicontinuous family of maps u ⊗ · which thus preserve those bounded sets of( D ′ Π i , Π i and thus also preserve equicontinuous sets in D ′− γ ci , − γ ci ( U i , ( O C i ) o ; E ∗ i ), as expected. For the second inclusion, we can translate what we are looking for through duality (recallnow j = 1 , D ′ γ,γ ( U × U , C ) , J j ) ֒ → [( D ′− γ c , − γ c ( U , ( O C ) o ) , I b ) ⊗ βe ( D ′− γ c , − γ c ( j ) ( U , ( O C ) o ) , J j = I ppp )] ′ . We used the computation of Arens duals summarized in Theorem 1.Our second injective map is thus built in dualizing the dense range tensor multiplicationmap of proposition 9 (iv). To check continuity, again, it suffices to see the tensor mul-tiplication map sends products of equicontinuous sets to equicontinuous sets. But fixing B equicontinuous in the first space (thus we can assume γ closed by our identification ofequicontinuous sets), u ⊗ . gives an equicontinuous family, and in the case j = 1 via theidentification of equicontinuous sets to those having wave front set in a closed subcone in theinductive limit, this reduces to the case j = 2 (with even closed γ ), in which case boundedsets and equicontinuous sets agree (at target partly from our reduction to γ closed) . Thisconcludes to our last result. (cid:3) Spaces of multilinear maps
Our description of tensor products gives us access to abstract description of spaces ofhypocontinuous multilinear maps on generalized H¨ormander spaces of distributions. Forconvenience we still use notation (8).We summarize the descriptions of the spaces of interests which are ǫ -products and giverelations with spaces controlled by wave front set conditions. All spaces of hypocontinuousmultilinear maps will be given their canonical topology as ǫ products, namely of uniformconvergence on products of equicontinuous sets, see [S]. Proposition 13.
Let λ i = − γ ci , λ = − γ c be cones, E i → U i , E → U vector bundles and k i ∈ { , , , } , i = 1 , ..., n ; k ∈ { , } L ( O βe,i ∈ [1 ,n ] ( D ′ γ i ,γ i ( k i ) ( U i , C i ; E i ) , J k i ); ( D ′ γ,γ ( U, C ; E ) , J k )) ≃ [ ε i ∈ [1 ,n ] ( D ′ λ i ,λ i ( k i ) ( U i , ( O C i ) o ; E ∗ i ) , J k i )] ε ( D ′ γ,γ ( U, C ; E ) , J k ) is nuclear. It is also complete when all k i ∈ { , } in which case it is also the space ofhypocontinuous multilinear maps, the corresponding completed projective tensor product andeven the space of bounded multilinear maps when k i = k = 4 . We also have the followingcontinuous inclusions for γ a = ( − γ ˙ × ... ˙ × − γ n ˙ × γ c ) c and C = (( O C ) oo × ... × ( O C n ) oo × C o ) o on V = U × ... × U n × U : ( D ′ γ a ,γ a ( K ) ( V, C ; E ∗ ⊗ ... ⊗ E ∗ n ⊗ E ) , J K ) ֒ → L ( O βe,i ∈ [1 ,n ] ( D ′ γ i ,γ i ( k i ) ( U i , C i ; E i ) , J k i ); ( D ′ γ,γ ( U, C ; E ) , J k ) , for k = ... = k n − equal in the cases k = k = k n = K ∈ { , } , or K = 3 with either k = k n = 4 , k = 2 , or k n ∈ { , , } , k = 3 , k = 2 and finally, the case K = 1 , k = 3 , k n ∈{ , } , k = 2 . We can also have the case k i ∈ { , } , k ∈ { , } , K = 3 partially coveredbefore.Proof. We computed in Theorem 1 the various Arens=Mackey duals. The identification withthe ǫ -product comes from [S, Prop 4 p 30], using only the completeness of the target space.Since from [S, Prop 11 p 46], the ǫ -product has for completion the completed injective tensorproduct, in the case where spaces have the approximation property, one deduces they are UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 19 all nuclear as their completion. They are complete when all the spaces of the ǫ product arecomplete [S, Prop 3 p 29]. The statement of hypocontinuity comes from the identificationof bounded and equicontinuous sets as explained in lemma 3 and in the bornological case,the identification of hypocontinuous linear maps and bounded linear maps.Applying [S, Prop 1 p 20, Prop 7 p 38] we get inductively from proposition 12 the variousstated injections. For instance in the case k = k = k n = K ∈ { , } we only have to userepeatedly the first injection, the point is that all the spaces of the ǫ -product involved arecomplete, so that we can apply associativity of ǫ product combined with the preservation ofinjectivity of ǫ products of continuous injections.When K ∈ { , } we first apply the second injection of proposition 12, once composed onthe right with the completion maps to go from case j ∈ { , } to j + 1 ∈ { , } we can thenapply again associativity freely and get the cases K = 3 with either k = k n = 4 , k = 2, or k n ∈ { , } , k = 3 , k = 2 and finally, the case K = 1 , k = 3 , k n = 3 , k = 2 , and withoutequalities of k i ’s, k i ∈ { , } , k ∈ { , } , K = 3.The cases K = 3 , k n = 2 , k = 3 , k = 2 and K = 1 = k , k n = 3 , k = 2 need a supplemen-tary remark since one space is never complete. Here, we can gather terms which are completein an ǫ -product. Indeed, this is a consequence of [S, Prop 7 p 38] with two applications of[S, Prop 4 p 30] which enables to gather complete spaces in the space of value, then applyassociativity, and then come back to the ǫ product with associativity applied. In this way,since there is only one term not complete in the ǫ product above, one gathers all others anduse the injections as above inductively. (cid:3) The easiest case is for the bornological complete topologies with all indexes j = 4. At firstreading, the reader should probably only consider this case where everything is straightfor-ward. Since it is not clear at this point if this will be sufficient for applications (especiallybecause the topology is quite tricky even to define in this case), we consider more generalsituations.Our next result makes explicit nice spaces with controlled (dual) wave front set includedin spaces of operators, and the various continuity of compositions on the full space and onrestrictions. As we will see the only technical assumption is made for the target space wherecomposition is made. We write ǫ eq the family of ǫ -equihypocontinuous [S, p 18] parts of L ( N βe,i ∈ [1 ,m ] ( D ′ γ ′ i ,γ ′ i ( k ′ i ) ( U, C ′ i ; E ′ i ) , J k ′ i ); ( D ′ γ ′ ,γ ′ ( U, C ′ ; E ′ ) , J k ′ )) . Proposition 14.
In the setting of our previous proposition, and with supplementary cones γ ′ i , γ ′ , i ∈ [1 , m ] , polar enlargeable families C ′ i , C ′ , vector bundles E ′ i → U ′ i , E ′ → U ′ suchthat for a fixed index j , γ ′ j = γ , C ′ j = C , E ′ j = E, U ′ j = U . Assume we write γ ′′ i = γ ′ i for i ∈ [1 , j − , γ ′′ i = γ i − j for i ∈ [ j, j + n − , γ ′′ i = γ ′ i − n +1 for i ∈ [ j + n, m + n − , and similarlyfor C ′′ j , E ′′ j . Consider also k i ∈ { , , , } , i = 1 ...n, k ′ i ∈ { , , , } , i = 1 ...m, k, k ′ ∈ { , } with k ′ j = k. Also write k ′′ i = k ′ i for i ∈ [1 , j − , k ′′ i = k i − j for i ∈ [ j, j + n − , k ′′ i = k ′ i − n +1 for i ∈ [ j + n, m + n − . Then the map below corresponding to composition in the j-th variable is hypocontinuouswhen k = 4 and ǫ eq − β -hypocontinuous (thus separately continuous) when k = 2 (always with the ǫ -product topologies of proposition ). ◦ j : L ( O βe,i ∈ [1 ,m ] ( D ′ γ ′ i ,γ ′ i ( k ′ i ) ( U ′ i , C ′ i ; E ′ i ) , J k ′ i ); ( D ′ γ ′ ,γ ′ ( U ′ , C ′ ; E ′ ) , J k ′ )) × L ( O βe,i ∈ [1 ,n ] ( D ′ γ i ,γ i ( k i ) ( U i , C i ; E i ) , J k j ); ( D ′ γ,γ ( U, C ; E )) , J k ))) → L ( O βe,i ∈ [1 ,n + m − ( D ′ γ ′′ i ,γ ′′ i ( k ′′ i ) ( U ′′ i , C ′′ i ; E ′′ i ) , J k ′′ i ); ( D ′ γ ′ ,γ ′ ( U ′ , C ′ ; E ′ ) , J k ′ )) which restricts also to ǫ -hypocontinuous maps to the (often continuously) embedded sub-spaces given in proposition . Explicitly if γ a = ( − γ ˙ × ... ˙ × − γ n ˙ × γ c ) c , γ b = ( − γ ′ ˙ × ... ˙ × − γ ′ m ˙ × ( γ ′ ) c ) c , γ c = ( − γ ′′ ˙ × ... ˙ × − γ ′′ n + m − ˙ × ( γ ′ ) c ) c , C = (( O C ) oo × ... × ( O C n ) oo × C o ) o , V = U × ... × U n × U, E = E ∗ ⊗ ... ⊗ E ∗ n ⊗ E → V and similarly for C ′ , C ′′ , V ′ , V ′′ = U ′′ × ... × U ′′ n + m − × U ′ , E ′ , E ′′ the following maps coming from restriction are ǫ -hypocontinuous : ◦ j : ( D ′ γ b ,γ b ( κ ) ( V ′ , C ′ ; E ′ ) , J κ ) ⊗ βe ( D ′ γ a ,γ a ( κ ) ( V, C ; E ) , J κ ) → ( D ′ γ c ,γ c ( κ ) ( V ′′ , C ′′ ; E ′′ ) , J κ ) , with either κ = κ = κ ∈ { , , } or κ = 2 , κ = κ ∈ { , } . We even get full hypoconti-nuity when κ ∈ { , } . Proof.
There is no problem to define the composition in any case.In order to check the spaceof value is indeed in ǫ -equicontinuous maps by composition, it suffices to note that a productof equicontinuous sets is sent by a map in the second argument to an equicontinuous set.For, note that evaluated in all but one variable the maps above gives an equicontinuousfamily of continuous linear maps, which thus sends bounded sets (as equicontinuous sets) tobounded sets (see [K2, § D ′ γ,γ ( U, C ; E ) , J k ) , k = 2 , ǫ − hypocontinuity of a composition A ◦ j B. Note that this boils down to getting equicontinuity of the family of maps obtainedby evaluating all but one variable of A ◦ j B to equicontinuous sets. But for this there are 2cases, the case where all the arguments of B are evaluated, in which case the equicontinuityof the image just proved concludes since the j -th argument of A then receives such anequicontinuous family. The second case is where all the arguments of A but the j -th oneare evaluated to equicontinuous sets, and then the equicontinuouity follows by (the obvious)composition of equicontinuous families.To prove hypocontinuity of ◦ j , take B i equicontinuous in ( D ′ γ ′′ i ,γ ′′ i ( k ′′ i ) ( U, C ′′ i ; E ′′ i ) , J k ′′ i ), A, A ′ bounded in L ( N βe,i ∈ [1 ,m ] ( D ′ γ ′ i ,γ ′ i ( k ′ i ) ( U, C ′ i ; E ′ i ) , J k ′ i ); ( D ′ γ ′ ,γ ′ ( U, C ′ ; E ′ ) , J k ′ )) and L ( N βe,i ∈ [1 ,n ] ( D ′ γ i ,γ i ( k i ) ( U, C i ; E i ) , J k j ); ( D ′ γ,γ ( U, C ; E ) , J k )) respectively, and finally C equicon-tinuous in ( D ′ γ ′ ,γ ′ ( U, C ′ ; E ′ ) , J k ′ ) ′ ≃ ( D ′− ( γ ′ ) c , − ( γ ′ ) c ( k ′ ) ( U, ( O C ′ ) o ; ( E ′ ) ∗ ) , I ibi ).We have to boundsup b i ∈ B i ,c ∈ C,u ∈ A |h ( u ◦ j v )( b , ...., b n + m − ) , c i| , sup b i ∈ B i ,c ∈ C,v ∈ A ′ |h ( u ◦ j v )( b , ...., b n + m − ) , c i| since without the sup over A, A ′ one gets a seminorm of the target space so that we have tobound this uniformly in A, A ′ by a seminorm for v, u respectively to get hypocontinuity.The second case is easy since A ′′ = A ′ ( B j , ..., B j + n − ) is bounded in the target space for v namely ( D ′ γ,γ ( U, C ; E ) , J k ) , k = 2 , A ′ bounded, but, as we said, in this UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 21 space, any bounded set is equicontinuous thus so is A ′′ , and :sup b i ∈ B i ,c ∈ C,v ∈ A ′ |h ( u ◦ j v )( b , ...., b n + m − ) , c i| ≤ sup b i ∈ B i ,v j ∈ A ′′ ,c ∈ C |h ( u ( b , ...., b j − , v j , b j + n , ..., b n + m − ) , c i| is the bound by a seminorm for u .The first case will require the restriction k = 4 to get full hypocontinuity. Indeed, like-wise A ( B , ...., B j − , · , B j + n , ..., B n + m − ) is, by [S, Prop 2 bis p 28], equicontinuous from( D ′ γ,γ ( U, C ; E ) , I ibi ) to ( D ′ γ ′ ,γ ′ ( U ′ , C ′ ; E ′ ) , J k ′ ) (where we used our computations of strong du-als ( D ′ γ,γ ( U, C ; E ) , I ibi ) of Arens duals ( D ′− ( γ ) c , − ( γ ) c ( k ) ( U, ( O C ′ ) o ; E ∗ ) , I ibi ) of ( D ′ γ,γ ( U, C ; E ) , J k )to formulate things under Schwartz hypothesis. The strong dual computation uses propo-sition 6. This is where we will need k = 4 so that the first space does not change). thus,by [K2, § C to a boundedset C ′ = [ A ( B , ...., B j − , · , B j + n , ..., B n + m − )] ′ ( C ) which is again equicontinuous when k = 4because of the target space has also topology I ibi where bounded sets are equicontinuous.Thus we have the expected boundsup b i ∈ B i ,c ∈ C,v ∈ A ′ |h ( u ◦ j v )( b , ...., b n + m − ) , c i| ≤ sup b i ∈ B i ,c ∈ C ′ ,v ∈ A ′ |h v ( b j , ...., b j + n − ) , c i| . In case k = 2 for the first argument fixed to a single element thus A = { u } is ǫ -equihypocontinuousby definition, or also more generally for A ǫ -equihypocontinuous, thus[ A ( B , ...., B j − , · , B j + n , ..., B n + m − )] is equicontinuous thus by [K2, § A ( B , ...., B j − , · , B j + n , ..., B n + m − )] ′ send equicontinuous sets in the dual toequicontinuous sets which enables to conclude as before.For the induced maps, we get continuity by realizing the restriction as a canonical compo-sition of 3 maps. The first is the tensor map t obtained in lemma 9 (ii) (case κ = 3 , κ = 1) and (iv) (case κ = 2, explaining the various classes of hypocontinuity ob-tained) with target space D ′ γ a × γ b , ( γ a ˙ × γ b )( κ ) ( V × V ′ , ( C × C ′ ) oo ; E ⊗ E ′ ) , J κ ) (we even got tothe completion of the target space obtained there to get a common target in all our cases.The second map is a pullback via the map (recall U ′ j = U ) f j : U := U ′ × ... × U ′ m × U ′ × U × ... × U n → V × V ′ with f j ( x , ...., x m , y, z , ..., z n ) = ( x , ...., x m , y, z , ..., z n , x j ), so that df ∗ j ( ξ , ...., ξ m , η, ζ , ..., ζ n , η ′ ) =( ξ , ...., ξ j + η ′ , ..., ξ m , η, ζ , ..., ζ n ) and thus df ∗ j ( γ a ˙ × γ b ) = { ( x , ...., x m , y, z , ..., z n , ξ , ...., ξ j + η ′ , ..., ξ m , η, ζ , ..., ζ n ) : ( x , ...., x m , y, z , ..., z n , x j , ξ , ...., ξ m , η, ζ , ..., ζ n , η ′ ) ∈ ( γ b ˙ × γ a ) ∩ ˙ T ∗ ( V × V ′ ) } . If X = ( x , ...., x m , y, z , ..., z n , ξ , ...., ξ j + η ′ , ..., ξ m , η, ζ , ..., ζ n ) ∈ df ∗ j ( γ b ˙ × γ a ) ∩ ˙ T ∗ ( V × V ′ ) and assume moreover ξ j + η ′ = 0 . Let us show that X ( − γ ′ ˙ × ... − γ ′ j − ˙ ×{ } ... ˙ × − γ ′ m ˙ × ( γ ′ ) c ˙ × − γ ˙ × ... ˙ × − γ n ). If it were, we would have (( x i , ξ i ) ∈ − γ ′ i or ξ i = 0) for all i = j ,and (( z i , ζ i ) ∈ − γ i or ζ i = 0) for all i and moreover ( y, η ) ∈ ( γ ′ ) c or η = 0. Thus if η ′ = ξ j = 0 one contradicts the assumption ( x , ...., x m , y, z , ..., z n , x j , ξ , ...., ξ m , η, ζ , ..., ζ n , η ′ ) ∈ ( γ b ˙ × γ a ) ∩ ˙ T ∗ ( U n + m +2 ) (since from the assumptions on ( z i , ζ i ) one cannot have (( x j , η ′ ) ∈ γ c or η ′ = 0) especially η ′ = 0 is forbidden because of the definition of γ a and also similarly, onecannot have (( x j , ξ j ) ∈ − γ j = γ or ξ ′ j = 0 from the definition of γ b ) or otherwise if η ′ = 0 andthus in order not to contradict ( z , ..., z n , x j , ζ , ..., ζ n , η ′ ) ∈ γ a one must have ( x j , η ′ ) ∈ γ andsimilarly since then ξ j = 0, in order not to contradict ( x , ...., x m , y, ξ , ...., ξ m , η ) ∈ γ b one must have ( x j , ξ j ) ∈ ( − γ ′ j ) c = ( − γ ) c and this contradicts ξ j = − η . Altogether we deduced df ∗ j ( γ a ˙ × γ b ) ∩ T ∗ U j − ×{ }× T ∗ U n + m +1 − j ⊂ ( − γ ′ ˙ × ... − γ ′ j − ˙ ×{ } ... ˙ ×− γ ′ m ˙ × ( γ ′ ) c ˙ ×− γ ˙ × ... ˙ ×− γ n ) c . Similarly, let us show that f − j (( C ′ × C ) oo ) ⊂ C ′′′ := (( O C ′ ) oo × ... × ( O C ′ j − ) oo × K o × ( O C ′ j +1 ) oo × ... × ( O C ′ m ) oo × C ′ o × ( O C ) oo × ... × ( O C n ) oo ) o Thus take O i ∈ ( O C ′ i ) oo , V i ∈ ( O C i ) oo , W ∈ K o , W ′ ∈ C ′ o and let A = O × ... × O j − × W × O j +1 × ...O m × W ′ × V × ... × V n , and let B ∈ C ′ , C ∈ C , then A ∩ f − j ( B × C ) ⊂ f − j ( f j ( A ) ∩ ( B × C )) ⊂ f − j [( A ′ ∩ B ) × ( A ′′ ∩ C )] with A ′ = O × ... × O j − × W × O j +1 × ...O m × W ′ , A ′′ = V × ... × V n × W . But note that if p j is the projection on the j -th coordinate,we have T := p j ( A ′ ∩ B ) ∈ ( O C ′ j ) o since if W ′′ ∈ ( O C ′ j ) oo , W ′′ ∩ p j ( A ′ ∩ B ) ⊂ p j ( B ′ ∩ B )which is compact, since B ′ := O × ... × O j − × W ′′ × O j +1 × ...O m × W ′ ∈ ( C ′ ) o . Likewise S := p n +1 ( A ′′ ∩ C ) ∈ C = C ′ j . Let U × U ′ j × U := U ′ × ... × U ′ m × U ′ , U := U × ... × U n . But from this and the definition of f j we obviously have f − j [( A ′ ∩ B ) × ( A ′′ ∩ C )] = f − j [( A ′ ∩ B ) ∩ ( U × T × U ) × ( A ′′ ∩ C ∩ ( U × S ))]= f − j [( A ′ ∩ B ) ∩ ( U × S × U ) × ( A ′′ ∩ C ∩ ( U × T ))] . thus A ∩ f − j ( B × C ) ⊂ f − j [( D ′ ∩ B ) × ( D ′′ ∩ C )] with D ′ = O × ... × O j − × Int ( S ǫ ) × O j +1 × ...O m × W ′ ∈ ( C ′ ) o , D ′′ = V × ... × V n × Int ( T ǫ ) ∈ C o ,for some ǫ i >
0, by en-largeability, so that ( D ′ ∩ B ) , ( D ′′ ∩ C ) are compact and thus since Im ( f j ) is closed and f j has a continuous inverse on its image, one deduces f − j [( D ′ ∩ B ) × ( D ′′ ∩ C )] and then A ∩ f − j ( B × C ) compact and thus the stated inclusion f − j (( C ′ × C ) oo ) ⊂ C ′′′ since by[D]Ex 5 any closed set in there is included in a set of the form above f − j ( B × C ). Again A ∩ ( f − j ( B × C )) ǫ ⊂ ( A δ ( ǫ ) ∩ ( f − j ( B × C ))) uǫ which is also compact for ǫ = ǫ ( B, C ) smallenough, thus we even have ( f j ) − e (( C ′ × C ) oo ) ⊂ C ′′′ , for some function ǫ given above (recallthe notation ( f j ) − e introduced in proposition 2 contains an implicit ǫ ).From the application of proposition 2, we thus got our second map after composition bya canonical map : f ∗ j : ( D ′ γ a × γ b , ( γ a ˙ × γ b )( κ ) ( V × V ′ , ( C × C ′ ) oo ; E ⊗ E ′ ) , J κ ) → ( D ′ γ d ,γ d ( κ ) ( V × U , C ′′′ ; f ∗ j ( E ⊗ E ′ )) , J κ )with γ d :=[( − γ ′ ˙ × ... − γ ′ j − ˙ ×{ } ... ˙ × − γ ′ m ˙ × ( γ ′ ) c ˙ × − γ ˙ × ... ˙ × − γ n ) c ∩ ( T ∗ U × { } × T ∗ ( U × U ))] ∪ T ∗ U × ˙ T ∗ ( U ′ j ) × T ∗ ( U × U )(our computation above implies the right condition in the hypothesis on the 0 section, γ d ⊂ ˙ T ∗ ( V × U )).The third map necessary to recover the restricted composition map is easier to build.Recall E ′′ = ( E ′ ) ∗ ⊗ ... ⊗ ( E ′ j − ) ∗ ⊗ ( E ) ∗ ⊗ ... ⊗ ( E n ) ∗ ⊗ ( E ′ j +1 ) ∗ ⊗ ... ⊗ ( E ′ n ) ∗ ⊗ E ′ → V ′′ anddefine E ′′′ = ( E ′ ) ∗ ⊗ ... ⊗ ( E ′ j − ) ∗ ⊗ ( lC) ∗ ⊗ ( E ′ j +1 ) ∗ ⊗ ... ⊗ ( E ′ n ) ∗ ⊗ E ′ ⊗ ( E ) ∗ ⊗ ... ⊗ ( E n ) ∗ → V × U . The third map is then defined from the projection map composed with a permutation
UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 23 g j ( x , ...., x m , y, z , ..., z n ) = ( x , ...., x j − , z , ..., z n , x j +1 , ...., x m , y ) , so that with the first partof proposition 2 again, we get g ∗ j : ( D ′− γ cc , − γ cc ( κ ) ( V ′′ , ( O C ′′ ) o ; ( E ′′ ) ∗ ) , J κ ) → ( D ′− γ cd , − γ cd ( κ ) ( V × U , ( O C ′′′ ) o ; g ∗ j (( E ′′ ) ∗ ) = ( E ′′′ ) ∗ ) , J κ )since by a computation already used in the proof of proposition 12 (using enlargeability)( O C ′′′ ) o = ( C ′ × ... × C ′ j − × F × C ′ j +1 × ... × C ′ m × O o C ′ × C × ... × C n ) oo ⊃ g − j ( O o C ′′ ) = g − j (( C ′ × ... × C ′ j − × C × ... × C n × C ′ j +1 × ... × C ′ m × O o C ′ ) oo )where we used again [D]Ex 5to see any element of ( C ′ × ... × C ′ j − × C × ... × C n × C ′ j +1 × ... × C ′ m × O o C ′ ) oo is included in some product and thus g − j (( C ′ × ... × C ′ j − × C × ... × C n ×C ′ j +1 × ... × C ′ m × O o C ′ ) oo ) ⊂ [ g − j (( C ′ × ... × C ′ j − × C × ... × C n × C ′ j +1 × ... × C ′ m × O o C ′ ))] oo . Taking the adjoint of g ∗ j continuous between Mackey duals, one gets the third map wewanted :( g j ) ∗ := ( g ∗ j ) ∗ ◦ T r : ( D ′ γ d ,γ d ( κ ) ( V × U , C ′′′ ; f ∗ j ( E ⊗ E ′ )) , J κ ) → ( D ′ γ c ,γ c ( κ ) ( V ′′ , C ′′ ; E ′′ ) , J κ )where T r is the map induced from the bundle map E ⊗ E ∗ → lC ∗ . Thus ◦ j = ( g j ) ∗ ◦ f ∗ j ◦ t and it suffices to check it agrees on a dense set thus everywhere with the restricted map. (cid:3) Beyond composition studied in the previous lemma, we will be interested in tensor productson spaces of multilinear maps and their continuity properties. The useful easy results aregathered in the next lemma which is a consequence of associativity of ǫ product [S, Prop7 p 38] and the canonical map between projective tensor product and ǫ -product (both inthe complete case (with approximation property for the second point requiring identificationwith completed injective product). Lemma 15.
In the setting above, and with supplementary cones γ ′ i , γ ′ , i ∈ [1 , m ] polar en-largeable families C ′ i , C ′ assume we write γ ′′ i = γ ′ i for i ∈ [1 , m ] , γ ′′ m +1 = ( − γ ′ ) c , γ ′′ i = γ i − m − for i ∈ [ m + 2 , m + n + 1] , and similarly for C ′′ j (especially C ′′ m +1 = O C ′ ) o ). Then we have acontinuous map corresponding to tensor multiplication at the level of distribution spaces . ⊗ . : L ( O βe,i ∈ [1 ,n ] ( D ′ γ ′ i ,γ ′ i ( U, C i ) , I b ); ( D ′ γ ′ ,γ ′ ( U, C ′ )) , I ppp ))) × L ( O βe,i ∈ [1 ,n ] ( D ′ γ i ,γ i ( U, C i ) , I b ); ( D ′ γ,γ ( U, C )) , I ppp ))) → L ( O βe,i ∈ [1 ,n + m − ( D ′ γ ′′ i ,γ ′′ i ( U, C ′′ i ) , I b ); ( D ′ γ,γ ( U, C ) I ppp )) Part Spaces of smooth functionals with wave front set conditions. Topology and algebraic structure
In this section, we want to study topological (and algebraic) properties of some spaces ofconveniently smooth [KM] (vector valued) functionals on an open U ⊂ E ( U, E ) (open in theusual Fr´echet space topology thus c ∞ -open, [KM, Th 4.11]) for E → U a bundle as above.These spaces will be variants of microcausal functionals (studied in this generality in [BFRi])when specified to specific cones. We will moreover study two classes of spaces. The goalis to solve easily an issue appeared there for defining Poisson (and retarded) brackets withfield dependent propagator. This uses crucially our study of multilinear maps on generalizedH¨ormander spaces of distributions. The first variant is the more similar to [BFRi]. Fix m ∈ IN ∗ . Let λ = ( λ n ) a family ofcones λ n ⊂ ( ˙ T ∗ U n ) J for some set J, λ n = ( λ n,j ) j ∈ J . We will often assume : Assumption 1 : λ n,j ˙ × λ N,j ⊂ λ n + N,j , ∀ j ∈ J ; n, N ≥ . We also consider C , C ′ polar enlargeable families of closed sets. Note that ( C n ) oo is thenalso enlargeable. We will fix E a separated locally convex space of value (as motivated forinstance by [FR, FR2, R, BFR], but note that our investigation at this stage is only reallypreliminary, and the assumption on E won’t be suitable to apply it as is to these papers, amore general treatment will require the investigation of the third part of this paper). Thislocally convex space will enable to treat field dependent retarded product in a uniform wayas functionals valued in a locally convex space.For a smooth map F : U → E it is known [KM, Corol 5.11] that the n -th differen-tial F ( n ) ∈ C ∞ ( U, L b ( E ( U, E ) ⊗ β n ; E )) is bounded (even symmetric) multilinear map. Butsince E ( U, E ) is a Fr´echet space it is known that E ( U, E ) ˆ ⊗ β n = E ( U, E ) ˆ ⊗ π n = E ( U n , E ⊗ n )(see e.g. [T, Cor of Th 34.1,Th 51.6] in the non-bundle case) which is bornological sothat F ( n ) ∈ C ∞ ( U, E ′ ( U n , ( E ⊗ n ) ∗ ) ǫ E ) ⊂ C ∞ ( U, D ′ ( U n , ( E ⊗ n ) ∗ ) ǫ E ) . Indeed smooth maps de-pend only of the bornology [KM, Corol 1.8] which is the same between E ′ ( U n , E ⊗ n ) ǫ E and L b ( E ( U, E ) ⊗ β n ; E ) (see [S, corol 2 p 34] with our identification above and since E ( U n , E ⊗ n )and its dual have their Mackey topology since they are bornological, we also use their equicon-tinuous sets are exactly their bounded set for the same reason).Note that from [BFRi, lemma 2.3.8] there is a natural notion of support of a functional(when E = lC), such that in the smooth case supp( F ) = S ϕ ∈ U supp( F (1) [ ϕ ]) . For the caseof more general E , we suppose given a bornology B on some subspace of E ′ . Then wecan define for B ∈ B , supp( F, B ) = S ϕ ∈ U ,ψ ∈ B supp( F (1) [ ϕ ]( ψ )) , where, as seen above, F (1) [ ϕ ]( ψ ) ∈ E ′ ( U, E ∗ ) . When not specified, we take the bornology B f of all subsets of thedual, especially in the case E = lC (in this case we only write supp( F )). In this subsection,this bornology can be ignored, it will be used in the next subsection. At this stage, forinstance, there is no reason to distinguish between scalarly compact support and real compactsupport (for B f ). We also fix a family of locally convex spaces with continuous injections E → E i , i ∈ J. We define a first space of functionals on an open U ⊂ E ( U, E ) : F λ ( E ( U, E ) , U , C , C ′ ; ( E , ( E i ) i ∈ J , B )) = { F : U → E | F smooth , ∀ n ∈ IN ∗ , ∀ j ∈ J,F ( n ) ∈ C ∞ ( U , ( D ′ λ n,j ,λ n,j ( U n , ( C n ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ǫ E i ); ∀ B ∈ B , supp( F, B ) ∈ C ′ } . The second variant using more explicitly spaces of multilinear maps will be called anoperadic variant, depending on Γ = { ( γ j , γ ′ j ) } j ∈ J a set of pairs of cones, then we define: F Γ ( U , C ; ( E , ( E i ) i ∈ J , B )) = { F : U → E | F smooth , ∀ B ∈ B , supp( F, B ) ∈ C , ∀ i ∈ J, ∀ n ∈ IN ∗ , F ( n ) ∈ C ∞ ( U , L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) n − ; ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi )] ǫ E i ) } . In most cases all E i = E and we only write ( E , B ) instead of ( E , ( E ) i ∈ J , B ), and E insteadof ( E , B f ) as we said.We will often assume : Assumption 2 : − γ i ⊂ ( γ ′ i ) c , ∀ i ∈ J. UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 25
Let us give examples :
Example 16.
Local functionals :
Let J = { } , and λ n = λ n, = C n = { ( x, ..., x ; ξ , ..., ξ n ) ∈ ˙ T ∗ ( U n ) : x ∈ U, ξ + ... + ξ n = 0 } the conormal bundle to the small diagonal and write λ = C. Note that C n ⊂ ( − γ ˙ × ( n − ˙ × γ c ) c as soon as γ is a semigroup for addition. Then F C ( E ( U, E ) , U , K , K ; lC ) has an interesting closed subspace (for the topologies defined be-low) : F µloc ( U ; lC ) = { F ∈ F C ( E ( U, E ) , U , K , K ; lC ) : F additive } is one of the possible definitions of microlocal functionals in [BFRi] , where F is said to beadditive if ∀ ϕ ∈ U and ϕ , ϕ ∈ E ( U, E ) such that ϕ + ϕ , ϕ + ϕ , ϕ + ϕ + ϕ ∈ U andsupp ( ϕ ) ∩ supp ( ϕ ) = ∅ then : F ( ϕ + ϕ + ϕ ) = F ( ϕ + ϕ ) + F ( ϕ + ϕ ) − F ( ϕ ) . We will thus see below it is complete nuclear for the topologies we will soon define. Note that C does not satisfy assumption 1. Example 17.
Microcausal functionals :
Let J = { , } , U = M a (time oriented)globally hyperbolic manifold (for a metric g ) and µ cn, = V + n = { ( x , ..., x n ; ξ , ..., ξ n ) ∈ ˙ T ∗ ( M n ) : ξ i ∈ V + ( x i ) } with V + ( x i ) the closure of V + ( x i ) the future light cone at x i and µ cn, = V − n analogously. Then the minimal variant of the space of microcausal functionals(see e.g. [BFRi] ) with DW F ( F ( n ) ) ⊂ µ n, ∩ µ n, (instead of the same condition on wavefront sets in the usual approach) is: F µc (( M, g ) , U ; E ) = F µ ( E ( U, E ) , U , K , K ; E ) . We have the operadic variant if we define γ mc, = V + , γ ′ mc, = ( V − ) c , γ mc, = V − , γ ′ mc, =( V + ) c , gathered in Γ mc = { ( γ mc,j , γ ′ mc,j ) } j =1 , which satisfies assumption 2. We can define : F µc (( M, g ) , U ; E ) ֒ → F mc (( M, g ) , U ; E ) := F Γ mc ( U , K ; E ) . The injection (which will soon be continuous) follows from proposition since for instance (( − γ mc, ) ˙ × ( n − ˙ × ( γ ′ mc, ) c ) = ( V − ˙ × n ) c = µ n, . Example 18.
Operadically Microcausal functionals :
The inconvenience in some applications of the last example is that γ = γ ′ so that it isnot possible to compose multilinear maps obtained via derivatives even in the case E = lC. For G a family of globally hyperbolic metrics on M , consider J = G × { , } . Wedefine γ omc, ( g, = γ ′ omc, ( g, = V + ( g ) , γ omc, ( g, = γ ′ omc, ( g, = V − ( g ) , gathered in Γ omc = { ( γ omc, ( g,j ) , γ ′ omc, ( g,j ) ) } ( g,j ) ∈ J , which satisfies assumption 2, and the space of operatorially mi-crocausal functionals : F omc ( M, G, U ; E ) := F Γ omc ( U , K ; E ) . We also have a variant with closed cones Γ omc = { ( γ omc, ( g,j ) , γ ′ omc, ( g,j ) ) } ( g,j ) ∈ J : F omc ( M, G, U ; E ) := F Γ omc ( U , K ; E ) ֒ → F mc (( M, g ) , U ; E ) , ( g ∈ G ) . To compare with examples already considered, let us make explicit the variants with wavefront set conditions.
Define oµ n, ( g,i ) = (( − γ omc, ( g,i ) ) ˙ × ( n − ˙ × ( γ ′ omc, ( g,i ) ) c ) = ( V ± ( g ) ˙ × ( n − ˙ × V ∓ ( g ) c ) c and then oµ n =( oµ n, ( g,i ) ) ( g,i ) ∈ J , with variant oµ ′ n, ( g,i ) = (( − γ omc , ( g,i ) ) ˙ × ( n − ˙ × ( γ ′ omc , ( g,i ) ) c ) = ( V ± ( g ) ˙ × ( n − ˙ × [ V ∓ ( g )] c ) c and oµ ′ n = ( oµ ′ n, ( g,i ) ) ( g,i ) ∈ J , and then deduce F oµc ( M, G, U ; E ) := F oµ ( E ( U, E ) , U , K , K ; E ) ֒ → F omc ( M, G, U ; E ) , F oµc ( M, G, U ; E ) := F oµ ′ ( E ( U, E ) , U , K , K ; E ) ֒ → F omc ( M, G, U ; E ) ∩ F µc (( M, g ) , U ; E ) , ( g ∈ G ) . Note that F oµc ( M, { g } , U ; E ) is the space introduced in [CH, p 14] (once considered thesymmetry of differentials that allows not to state symmetric conditions with respect to whichvariable is the last variable). Note that F µloc ( U ; lC ) ⊂ F oµc ( M, G, U ; lC ) using as explainedin the example for multilocal functionals that V ± ( g ) are semigroups for addition. In the case C ′ = F , there are 2 natural topologies. We can put the topology of uniformconvergence on images of finite dimensional compacts by smooth functions I s or the topologyof uniform convergence on images of compacts by smooth curves I c . I s (resp. I c ) is theinitial topology generated, for any f φ : IR n → E ( U, E ) for n ≥ n = 1) by themaps f ∗ ( k,i ) φ : F λ ( E ( U, E ) , U , C , C ′ ; ( E , B )) → E [IR n , ( D ′ λ k,i ,λ k,i ( U k , ( C k ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ǫ E ](this last space being given the usual topology of uniform convergence on compacts of allderivatives). We have E [IR n , ( D ′ λ k,i ,λ k,i ( U k , ( C k ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ǫ E ] ≃ E [IR n ] ǫ [( D ′ λ k,i ,λ k,i ( U k , ( C k ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ǫ E ]even if E is not quasi-complete , and the second bracket can be removed in the quasi-completecase ([S] for associativity of ǫ product). Of course f ∗ ( k,i ) φ ( F ) = F ( k ) ◦ f φ . For the second type of spaces, we likewise define I s , I c using : f ∗ ( k,i ) φ : F Γ ( U , C ; ( E , B )) → E (IR n , L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) n − ; ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi )] ǫ E ) . The topologies above are also defined by the following families of seminorms. For K ⊂ IR m a compact and f : IR m → U ⊂ E ( U ) a smooth map, C an equicontinuous set in( D ′− λ cn , − λ cn ( U n , ( C n ) o ; E ⊗ n )) , I ibi ), CE an equicontinuous set in E ′ c , one defines for the firstcase: p f,K,CE ( F ) = sup φ ∈ f ( K ) ,e ∈ CE |h F ( φ ) , e i| ,p n,f,K,C,CE ( F ) = sup φ ∈ f ( K ) sup v ∈ C,e ∈ CE |h F ( n ) ( φ ) , v ⊗ e i| . For the second case, we keep the seminorms p f,K,CE and replace the second family, usingequicontinuous=bounded sets C ⊂ ( D ′ γ i ,γ i ( U ) , I ibi ) and D ⊂ ( D ′− ( γ ′ i ) c , − ( γ ′ i ) c ( U ) , I ibi ) (cf lemma3) and consider the seminorms : Indeed it is enough to deal with the case k = 0 and E [IR n , E ] defines an element of E [IR n ] ǫ E = L ǫ ( E ′ c , E [IR n ]) but conversely, an element of the ǫ product defines a scalarly smooth map, thus composedwith curves, a scalarly smooth curve, but from [KM, Corol 1.8] smoothness of curves depends only of thebornology, thus our element of E [IR n ] ǫ E defines a conveniently smooth map (with the original topology of E )and, from [KM, Corol 3.14], this is the usual notion of smooth maps. Then the identification of topologiesfollows as in [T, Th 44.1] UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 27 p n,f,K,C,D ( F ) = sup φ ∈ f ( K ) sup v ∈ D,u ,...,u n − ∈ C,e ∈ CE |h F ( n ) ( φ )[ e ] , u ⊗ ... ⊗ u n − ⊗ v i| . It is crucial in the case E not necessarily quasi-complete that we see L βe [( D ′ γ i ,γ i ( U ; E )) n − ; E ′ γ ′ i ,γ ′ i ( U ; E ∗ )] ǫ E ≃ L ǫ ( E ′ c ; L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) n − ; ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi )])thanks to [S, Corol 2 p 34].For cases with specified support, C ′ one of course uses inductive and projective limits, for α = s, c (with the same name for induced topologies on a subspace)( F λ ( E ( U, E ) , U , C , C ′ ; ( E , B )) , I αi ) = lim −→ C ∈C ′ ( F λ ( E ( U, E ) , U , C , { F ∈ F , F ⊂ C } ; ( E , B )) , I α ) → ( F λ ( E ( U, E ) , U , C , C ′ ; ( E , B )) , I αp ) := lim ←− O ∈ ( C ′ ) o ( F λ ( E ( U, E ) , U , C , { O } o ; ( E , B )) , I αi ) , ( F Γ ( U , C ; ( E , B )) , I αi ) = lim −→ C ∈C ( F Γ ( U , { F ∈ F , F ⊂ C } ; ( E , B )) , I α ) → ( F Γ ( U , C ; ( E , B )) , I αp ) := lim ←− O ∈C o ( F Γ ( U , { O } o ; ( E , B )) , I αi ) . The spaces above are related in using proposition 12 and its consequence explained inproposition 13. We recover in special cases statements of our previous examples.
Example 19.
Wave-Front variant of operadically controlled functionals
If for Γ as above, one defines λ (Γ) by λ n,i (Γ) = (( − γ i ) ˙ × ( n − ˙ × ( γ ′ i ) c ) c ; i ∈ J. Note that we easily checkassumption namely λ n,i (Γ) ˙ × λ N,i (Γ) ⊂ ( λ n,i (Γ) c ˙ × λ N,i (Γ) c ) c ⊂ λ n + N,i (Γ) as soon as − γ i ⊂ ( γ ′ i ) c , namely under assumption 2. One gets continuous embeddings for α ∈ { si, ci, sp, cp } : ( F λ (Γ) ( E ( U, E ) , U , K , C ; ( E , B )) , I α ) ֒ → ( F Γ ( U , C ; ( E , B )) , I α ) , where we used proposition and [S, Prop 1 p 20] for ǫ product of injective continuous maps. Our first main general result generalizing and improving the known properties of micro-causal functionals is the following :
Theorem 20.
The spaces F λ ( E ( U, E ) , U , K , C ; ( E , B )) , F Γ ( U , C ; ( E , B )) with topologies I sp or I cp (or I si or I ci if C is countably generated) are complete locally convex spaces(resp. nuclear) as soon as E is. Assume moreover E is a quasi-complete locally con-vex algebra with continuous product and assume assumption 1 (resp. assumption 2), then F λ ( E ( U, E ) , U , K , C ; ( E , B )) is an algebra with hypocontinuous multiplication defined point-wise for I si or I ci (resp. F Γ ( U , C ; ( E , B )) an algebra with continuous multiplication if C = F for I s or I c and otherwise hypocontinuous for I si or I ci ). The reader will note in the proof that, in order to use [S2], it is crucial that in all cases E has a continuous and not only hypocontinuous multiplication map. Proof.
Step 1:
Nuclearity.By definition, In the case C ′ = F , I s , I c are locally convex kernels of nuclear spaces E [IR n , ( D ′ λ k,i ,λ k,i ( U k , ( C k ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ǫ E ] , (including E [IR n , E ] , in case k = 0) E (IR n , L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) n − ; ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi )] ǫ E ). From [S] and our identification ofthese spaces with ǫ products, the completion of these spaces are the same with E replaced by its completion which is still nuclear. Then ǫ -products are completed projective products ofnuclear spaces thus nuclear.Then from projective limits, subspaces and countable projectivelimits, one deduces all the support cases. Step 2:
Completeness.In the case C ′ = F all topologies are defined as locally convex kernels. As made explicitbelow,it remains to check those locally convex kernels are closed subspaces of correspondingproducts (then they will be complete as closed subspaces of complete locally convex spaces,see [T, Prop 5.4], since continuous and smooth functions considered valued in completelocally convex spaces are complete, see Tr`eves [T, Prop 44.1]). Said otherwise the proof isthe same as for the classical smooth structure in convenient analysis (cf [KM, Def 3.11]). Step 3: (Hypo)continuity.We only consider the cases with support F , the general case follows by regular (since count-able strict) inductive limits and [M].Using composition, product [KM, corol 3.13] and smoothness of bounded multilinear maps[KM, lemma 5.5], one gets pointwise product of smooth maps are smooth and the derivativegiven by ordinary Leibniz rule.From Schwartz’ crossing theorems [S2, Prop 2 p 18], and replacing by completeness andnuclearity an ǫ product by a quasi-completed projective tensor product (here any ˆ ⊗ is sucha quasi-completion), on gets a canonical map :Γ β,π : [( D ′ λ n,j ,λ n,j ( U n , ( C n ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ˆ ⊗ π E ] ⊗ β ( D ′ λ m,j ,λ m,j ( U m , ( C m ) oo ; ( E ⊗ m ) ∗ ) , I ibi ) ǫ E ) → [ D ′ λ n,j ,λ n,j ( U n , ( C n ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ˆ ⊗ β ( D ′ λ m,j ,λ m,j ( U m , ( C m ) oo ; ( E ⊗ m ) ∗ ) , I ibi )] ǫ [ E ˆ ⊗ π E ]Note that the extension to the hypocontinuous product follows from 1 , sup ( β, π ) = β and because any bounded set in ( D ′ λ n,j ,λ n,j ( U n , ( C n ) oo ; ( E ⊗ n ) ∗ ) , I ibi ) ˆ ⊗ π E is a β − π decomposable (even γ − π -decomposable) by [S2, Prop 1.4 p 16] by nuclearity ofthe strong=Arens dual of ( D ′ λ n,j ,λ n,j ( U n , ( C n ) oo ; ( E ⊗ n ) ∗ ) , I ibi ). Now if M is the multiplicationmap on E , we thus obtain a map [ ι ◦ ( . ⊗ . ) ǫM ] ◦ Γ β,π from the tensor multiplication map of9 and a canonical injection from assumption 1, with value in( D ′ λ n,j ˙ × λ m,j ,λ n,j ˙ × λ m,j ( U n + m , ( C n + m ) oo ; ( E ⊗ ( n + m ) ) ∗ ) , I ibi ) ǫ E → ( D ′ λ n + m,j ,λ n + m,j ( U n + m , ( C n + m ) oo ; ( E ⊗ ( n + m ) ) ∗ ) , I ibi ) ǫ E . This is the expected space of value. Using Leibniz rule, in order to get the derivative of(
F G ) ( n + m ) and reasoning in composing the above map with φ ( F ( n ) ( φ ) ⊗ G ( m ) ( φ )) onededuces the stated hypocontinuity in the first case. UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 29
In the second case, one reasons similarly with Γ π,π which is continuous instead [S2,Prop 2.3 p 18-19], the tensor multiplication map of lemma 15 and the canonical injec-tion one needs to use with assumption 2 is the composition with ( D ′ γ i ,γ i ( U ; E ) , I ibi ) → ( D ′ ( − γ ′ i ) c , ( − γ ′ i ) c ( U ; E ) , I ibi ) . (cid:3) Supplementary algebraic structure on spaces of smooth functionals
Our final task is to get a general result to recover Poisson brackets or retarded brackets onour spaces of functionals. We especially want to cover field dependent variants as in [BFRi].The reader not familiar with vector valued distributions should assume E = lC later at firstreading.If E is a quasi-complete algebra with continuous multiplication M , then one can extendduality pairings to an hypocontinuous E -valued pairing defining for f ∈ ( E ′ γ ′ i ,γ ′ i ( U ; E ) , I ibi ) ǫ E , g ∈ ( D ′ ( − γ ′ i ) c , − ( γ ′ i ) c ( U ; E ) , I ibi ) ǫ E a pairing h f, g i ∈ E . This comes from [S2] (using ( h ., . i ǫM ) ◦ Γ β,π ) as in the proof of our previous theorem.Assume moreover given an action A : F ˆ ⊗ π E → E with F a quasi-complete locally convexspace.One can extend ◦ j from lemma 14 to : ◦ j : L ( O βe,i ∈ [1 ,m ] ( D ′ γ ′ i ,γ ′ i ( U ′ i , C ′ i ; E ′ i ) , I ibi ); ( D ′ γ ′ ,γ ′ ( U ′ , C ′ ; E ′ ) , I ibi )) ǫ F × L ( O βe,i ∈ [1 ,n ] ( D ′ γ i ,γ i ( U i , C i ; E i ) , I ibi ); ( D ′ γ,γ ( U, C ; E )) , I ibi )) ǫ E → L ( O βe,i ∈ [1 ,n + m − ( D ′ γ ′′ i ,γ ′′ i ( U ′′ i , C ′′ i ; E ′′ i ) , I ibi ); ( D ′ γ ′ ,γ ′ ( U ′ , C ′ ; E ′ ) , I ibi )) ǫ E using again Γ β,π of [S2, Prop 2 p 18] to obtain the above map as ([ ◦ j ] ǫA ) ◦ Γ β,π . One provesthat this map is hypocontinuous by composition and using Γ β,π hypocontinuous as provedbefore.Note also that L βe [( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ γ i ,γ i ( U ; E ) , I ibi )] ǫ F ⊂ L βe [ E ( U ; E ∗ ); D ′ ( U ; E )] ǫ F and one can define F (Γ) = [ ∩ i ∈ J L βe [( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ γ i ,γ i ( U ; E ) , I ibi )] ǫ F ] ∩ [ ∩ i ∈ J L βe [( E ′ ( − γ ci ) , ( − γ ci ) ( U ; E ∗ ) , I ibi ); ( D ′ ( − γ ′ i ) c , ( − γ ′ i ) c ( U ; E ) , I ibi )] ǫ F ] ∩ [ ∩ i ∈ J L βe [( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ ( − γ ′ i ) c , − ( γ ′ i ) c ( U ; E ) , I ibi )] ǫ F ]in this space with the projective kernel topology. We can associate continuous maps for j ∈ J F (Γ) → F ( j, (Γ) := [ L βe [( E ′ γ ′ i ,γ ′ i ( U ; E ∗ )); ( D ′ γ i ,γ i ( U ; E ))] ǫ F ] ∩ [ L βe [( E ′ ( − γ ci ) , ( − γ ci ) ( U ; E ∗ )); ( D ′ ( − γ ′ i ) c , ( − γ ′ i ) c ( U ; E ))] ǫ F ] , F (Γ) → F ( j, (Γ) := L βe [( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ ( − γ ′ i ) c , − ( γ ′ i ) c ( U ; E ) , I ibi )] ǫ F . We reach the point where we need to introduce a bornology B (Γ , B , C ) on a subspace of F and depending on ( E , B ). For sets A ⊂ E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) ǫ E , A ′ ⊂ E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) ǫ E uniformly supported in C for B , namely e.g. for any B ∈ B , S a ∈ A,b ∈ B supp( a ( b )) ∈ C . Then thebornology B (Γ , B , C ) is by definition generated by sets C of form { F ψ ( h F ◦ ( a ) , a ′ i ) , a ∈ A, a ′ ∈ A ′ , ψ ∈ B } for A, A ′ as above and B ∈ B . Those bornologies have only beenintroduced to state a natural condition for preservation of support conditions.
Theorem 21.
Assume E , F quasi-complete with continuous multiplications F ˆ ⊗ π E → E , E ˆ ⊗ π E → E . Let Γ ′ = { ( γ i , − γ ci ) , ( γ i , γ ′ i ) } i ∈ J indexed by J × { , } and R ∈ F Γ ′ ( U , C ; ( F (Γ) , ( F g (Γ)) g ∈ J ×{ , } , B (Γ , B f , C ))) . Then the product defined by [ R ( F, G )]( ϕ ) = h R ( ϕ )[ F (1) ( ϕ )] , G (1) ( ϕ ) i is an hypocontinuous product on F Γ ( U , C ; ( E , B f )) with topologies I si or I ci . We of course wrote R ( ϕ )[ F (1) ( ϕ )] for R ( ϕ ) ◦ [ F (1) ( ϕ )] . Note we could formulate an obviousvariant for F λ (Γ) ( E ( U, E ) , U , K , C ; ( E , B f )) (but notationally less convenient since ǫ productsdon’t recover wave front set conditions and we could not formulate the assumption on R asa vector valued statement). Proof.
The smoothness of R ( F, G ) comes from composition of bounded(= smooth) mul-tilinear maps and smooth maps. By Leibniz rule, the derivatives are expressed using h R ( l ) ( ϕ )[ F ( m +1) ( ϕ )] , G ( n +1) ( ϕ ) i . Let us explain the meaning of this expression in more detail.First note that R ( l ) ( ϕ ) ∈ L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l − ; ( E ′ ( − γ i ) c , ( − γ i ) c ( U ; E ∗ ) , I ibi )] ǫ F ( i, (Γ) ֒ → L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l − ; ( E ′ ( − γ i ) c , ( − γ i ) c ( U ; E ∗ ) , I ibi )] ǫ [ L βe [( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ γ i ,γ i ( U ; E ) , I ibi )] ǫ F ] ≃ L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l × ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ γ i ,γ i ( U ; E ) , I ibi )] ǫ F Thus from the extended map before the proof R ( l ) ( ϕ )[ F ( m +1) ( ϕ )] = R ( l ) ( ϕ ) ◦ l +1 [ F ( m +1) ( ϕ )] ∈ L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l + m ; ( D ′ γ i ,γ i ( U ; E ) , I ibi )] ǫ E and from the way it is composed from smoothand bounded multilinear maps, it is again (conveniently) smooth in ϕ. Since (using a partial transpose which gives the same map by symmetry of derivatives) wehave G ( n +1) ( ϕ ) ∈ L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) n − × ( D ′ ( − γ ′ i ) c , − ( γ ′ i ) c ( U ; E ) , I ibi ); ( E ′ ( − γ i ) c , ( − γ i ) c ( U ; E ∗ ))] ǫ E . This shows one can also pair h R ( l ) ( ϕ )[ F ( m +1) ( ϕ )] , G ( n +1) ( ϕ ) i with the E -valued dualitypairing to get a smooth map with value in L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l + m + n − ; ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi )] ǫ E if the last variable (evaluated by duality) is a variable below G ( n +1) ( ϕ ) (one uses multipletimes associativity of ǫ -products to do this).If the last variable is below F , we have a partial transpose on the term with F ( m +1) ( ϕ ) asabove for G , and uses then a R ( l ) ( ϕ ) ∈ L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l × ( E ′ ( − γ ci ) , ( − γ i ) c ( U ; E ∗ ) , I ibi ); ( D ′ ( − γ ′ i ) c , − ( γ ′ i ) c ( U ; E ) , I ibi )] ǫ F and the reasoning is similar. Finally, if the last variable is below R , one finally uses F ( i, (Γ)to get : R ( l ) ( ϕ ) ∈ L βe [( D ′ γ i ,γ i ( U ; E ) , I ibi ) l × D ′ ( − γ ′ i ) c , ( − γ ′ i ) c ( U ; E ) , I ibi ) l × ( E ′ γ ′ i ,γ ′ i ( U ; E ∗ ) , I ibi ); ( D ′ ( − γ ′ i ) c , − ( γ ′ i ) c ( U ; E ) , I ibi )] ǫ F . UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 31
From all hypocontinuities of our building maps, it is then obvious to get the stated hypoconti-nuity in the full support case. It remains to check the support condition for the distributionsgiven for ψ ∈ B ⊂ B h ψ ([ R ( F, G )] (1) ( ϕ )[ h ])= ψ ( h R (1) ( ϕ )[ h, F (1) ( ϕ )] , G (1) ( ϕ ) i ) + ψ ( h R ( ϕ )[ F (2) ( ϕ )[ h ]] , G (1) ( ϕ ) i ) + ψ ( h R ( ϕ )[ F (1) ( ϕ )] , G (2) ( ϕ )[ h ] i )The two last terms are supported respectively in supp( F, E ′ ) , supp( G, E ′ ). Indeed let h ∈D ( U ) with supp( h ) ∩ supp( F, E ′ ) = ∅ , and any g ∈ D ( U ) , ψ ∈ E ′ then ψ ( F (1) ( ϕ )[ h ]) = 0 thusif we differentiate in φ h F (2) ( ϕ )[ h ] , g ⊗ ψ i = ψ ( F (2) ( ϕ )[ h, g ]) = 0 where F (2) ( ϕ )[ h ] is seen in( E ′ ( − γ i ) c , ( − γ i ) c ( U ; E ∗ ))] ǫ E and thus by definition and density of D ( U ) ⊂ ( E ′ ( − γ i ) c , ( − γ i ) c ( U ; E ∗ )) ′ c one gets F (2) ( ϕ )[ h ] = 0 in this space and thus ψ ( h R ( ϕ )[ F (2) ( ϕ )[ h ]] , G (1) ( ϕ ) i ) = 0 . It remains to consider the first term.Let C = { A ψ ( h A ◦ [ F (1) ( ϕ )] , G (1) ( ϕ ) i ) , ϕ i ∈ U , ψ ∈ B } then C ∈ B (Γ , B , C ) bydefinition since { F (1) ( ϕ ) , ϕ i ∈ U } ⊂ ( E ′ ( − γ i ) c , ( − γ i ) c ( U ; E ∗ )) ǫ E and is uniformly supporteduniformly in elements of B in C . Thus for ψ ′ ∈ C , h ∈ D ( U ) , supp( h ) ∩ supp( R, C ) = ∅ we have ψ ′ ( R (1) ( ϕ )[ h ]) =[ ψ ′ ( R ( ϕ ))] (1) [ h ] = 0 and especially ψ ( h R (1) ( ϕ )[ h, F (1) ( ϕ )] , G (1) ( ϕ ) i = 0. As a conclusion ,we havesupp( R ( F, G ) , B ) = [ ϕ ∈ U ,ψ ∈ B supp([ R ( F, G )] (1) ( ϕ )[ ψ ]) ⊂ C ∪ supp( F, E ′ ) ∪ supp( G, E ′ ) ∈ C , since C is polar thus stable by finite unions and this concludes to the expected supportcondition. This also shows compatibility with the strict inductive limits and thus by [M],hypocontinuity needs only be checked in the case C = F and this is obvious by our formulasfor derivative in terms of our various hypocontinuous products. (cid:3) Application to Retarded and advanced products
As we will explain in the next remark, the construction of retarded products explained in[BFRi] fails, as stated there, on microcausal functionals. Let us apply our result in the case E = F = lC for simplicity. We mostly follow their physical setting.We stick to the setting of [BGF] (but with reversed convention on what is called re-tarded/advanced to follow [BFRi]) to which we refer for terminology, for the first resultwhich gathers in our functional analytic language known results similar to [BFRi, Corol3.2.5]. Especially, E ∗ is now obtained using a volume form for any (fixed) Riemanian metricon M (cf. p 167) and a vector bundle isomorphism ∗ : E ∗ → E for some scalar productidentifying sections of E ∗ and E .We call L ( g ) = [ V + ( g ) ∩ ( V + ( g )) c ] ∪ [ V − ( g ) ∩ ( V − ( g )) c ] ⊂ ˙ T ∗ M the light cone bundle (withthe notation of example 18). Proposition 22.
Let ( M, g ) a globally hyperbolic Lorentzian manifold, P a normally hy-perbolic operator acting on sections of a vector bundle E → M (especially its principalsymbol is related as usual to ˆ g ⊗ Id E for ˆ g globally hyperbolic). We assume V ∓ (ˆ g ) ⊂ V ∓ ( g ) . Then the retarded Green’s operator ∆ retP (resp. the advanced Green’s operator ∆ advP )from D ( M, E ) → E ( M, E ) with supp (∆ retP ( v )) ⊂ J + , ˆ g ( supp ( v )) ⊂ J + ,g ( supp ( v )) (resp.supp (∆ advP ( v )) ⊂ J − , ˆ g ( supp ( v )) ) extends continuously to D ′ ( M, K P ( g ) , E ) → D ′ ( M, K P ( g ); E ) (resp. D ′ ( M, K F ( g ) , E ) → D ′ ( M, K F ( g ); E ) with the notation of [D, Ex 16] . Moreover forany γ ⊂ L (ˆ g ) c or γ ⊃ L (ˆ g ) , any γ ⊂ Λ ⊂ γ, either equal to γ or satisfying the same conditionas γ , any C P ∈ {K P ( g ) , SK P ( g ) } , C F ∈ {K F ( g ) , SK F ( g ) } and finally any I among I ppp , I ibi it can also be extended to : ∆ retP : ( D ′ γ, Λ ( M, C P , E ) , I ) → ( D ′ γ, Λ ( M, C P ; E ) , I ) , ∆ advP : ( D ′ γ, Λ ( M, C F , E ) , I ) → ( D ′ γ, Λ ( M, C F ; E ) , I ) . Finally, assume a family of P as above is continuously parametrized by a compact set as seenin L ( D ( M, K ; E ) , D ( M, K ; E )) with the weak topology and gives bounded set of ∆ retP ∈ L b ( D ( M, E ) , E ( M, E )) . If moreover the inclusion of light cones are uniform for the symbols of P in the family thenthe corresponding extensions form also bounded sets in the spaces they are stated to live in,for the topology of convergence on bounded sets. Note that our statement implies that the if part of [BFRi, Corol 3.2.5 (c)] is wrong intaking γ = V + there may be points in the wave front set in V + (with DW F in V + ) withoutany point in the light cone in W F (∆ retP ( v )) outside of V + . Proof.
From [BGF, Prop 3.4.8 p 91], we know sequential continuity ∆ retP : D ( M, E ) →D ( M, O SK P (ˆ g ) , E ) → D ( M, O SK P ( g ) , E ) (resp. ∆ advP : D ( M, E ) → D ( M, O SK F ( g ) , E ), asstated we only know with value in SK (ˆ g ) but the defining support condition implies this au-tomatically). Note that in a bounded family case one gets something bounded here since thesupport control by g gives a uniform control on support. Since the spaces involved are (quasi)-complete separated locally convex space whose strong dual is a Schwartz space using [D, Prop8], an application of [BD, lemma 21] implies they are Mackey-sequentially continuous, thusbounded, thus continuous since the spaces are (ultra)bornological. Thus we have well definedcontinuous adjoint maps (on Mackey=strong duals) (∆ retP ) ∗ : D ′ ( M, K F ( g ) , E ∗ ) → D ′ ( M, E ∗ )(resp. (∆ advP ) ∗ : D ′ ( M, K P ( g ) , E ∗ ) → D ′ ( M, E ∗ )). But from [BGF, lemma 3.4.4 p 89], theycorrespond to extensions of ∆ advP ∗ (resp. ∆ retP ∗ ). Exchanging P, P ∗ and using again the supportcondition on the dense space of smooth maps one deduces the first extension with modifiedtarget. Moreover, since J ± , ˆ g ( J ± ,g ( . )) ⊂ J ± ,g ( . ), evaluated at compacts or for Σ Cauchy sur-face for g thus ˆ g , SK P (ˆ g ) ⊂ SK P ( g ) and thus by duality K P ( g ) ⊂ K P (ˆ g ) and both supportconditions are stable by ∆ retP . Thus one gets the stated spaces of value of the extension. Thebounded family yields bounded family of extensions as above.From propagation of singularity theorems for wave front set of solutions (cf. e.g. [Du] or[St, Th 13,15]) and the statement on support, it is well known that the spaces claimed to beleft stable by ∆ ret/advP are indeed stable in case γ = Λ . It remains to see continuity, first for I ppp . Since open cones containing γ can follow the same constraint (since L (ˆ g ) closed) weare reduced using (1) to the case γ open (using also the completion to go to Λ = γ. ) But inthe open case I pmp = I iii is both ultrabornological by [D, Th 23]and quasi-LB thus strictlywebbed by Theorem 1 (since either the support condition or its dual is countably generated).Applying De Wilde’s closed graph theorem [K2, § D ′ and the continuous injection, this isobvious. Going to Λ = γ by completion, it only remains to check a known wave front set UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 33 condition to get general Λ (since the topology in this case is induced). The case I ibi is aconsequence by bornologification.For the boundedness statement, it only remains to consider the extensions with wave frontset conditions. We start again from the case I ppp for γ = Λ ⊃ L (ˆ g ) (for all symbols in thefamily of P ) open. By barrelledness and a standard result [K2, § u with W F ( u ) = Γ ⊂ γ , we want to prove boundednessof { ∆ retP ( u ) } ⊂ D ′ Γ , Γ ( M, F ; E ) (since from the known support conditions and the inductivelimit definition in the γ open case this will be enough). But our family of P are contin-uously parametrized on a compact in the sense stated above which implies that describedin coordinates, so are the coefficients of the differential operators and all their derivativesin positions. But now, a cone in γ c has been assumed not to intersect any char ( P ) in thefamily, thus from the continuity in parameters, we can apply the argument in [H, Th 8.3.1],get (8.3.5) uniformly on parameters the boundedness of coefficients of R j ’s also uniformlyon our compact set of parameters, and thus also the conclusion, which gives exactly theexpected boundedness D ′ Γ , Γ ( M, F ; E ) (the seminorms of D ′ have been treated before).We thus got { ∆ retP } ⊂ L b ( D ′ γ,γ ( M, C P ; E ); D ′ γ,γ ( M, C P ; E )) is bounded. From the descrip-tion of bounded sets in the completion in lemma 6 one obtains the case Λ = γ by completion,then by the projective description (1) for any γ in the case Λ = γ and then as above byinduced topology for any Λ . For the topology I ibi , a bornologification argument doesn’t work here but we reason asfollows. Consider first the case Λ = γ and look at adjoint maps (∆ retP ) ∗ thus defined forultrabonological (thus barrelled) topologies I b . The boundedness proved for the adjointsprove strong pointwise boundedness (and the strong topology is known to be I b by [D, Prop33]). From [K2, § retP ) ∗ is thus equicontinuous thus bounded,thus bounded as above at the completion level and since in this case equicontinuous sets andbounded sets coincide, we deduce the adjoint are bounded for the convergence on boundedsets for I ibi in the case Λ = γ. The general Λ case follows since the topology is induced asbefore. This finishes all the cases in the case γ ⊃ L (ˆ g ) . By duality and exchanging advancedand retarded propagators, one gets the case γ ⊂ L (ˆ g ) c in considering λ = − γ c ⊃ L (ˆ g ) . The assumption on P gives the boundedness in λ of coefficients of the adjoint differentialoperators and their derivatives that is what we used for the support case, and since we alreadyknow the case of full wave front set, one gets ∆ retP ∈ L b ( E ′ ( M, E ) , D ′ ( M, E )) bounded andthus by duality the assumed boundedness at the adjoint level. As in the proof before onegets boundedness for I ibi of the adjoints, thus of ∆ retP and one goes from this to I ppp usingthe previous was its bornologification. (cid:3) The next result is a variant adapted to our setting of [BFRi, Prop 3.2.8]. Since the proofis not included there and would need, in our view, to go back to parameter dependence of∆ retP along smooth curve, at least at the level of smooth functions that can in principal beobtained by using parameter dependent variants of all local geometric objects (exponentialmaps, parallel transport) used in [BGF] and controlling global geometric data uniformly bygiven controlling globally hyperbolic metrics, we don’t enter in these details and only assumea reasonable minimal result needed to make work functional analytic argument. Going much further in the construction of retarded Green’s operator is clearly not the purpose of thisarticle.Following them and [BDF], we call generalized Lagrangian a map L : D ( U ) → F µloc ( U , lC)(this space is defined in example 16) with supp( L ( f )) ⊂ supp( f ) and L additive in f asdefined in the cited example.From the definition we know that L ( f ) (2) ( ϕ ) ∈ D ′ C ,C ( M , K , ( E ⊗ ) ∗ ) and is smooth in ϕ. Noting that C ⊂ ( − γ ˙ × γ c ) c for any cone γ ⊂ ˙ T ∗ M , and composing with the canonicalmap built in proposition 13 one gets an image P L ( f ) ,ϕ ∈ L (( D ′ γ,γ ( M, C ; E ) , I ibi ) , ( D ′ γ,γ ( M, C ; E ∗ ) , I ibi )) . Noting that if f, g ∈ D ( M ) equal 1 on a neighborhood of supp( u ) ∈ K , P L ( f ) ,ϕ ( u ) = P L ( g ) ,ϕ ( u ) starting with u smooth, since by [BFRi, lemma 3.1.2] supp( L (1) ( f ) − L (1) ( g )) ⊂ supp( f − g ) so that just after one derivative evaluating it to u make vanish the difference.As a consequence, take now u ∈ D ′ γ,γ ( M, C ; E ) , v ∈ [ D ′ γ,γ ( M, C ; E )] ′ thus supported respec-tively in C ∈ C , D ∈ ( O C ) o with C ǫ ∩ D ǫ compact, and approximating by u, v smooth onegets, assuming this time f, g equal to 1 on a neighborhood of C ǫ ∩ D ǫ h P L ( f ) ,ϕ ( u ) , v i = h P L ( f ) ,ϕ ( uh ) , v i = h P L ( g ) ,ϕ ( uh ) , v i = h P L ( g ) ,ϕ ( u ) , v i , using h equal to 1 on a neighborhoodof supp( v ) supported in D ǫ and the fact that additivity of L ( f ) , L ( g ) implies its secondderivative is supported on the diagonal [BFRi, prop 2.3.11].Arguing with the inductive limit definition of the topologies (also on the dual) one can thusdefine h P L ,ϕ ( u ) , v i = h P L ( f ) ,ϕ ( u ) , v i , f ∈ D ( M ) equal 1 on a sufficiently huge neighborhoodof supp( u ) ∩ supp( v ) and one gets for any cone γ a continuous map (agreeing for different γ and polar enlargeable families C ): P L ,ϕ ∈ L (( D ′ γ,γ ( M, C ; E ) , I ibi ) , D ′ γ,γ ( M, C ; E ∗ ) , I ibi )) . Proposition 23.
Let ( M, g ) , ( M, g ′ ) globally hyperbolic. Let L a generalized Lagrangian(on U ⊂ E ( M, E ) smooth sections of a bundle E → M ). Assume given a smooth curve onIR, λ ϕ λ ∈ U and the associated P : λ P λ := P L ,ϕ λ and assume ∗ P λ are normallyhyperbolic differentials operators on E → M with metric ˆ g λ globally hyperbolic and with V ± ( g ′ ) ⊃ V ± (ˆ g λ ) ⊃ V ± ( g ) . Then for any γ ∈ { V ± ( g ) , V ± ( g ) c } , we have the smoothmness of P ∈ C ∞ [ IR ; L βe ( D ′ γ,γ ( M, C ; E ); D ′ γ,γ ( M, C ; E ∗ ))] Also assume ∆ ret ∗ P λ ∈ L b ( D ( M, E ); E ( M, E )) is locally bounded in λ , then for any γ as above C P ∈ {K P ( g ′ ) , SK P ( g ′ ) } , the extension ∆ retP λ = (∆ ret ∗ P λ ) ∗ ∈ L βe [( D γ,γ ( M, C P , E ∗ ) , I ibi ) → ( D ′ γ,γ ( M, C P ; E ) , I ibi )] of the previous proposition is smooth in λ too with : ∂∂λ ∆ retP λ = − ∆ retP λ ˙ P λ ∆ retP λ . There are corresponding results for advanced propagators.Proof.
The smoothness of P follows from the construction above and the definition of semi-norms that only involves evaluating on sets uniformly supported in C and ( O C ) o . Moreprecisely, one defines the candidate derivatives as above by inserting f and uses a weakcharacterization as in [KM, corol 1.9 p 14] but instead of evaluating boundedness on allelements of the dual, one evaluates it by the definition for the ǫ product on tensor product UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 35 of equicontinuous sets for which the f could be chosen uniformly. To get smoothness of ∆ retP λ it suffices to prove the formula for the first derivative and apply induction using smoothness(boundedness, even hypocontinuity) of the composition map by proposition 14 between ourspaces.We first note that on compact sets of λ ’s we have uniform boundedness in the stated spacesof ∆ retP λ since V ± ( g ) ⊂ L (ˆ g λ ) c for all λ and the corresponding inclusion for complements thatenable to use the boundedness statement in proposition 22.To compute the first derivative, one also wants to check a resolvent equation :∆ retP λ = ∆ retP µ + ∆ retP λ ( P µ − P λ )∆ retP µ . Indeed from the continuity of the operators used one can extend the defining relation∆ retP λ P λ = Id to functions with support in C P so that we have ∆ retP µ = ∆ retP λ P λ ∆ retP µ and also(the easier relation when applied to compact) ∆ retP λ = ∆ retP λ P µ ∆ retP µ . Taking the difference, onegets the relation, first on compactly supported function, and then by continuity and densityon distributions supported on C P . Then from the equation, and continuity of composition, one checks ∆ retP λ is continuous(for Mackey convergence since λ − µ (∆ retP µ − ∆ retP λ ) is bounded, using the uniform boundednessderived above from proposition 22 and some boundedness from [KM, p 9] since P λ smooth)and then from the usual computation of difference quotients, one gets the usual derivativesagain by hypocontinuity of compositions and Mackey convergence argument using the proof[KM, Corol 1.9] to get that after dividing by λ − µ the next expression is again bounded :1 λ − µ (∆ retP λ − ∆ retP µ ) + ∆ retP µ ˙ P µ ∆ retP µ = ∆ retP µ [ 1 λ − µ ( P µ − P λ ) + ˙ P µ ]∆ retP µ + (∆ retP λ − ∆ retP µ ) 1 λ − µ ( P µ − P λ )∆ retP µ . (cid:3) We are now ready to get retarded products from the results of our previous section. Someof the assumptions are probably redundant but we don’t want to go back to the constructionof retarded products in this paper to make them minimal, even though this is probably easyfollowing [BGF].
Proposition 24.
Let ( M, g ) , ( M, g ′ ) globally hyperbolic. Let L a generalized Lagrangian(on U ⊂ E ( M, E ) smooth sections of a bundle E → M ). Assume that for any smoothcurve λ ϕ λ ∈ U and the associated ∗ P L ,ϕ λ are normally hyperbolic differential operatorson E → M with metric ˆ g λ globally hyperbolic and with V ± ( g ′ ) ⊃ V ± (ˆ g λ ) ⊃ V ± ( g ) . Finallyassume that for any curve as above ∆ ret ∗ P λ ∈ L b ( D ( M, E ); E ( M, E )) is locally bounded in λ .Let Γ omc be defined as in example for G = { g } and define R ( ϕ ) = ∆ ret ∗ P L ,ϕ ∗ andrecall Γ ′ omc = { ( V + ( g ) , ( V − ( g )) c ) , ( V + ( g ) , V + ( g )) , ( V − ( g ) , ( V + ( g )) c ) , ( V − ( g ) , V − ( g )) } indexed by { + , −} × { , } Then R ∈ F Γ ′ omc ( U , K ; ( lC (Γ omc ) , ( lC g (Γ omc ) g ∈{ + , −}×{ , } , B (Γ omc , B f , K ))) and thus defines an hypocontinuous retarded bracket on F Γ omc ( U , K ; lC ) with topologies I si or I ci . Assuming the corresponding assumption for an advanced bracket giving a map A , F Γ omc ( U , C ; ( E , B f )) is thus a Poisson algebra (with hypocontinuous multiplication maps)with Poisson bracket { ., . } = R ( ., . ) − A ( ., . ) . Proof.
Note that, using [ V + ( g )] c ⊃ V − ( g )we have the simplified formulalC(Γ omc ) = L βe [( E ′ V + ( g ) ,V + ( g ) ( U ; E ∗ ) , I ibi ); ( D ′ V + ( g ) ,V + ( g ) ( U ; E ) , I ibi )] ∩ L βe [( E ′ [ V + ( g )] c , [ V + ( g )] c ( U ; E ∗ ) , I ibi ); ( D ′ [ V + ( g )] c , [ V + ( g )] c ( U ; E ) , I ibi )] ∩ L βe [( E ′ V − ( g ) ,V − ( g ) ( U ; E ∗ ) , I ibi ); ( D ′ V − ( g ) ,V − ( g ) ( U ; E ) , I ibi )] ∩ L βe [( E ′ [ V − ( g )] c , [ V − ( g )] c ( U ; E ∗ ) , I ibi ); ( D ′ [ V − ( g )] c , [ V − ( g )] c ( U ; E ) , I ibi )]and recall the notation (we now often implicitly assume topology I ibi ):lC ( ± , (Γ omc ) = L βe [ E ′ V ± ( g ) ,V ± ( g ) ( U ; E ∗ ); D ′ V ± ( g ) ,V ± ( g ) ( U ; E )] ∩ L βe [ E ′ [ V ∓ ( g )] c , [ V ∓ ( g )] c ( U ; E ∗ ); D ′ [ V ∓ ( g )] c , [ V ∓ ( g )] c ( U ; E )] , lC ( ± , (Γ omc ) = L βe [ E ′ [ V ± ( g )] , [ V ± ( g )] ( U ; E ∗ ); D ′ [ V ∓ ( g )] c , [ V ∓ ( g )] c ( U ; E )] . Thus our previous proposition 23 already explains why R is (conveniently) smooth withvalue lC(Γ omc ) . Then we have to control the space of value of R ( n ) ( ϕ ) . From the computationof R (1) and induction, one sees that it is a linear combination of compositions of R ( ϕ ) anddifferentials of h d ( k ) P L ,ϕ ( h , ..., h k )( u ) , v i = L ( f ) ( k +2) ( ϕ )[ h , ....h k , u, v ]with f chosen depending only of the supports of u, v as above. Using the condition on L , d ( k ) P L ,ϕ is smooth in ϕ with value in( D ′ C k +2 ,C k +2 ( M k +2 , K ;( E ⊗ ( k +2) ) ∗ ) , I ibi ) ֒ → L ( O βe,i ∈ [1 ,k ] ( D ′ V ± ( g ) ,V ± ( g ) ( M, F ; E ) , I ibi ) ⊗ βe ( D ′ V ± ( g ) ,V ± ( g ) ( M, SK P ( g ′ ); E ) , I ibi ); ( D ′ V ± ( g ) ,V ± ( g ) ( M, SK P ( g ′ ); E ∗ ) , I ibi )) . (9)The first space of value is only with f fixed, but the second one obtained via proposition 13also holds by commutation of (regular) inductive limit on support and hypocontinuous mapfor general u, v and also using locality of L ( f ) as for the definition of P .Now, as explained above a typical derivative is a linear combination of composition ofthe form [ R ( ϕ ) ◦ [ d ( k ) P L ,ϕ ◦ k +1 R ( ϕ )] ◦ k +1 d ( l ) P L ,ϕ ] ◦ l + k +1 R ( ϕ ) with an arbitrary length ofcomposition (the term above appears in the ( k + l )-th differential. From what we stated andproposition 14 (especially boundedness of composition), the differential above is smooth in ϕ with value in a space of the form L ( O βe,i ∈ [1 ,k + l ] ( D ′ V ± ( g ) ,V ± ( g ) ( M, F ; E )) ⊗ βe ( D ′ V ± ( g ) ,V ± ( g ) ( M, SK P ( g ′ ); E )); ( D ′ V ± ( g ) ,V ± ( g ) ( M, SK P ( g ′ ); E ∗ ))) . Note it is here that it is crucial for composition that we developed a theory for general supportleft stable by propagators like SK P ( g ′ ) . Using a partial transpose in the last coordinate, wealso have d ( k ) P L ,ϕ ∈ L ( O βe,i ∈ [1 ,k ] ( D ′ V ± ( g ) ,V ± ( g ) ( M, F ; E )) ⊗ βe ( D ′ ( V ∓ ( g )) c , ( V ∓ ( g )) c ( M, SK P ( g ′ ); E )); ( D ′ ( V ∓ ( g )) c , ( V ∓ ( g )) c ( M, SK P ( g ′ ); E ∗ ))) . (10)Arguing for compositions as before, this gives the smoothness of derivatives valued inlC ( ± , (Γ omc ) corresponding to cones ( V ± ( g ) , ( V ∓ ( g )) c ) ∈ Γ ′ omc . UNCTIONAL PROPERTIES OF GENERALIZED H ¨ORMANDER SPACES OF DISTRIBUTIONS II 37
Arguing as for the first map, we also know that d ( k ) P L ,ϕ ∈ L ( O βe,i ∈ [1 ,k − ( D ′ V ± ( g ) ,V ± ( g ) ( M, F ; E )) ⊗ βe ( D ′ ( V ± ( g ) ,V ± ( g ) ( M, K F ( g ′ ); E )) ⊗ βe ( D ′ ( V ± ( g )) , ( V ± ( g )) ( M, SK P ( g ′ ); E )); ( D ′ V ± ( g ) ,V ± ( g ) ( M, K ; E ∗ ))) , and thus again by partial transpose but with respect to the next-to-last coordinate, weget : d ( k ) P L ,ϕ ∈ L ( O βe,i ∈ [1 ,k − ( D ′ V ± ( g ) ,V ± ( g ) ( M, F ; E )) ⊗ βe ( D ′ ( V ∓ ( g )) c , ( V ∓ ( g )) c ( M, F ; E )) ⊗ βe ( D ′ ( V ± ( g )) , ( V ± ( g )) ( M, SK P ( g ′ ); E )); ( D ′ ( V ∓ ( g )) c , ( V ∓ ( g )) c ( M, SK P ( g ′ ); E ∗ )) . (11)Then by composing (9),(11) and (10) in this order, one gets the smoothness of derivativesof R in the space valued in lC ( ± , (Γ omc ) corresponding to cones ( V ± ( g ) , ( V ± ( g ))) ∈ Γ ′ omc . It only remains to check the global support condition namely for B ∈ B (Γ omc , B f , K ),supp( R, B ) ∈ K . Without loss of generality, one can assume B = { F ( h F ◦ ( a ) , a ′ i ) , a ∈ A, a ′ ∈ A ′ } for A ⊂ E ′ V ± ( g ) ,V ± ( g ) ( U ; E ∗ ) , A ′ ⊂ E ′ V ± ( g ) ,V ± ( g ) ( U ; E ∗ ) uniformly supported in K , K ∈ K .From the computation of the first derivative of R ( ϕ ), one considers the support of thedistribution for a ∈ A, a ′ ∈ A ′ h
7→ h R ( ϕ ) ◦ dP L ,ϕ ( h ) ◦ R ( ϕ )[ a ] , a ′ i = L ( f ) (3) ( ϕ )( h, R ( ϕ )[ a ] , A ( ϕ )[ a ′ ]) . But by support property of retarded and advanced propagators, supp( R ( ϕ )[ a ]) ⊂ J + ,g ′ ( K ) , supp( A ( ϕ )[ a ′ ]) ⊂ J − ,g ′ ( K ) , and by locality of L ( f ) one deduces the support above is in J − ,g ′ ( K ) ∩ J + ,g ′ ( K ) which is compact by global hyperbolicity of g ′ . Once the varioushypocontinuity proved, we can reason as in [BFRi] to check one obtains a Poisson algebrastructure. (cid:3) Remark . The argument to prove microcausal functionals form a Poisson algebra evenstable by a retarded product R in [BFRi] have the following flaw. The stability of theretarded product is not convincingly checked. As we pointed out to the authors, in the for-mula for R ( F, G ) (3) [ φ ]( ψ , ψ , ψ ) there is a term − F (2) [ φ ]( ψ , D ∆ R [ φ ]( ψ , G (2) [ φ ]( ψ ))) = L ( f ) (3) (∆ A [ φ ] F (2) [ φ ]( ψ ) , ψ , ∆ R [ φ ]( G (2) [ φ ]( ψ ))) . The distribution for this term in R ( F, G ) (3) [ φ ]is the multiplication of L ( f ) (3) with ∆ A [ φ ] F (2) [ φ ] ⊗ ⊗ ∆ R [ φ ] G (2) [ φ ] . Of course one can take ξ , ξ space-like vectors such that ξ + ξ = ξ ∈ V + ( x ). Becauseof the ( x, x, ξ, − ξ ) terms in the wave front set of ∆ A , ∆ R , the wave front set of ∆ A [ φ ] F (2) [ φ ]contains a priori all the wave front set of F (2) [ φ ], which, in the microcausal functional case,can contain ( x, x, ξ , ξ ), and respectively G (2) [ φ ] can contain ( x, x, ξ , ξ ). From there onecan find a point ( x, x, x, ξ , ξ + ξ , ξ ) ∈ V + ( x ) in the wave front set of the term above (the ξ + ξ appearing because of locality of L ). The motivation of the study in this last sectioncomes from the observation of this issue in the argument. One can of course take polynomialfunctionals with exactly distributional kernels of derivatives given by H¨ormander’s examplewith only one point above in the wave front set in order to show this argument indeedprevents microcausal functionals to be stable by R . References [A] C.
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