Inverse Function Theorems for Generalized Smooth Functions PaoloGiordanoandMichaelKunzinger 6 1 0 2 n u J 3 1 1 Introduction ] A Sinceitsinception,categorytheoryhasunderscoredtheimportanceofunrestricted F compositionof morphismsformanypartsof mathematics.The closureof a given . h spaceof“arrows”withrespecttocompositionprovedtobeakeyfoundationalprop- t erty.ItisthereforeclearthatthelackofthisfeatureforSchwartzdistributionshas a m considerableconsequencesinthestudyofdifferentialequations[28,14],inmath- ematical physics [4, 6, 8, 10, 18, 21, 25, 41, 42, 43, 46], and in the calculus of [ variations[30],tonamebutafew. 4 On the other hand, Schwartz distributions are so deeply rooted in the linear v frameworkthat onecan even isomorphicallyapproachthem focusingonlyon this 3 aspect, opting for a completely formal/syntactic viewpoint and without requiring 1 0 any functional analysis, see [49]. So, Schwartz distributions do not have a notion 0 of pointwiseevaluationin general,and donotforma category,althoughit is well 0 knownthatcertainsubclassesofdistributionshavemeaningfulnotionsofpointwise 2. evaluation,seee.g.[35,36,47,45,16,15,51]. 0 Thisisevenmoresurprisingifonetakesintoaccounttheearlierhistoricalgene- 6 sisofgeneralizedfunctionsdatingbacktoauthorslikeCauchy,Poisson,Kirchhoff, 1 Helmholtz,Kelvin,Heaviside,andDirac,see[29,33,34,50].Forthem,this“gen- : v eralization”issimplyaccomplishedbyfixinganinfinitesimalorinfiniteparameter i inanordinarysmoothfunction,e.g.aninfinitesimalandinvertiblestandarddevia- X tioninaGaussianprobabilitydensity.Therefore,generalizedfunctionsarethought r a ofassomekindofsmoothset-theoreticalfunctionsdefinedandvaluedinasuitable PaoloGiordano University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien e-mail: [email protected] MichaelKunzinger University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien e-mail: [email protected] 1 2 PaoloGiordanoandMichaelKunzinger non-Archimedeanringofscalars.Fromthisintuitivepointofview,theyclearlyhave pointvaluesandformacategory. This aspect also bears upon the conceptof (a generalized)solution of a differ- ential equation.In fact, any theoryof generalizedfunctionsmusthave a link with the classical notion of (smooth) solution. However, this classical notion is deeply grounded on the concept of composition of functions and, at the same time, it is oftentoonarrow,asisamplydemonstratede.g.inthestudyofPDEinthepresence of singularities. In our opinion, it is at least not surprising that also the notion of distributionalsolutiondidnotleadtoasatisfyingtheoryofnonlinearPDE(noteven ofsingularODE).Wehavehenceawildgardenofflourishingequation-dependent techniquesandazooofcounter-examples.Thewell-knowndetachingbetweenthese techniquesandnumericalsolutionsofPDEisanothersideofthesamequestion. One can say that this situation presents several analogies with the classical compass-and-straightedgesolutionofgeometricalproblems,orwiththesolutionof polynomialequationsbyradicals.Thedistinctionbetweenalgebraicandirrational numbers and the advent of Galois theory were essential steps for mathematics to startfocusingonadifferentconceptofsolution,frequentlynearertoappliedprob- lems.Intheend,theseclassicalproblemsstimulatedmoregeneralnotionsofgeo- metricaltransformationand numericalsolution,which nowadayshave superseded theirorigins.Theanalogiesareevengreaterwhenobservingthatfirststepstoward aGaloistheoryofnonlinearPDEarearising,see[5,7,38,39]. Generalizedsmoothfunctions(GSF)areapossibleformalizationoftheoriginal historicalapproachoftheaforementionedclassicalauthors.We extendthefieldof realnumbersintoanaturalnonArchimedeanringr Randweconsiderthesimplest notion of smooth function on the extended ring of scalars r R. To define a GSF e f :X −→Y,X ⊆r Rn,Y ⊆r Rd,wesimplyrequiretheminimallogicalconditions sothatanetofordinarysmoothfunctions fe ∈C¥ (W e ,Rd),W ee⊆Rn,definesaset- e e theoreticalmapX −→Y whichisinfinitelydifferentiable;seebelowforthedetails. This freedom in the choice of domains and codomainsis a key propertyto prove that GSF are closed with respect to composition.As a result, GSF share so many propertieswithordinarysmoothfunctionsthatfrequentlyweonlyhavetoformally generalizeclassicalproofstothenewcontext.Thisallowsaneasierapproachtothis newtheoryofgeneralizedfunctions. It is important to note that the new framework is richer than the classical one becauseofthepossibilitytoexpressnon-Archimedeanproperties.So,e.g.,twodif- ferentinfinitesimalstandarddeviationsinaGaussianresultininfinitelycloseDirac- delta-like functionalsbut, generallyspeaking,these two GSF could have different infinitevaluesatinfinitesimalpointsh∈r R.Forthisreason,Schwartzdistributions are embeddedas GSF, but this embeddingis not intrinsic and it has to be chosen e dependingonthephysicalproblemorontheparticulardifferentialequationweaim tosolve. Inthepresentwork,weestablishseveralinversefunctiontheoremsforGSF.We prove both the classical local and also some global versions of this theorem. It is remarkabletonotethatthelocalversionisformallyverysimilartotheclassicalone, but with the sharp topology instead of the standard Euclidean one. We also show InverseFunctionTheoremsforGeneralizedSmoothFunctions 3 the relationsbetweenour results and the inversefunctiontheoremfor Colombeau functionsestablishedbyusingthediscontinuouscalculusof[2,3]. The paper is self-contained in the sense that it contains all the statements of results required for the proofs of the new inverse function theorems. If proofs of preliminariesareomitted,wegivereferencestowheretheycanbefound. 1.1 Basicnotions Theringofgeneralizedscalars Inthiswork,I denotestheinterval(0,1]⊆Randwewillalwaysusethevariablee forelementsofI;wealsodenotee -dependentnetsx∈RI simplyby(xe ).ByNwe denotethesetofnaturalnumbers,includingzero. We start by defining the non-Archimedeanring of scalars that extends the real fieldR.Foralltheproofsofresultsinthissection,see[19,18]. Definition1. Letr =(r e )∈RI beanetsuchthatlime→0+r e =0+,then (i) I(r ):={(r e−a)|a∈R>0}iscalledtheasymptoticgaugegeneratedbyr . (ii) If P(e ) is a property of e ∈I, we use the notation ∀0e : P(e ) to denote ∃e ∈I∀e ∈(0,e ]: P(e ).Wecanread∀0e asfore small. 0 0 (iii) Wesaythatanet(xe )∈RI isr -moderate,andwewrite(xe )∈Rr if∃(Je )∈ I(r ): xe =O(Je )ase →0+. (iv) Let (xe ), (ye )∈RI, then we say that (xe )∼r (ye ) if ∀(Je )∈I(r ): xe = ye +O(Je−1) as e →0+. This is an equivalence relation on the ring Rr of moderatenetswithrespecttopointwiseoperations,andwecanhencedefine r R:=Rr /∼r , e whichwecallRobinson-Colombeauringofgeneralizednumbers,[48,8].We denotetheequivalenceclassx∈r Rsimplybyx=:[xe ]:=[(xe )]∼∈r R. Inthefollowing,r willalwaysdenoteaenetasinDef.1,andwewillusetheseimpler notationRforthecaser e =e .Theinfinitesimalr canbechosendependingonthe classofdifferentialequationsweneedtosolveforthegeneralizedfunctionsweare e going to introduce,see [20]. For motivationsconcerningthe naturality of r R, see [18].Wealsousethenotation dr :=[r e ]∈r Rand de :=[e ]∈(e)R. e We can also define an orderrelation on r R by saying [xe ]≤[ye ] if there exists e e (ze )∈RI suchthat(ze )∼r 0(wethensaythat(ze )isr -negligible)andxe ≤ye +ze e fore small.Equivalently,wehavethatx≤yifandonlyifthereexistrepresentatives (xe ),(ye )ofx,ysuchthatxe ≤ye foralle .Clearly,r Risapartiallyorderedring. Theusualrealnumbersr∈Rareembeddedinr Rconsideringconstantnets[r]∈r R. e Eveniftheorder≤isnottotal,westillhavethepossibilitytodefinetheinfimum e e [xe ]∧[ye ]:=[min(xe ,ye )], and analogously the supremum function [xe ]∨[ye ]:= [max(xe ,ye )]andtheabsolutevalue|[xe ]|:=[|xe |]∈r R.Ournotationsforintervals e 4 PaoloGiordanoandMichaelKunzinger are:[a,b]:={x∈r R|a≤x≤b},[a,b]R:=[a,b]∩R,andanalogouslyforsegments [x,y]:={x+r·(y−ex)|r∈[0,1]}⊆r Rnand[x,y]Rn =[x,y]∩Rn.Finally,wewrite x≈ytodenotethat|x−y|isaninfinitesimalnumber,i.e.|x−y|≤rforallr∈R . e >0 Thisisequivalenttolime→0+|xe −ye|=0forallrepresentatives(xe ),(ye )ofx,y. Topologiesonr Rn e On the r R-module r Rn, we can consider the natural extension of the Euclidean norm, i.e. |[xe ]| := [|xe |] ∈r R, where [xe ] ∈ r Rn. Even if this generalized norm e e takesvaluesinr R,itsharesseveralpropertieswithusualnorms,likethetriangular e e inequalityortheproperty|y·x|=|y|·|x|.Itisthereforenaturaltoconsideronr Rn e topologiesgeneratedbyballsdefinedbythisgeneralizednormandsuitablenotions e ofbeing“strictlylessthanagivenradius”: Definition2. Letc∈r Rnandx,y∈r R,then: (i) Wewritex<yife∃r∈r R≥0: reisinvertible,andr≤y−x (ii) Wewritex<Ryif∃r∈eR>0: r≤y−x. (iii) B (c):= x∈r Rn||x−c|<r foreachr∈r R . r >0 n o (iv) BFr(c):= x∈r Ren||x−c|<Rr foreachr∈eR>0. (v) BE(c):=n{x∈Rn ||x−c|<r},ofor each r ∈R , denotes an ordinary Eu- r e >0 clideanballinRn. The relations <, <R have better topological properties as compared to the usual strictorderrelationa≤banda6=b(thatwewillneveruse)becauseboththesets ofballs B (c)|r∈r R , c∈r Rn and BF(c)|r∈R , c∈r Rn arebasesfor r >0 r >0 n o n o two topologieson r Ren. The formeeris called sharp topology,wheereas the latter is called Fermat topology. We will call sharply open set any open set in the sharp e topology,andlargeopensetanyopensetintheFermattopology;clearly,thelatter iscoarserthantheformer.Theexistenceofinfinitesimalneighborhoodsimpliesthat the sharp topology induces the discrete topology on R. This is a necessary result when one has to deal with continuous generalized functions which have infinite derivatives.Infact,if f′(x )isinfinite,onlyforx≈x wecanhave f(x)≈ f(x ). 0 0 0 The following result is useful to deal with positive and invertible generalized numbers(cf.[24,40]). Lemma1.Letx∈r R.Thenthefollowingareequivalent: (i) xisinvertibleaendx≥0,i.e.x>0. (ii) Foreachrepresentative(xe )∈Rr ofxwehave∀0e : xe >0. (iii) Foreachrepresentative(xe )∈Rr ofxwehave∃m∈N∀0e : xe >r em InverseFunctionTheoremsforGeneralizedSmoothFunctions 5 Internalandstronglyinternalsets A naturalway to obtainsharplyopen,closedandboundedsets in r Rn isby using a net (Ae ) of subsets Ae ⊆Rn. We have two ways of extending the membership e relationxe ∈Ae togeneralizedpoints[xe ]∈r R: Definition3. Let(Ae )beanetofsubsetsofRen,then (i) [Ae ]:= [xe ]∈r Rn|∀0e : xe ∈Ae iscalledtheinternalsetgeneratedbythe n o net(Ae ).See[44e]fortheintroductionandanin-depthstudyofthisnotion. (ii) Let(xe )beanetofpointsofRn,thenwesaythatxe ∈e Ae ,andwereaditas (xe )stronglybelongsto(Ae ),if∀0e : xe ∈Ae andif(xe′)∼r (xe ),thenalso x′e ∈Ae for e small. Moreover,we set hAe i:= [xe ]∈r Rn|xe ∈e Ae , and n o wecallitthestronglyinternalsetgeneratedbythenet(Aee). (iii) Finally,wesaythattheinternalsetK=[Ae ]issharplyboundedifthereexists r∈r R>0 suchthatK⊆Br(0).Analogously,anet(Ae )issharplyboundedif thereeexistsr∈r R>0 suchthat[Ae ]⊆Br(0). Therefore,x∈[Ae ]iftehereexistsarepresentative(xe )ofxsuchthatxe ∈Ae fore small, whereas this membership is independentfrom the chosen representative in thecaseofstronglyinternalsets.Noteexplicitlythataninternalsetgeneratedbya constantnetAe =A⊆Rn issimplydenotedby[A]. The following theoremshows that internaland stronglyinternalsets have dual topologicalproperties: Theorem1.Fore ∈I,letAe ⊆Rnandletxe ∈Rn.Thenwehave (i) [xe ]∈[Ae ]ifandonlyif∀q∈R>0∀0e : d(xe ,Ae )≤r eq.Therefore[xe ]∈[Ae ] ifandonlyif[d(xe ,Ae )]=0∈r R. (ii) [xe ]∈hAe iifandonlyif∃q∈R>e0∀0e : d(xe ,Ace )>r eq,whereAec :=Rn\Ae. Therefore,if(d(xe ,Aec))∈Rr ,then[xe ]∈hAe iifandonlyif[d(xe ,Aec)]>0. (iii) [Ae ]issharplyclosedandhAe iissharplyopen. (iv) [Ae ]=[cl(Ae )],wherecl(S)istheclosureofS⊆Rn.OntheotherhandhAe i= hint(Ae )i,whereint(S)istheinteriorofS⊆Rn. Wewillalsousethefollowing: Lemma2.Let(W e )beanetofsubsetsinRn foralle ,and(Be )asharplybounded netsuchthat[Be ]⊆hW e i,then ∀0e : Be ⊆W e . Sharply bounded internal sets (which are always sharply closed by Thm. 1 (iii)) serveascompactsetsforourgeneralizedfunctions.Foradeeperstudyofthistype ofsetsinthecaser =(e )see[44,17];inthesameparticularsetting,see[19]and referencesthereinfor(strongly)internalsets. 6 PaoloGiordanoandMichaelKunzinger Generalizedsmoothfunctions Fortheideaspresentedinthissection,seealsoe.g.[19,18]. Usingtheringr R,itiseasytoconsideraGaussianwithaninfinitesimalstandard deviation.Ifwedenotethisprobabilitydensityby f(x,s ),andifwesets =[s e ]∈ e r R>0, where s ≈0, we obtain the net of smooth functions(f(−,s e ))e∈I. This is thebasicideawedevelopinthefollowing e Definition4. LetX ⊆r Rn andY ⊆r Rd bearbitrarysubsetsofgeneralizedpoints. Thenwesaythat e e f :X −→Y isageneralizedsmoothfunction ifthereexistsanetoffunctions fe ∈C¥ (W e ,Rd)defining f inthesensethatX ⊆ hW e i, f([xe ])=[fe (xe )]∈Y and(¶ a fe (xe ))∈Rdr forallx=[xe ]∈Xandalla ∈Nn. ThespaceofGSFfromX toY isdenotedbyr GC¥ (X,Y). Letusnoteexplicitlythatthisdefinitionstatesminimallogicalconditionstoobtain a set-theoretical map from X intoY and defined by a net of smooth functions. In particular,thefollowingThm.2statesthattheequality f([xe ])=[fe (xe )]ismean- ingful, i.e. that we have independence from the representatives for all derivatives [xe ]∈X 7→[¶ a fe (xe )]∈r Rd,a ∈Nn. Theorem2.LetX ⊆r Rn aendY ⊆r Rd bearbitrarysubsetsofgeneralizedpoints. Let fe ∈C¥ (W e ,Rd)beanetofsmoothfunctionsthatdefinesageneralizedsmooth e e mapofthetypeX −→Y,then (i) ∀a ∈Nn∀(xe ),(xe′)∈Rrn : [xe ]=[xe′]∈X ⇒ (¶ a ue (xe ))∼r (¶ a ue(xe′)). (ii) ∀[xe ]∈X∀a ∈Nn∃q∈R>0∀0e : supy∈BeEq(xe)|¶ a ue(y)|≤e −q. (iii) For all a ∈Nn, the GSF g:[xe ]∈X 7→[¶ a fe (xe )]∈Rd is locally Lipschitz inthesharptopology,i.e.eachx∈X possessesasharpneighborhoodU such e that|g(x)−g(y)|≤L|x−y|forallx,y∈U andsomeL∈r R. (iv) Each f ∈r GC¥ (X,Y)iscontinuouswithrespecttothesharptopologiesin- e ducedonX,Y. (v) Assume that the GSF f is locally Lipschitz in the Fermat topology and that its Lipschitz constants are always finite: L∈R. Then f is continuousin the Fermattopology. (vi) f :X−→Y isaGSFifandonlyifthereexistsanetve ∈C¥ (Rn,Rd)defining ageneralizedsmoothmapoftypeX −→Y suchthat f =[ve (−)]|X. (vii) SubsetsS⊆r Rs withthetraceofthesharptopology,andgeneralizedsmooth mapsasarrowsformasubcategoryofthecategoryoftopologicalspaces.We willcallthisceategoryr GC¥ ,thecategoryofGSF. ThedifferentialcalculusforGSFcanbeintroducedshowingexistenceandunique- ness of another GSF serving as incremental ratio. For its statement, if P(h) is a property of h∈r R, then we write ∀sh: P(h) to denote ∃r∈r R ∀h∈B (0): >0 r P(h)and∀Fh: P(h)for∃r∈R ∀h∈BF(c): P(h). e >0 r e InverseFunctionTheoremsforGeneralizedSmoothFunctions 7 Theorem3. Let U ⊆ r Rn be a sharply open set, let v = [ve ] ∈ r Rn, and let f ∈r GC¥ (U,r R) be a generalized smooth map generated by the net of smooth e e functions fe ∈C¥ (W e ,R).Then e (i) There exists a sharp neighborhood T ofU×{0} and a generalized smooth mapr∈r GC¥ (T,r R),calledthegeneralizedincrementalratioof f alongv, suchthat e ∀x∈U∀sh: f(x+hv)= f(x)+h·r(x,h). (ii) Ifr¯∈r GC¥ (S,r R)isanothergeneralizedincrementalratioof f alongvde- finedonasharpneighborhoodSofU×{0},then e ∀x∈U∀sh: r(x,h)=r¯(x,h). (iii) Wehaver(x,0)= ¶¶ vfee (xe ) foreveryx∈U andwecanthusdefine ¶¶ vf(x):= h i r(x,0),sothat ¶ f ∈r GC¥ (U,r R). ¶ v IfU is a large open set, then an anaelogousstatement holds replacing ∀sh by ∀Fh andsharpneighborhoodsbylargeneighborhoods. Notethatthisresultpermitstoconsiderthepartialderivativeof f withrespecttoan arbitrarygeneralizedvectorv∈r Rnwhichcanbe,e.g.,infinitesimalorinfinite. Usingthisresultweobtaintheusualrulesofdifferentialcalculus,includingthe chainrule.Finally,wenotethatfoereachx∈U,themapDf(x).v:= ¶ f(x)∈r Rd is ¶ v r R-linearinv∈r Rn.Thesetofallther R-linearmapsr Rn−→r Rd willbedenoted e byL(r Rn,r Rd).ForA=[Ae (−)]∈L(r Rn,r Rd),weset|A|:=[|Ae |],thegeneralized e e e e e numberdefinedbytheoperatornormsofthematricesAe ∈L(Rn,Rd). e e e e EmbeddingofSchwartzdistributionsandColombeaufunctions Wefinallyrecalltworesultsthatgiveacertainflexibilityinconstructingembeddings ofSchwartzdistributions.Notethatboththeinfinitesimalr andtheembeddingof Schwartzdistributionshavetobechosendependingontheproblemweaimtosolve. A trivial example in this directionis the ODE y′ =y/de , which cannotbe solved forr =(e ),butithasasolutionforr =(e−1/e ).Asanothersimpleexample,ifwe needthepropertyH(0)=1/2,whereH istheHeavisidefunction,thenwehaveto choosetheembeddingofdistributionsaccordingly.Thiscorrespondstothephilos- ophyfollowedin[26].Seealso[20]forfurtherdetails. If j ∈D(Rn), r ∈R and x∈Rn, we use the notations r⊙j for the function >0 x∈Rn7→ 1 ·j x ∈Randx⊕j forthefunctiony∈Rn7→j (y−x)∈R.These rn r notations permit(cid:0)to(cid:1) highlight that ⊙ is a free action of the multiplicative group (R ,·,1) on D(Rn) and ⊕ is a free action of the additive group (R ,+,0) on >0 >0 D(Rn).Wealsohavethedistributivepropertyr⊙(x⊕j )=rx⊕r⊙j . Lemma3.Let b∈Rr be a net such that lime→0+be =+¥ . Let d ∈(0,1). There existsanet(y e )e∈I ofD(Rn)withtheproperties: 8 PaoloGiordanoandMichaelKunzinger (i) supp(y e )⊆B1(0)foralle ∈I. (ii) y e =1foralle ∈I. (iii) ´∀a ∈Nn∃p∈N: supx∈Rn|¶ a y e (x)|=O(bep)ase →0+. (iv) ∀j∈N∀0e : 1≤|a |≤ j⇒ xa ·y e(x)dx=0. (v) ∀h ∈R>0∀0e : |y e |≤1+´h . (vi) Ifn=1,thenthe´net(y e )e∈I canbechosensothat −0¥ y e =d. ´ Ify e satisfies(i)–(vi)theninparticulary eb:=b−e 1⊙y e satisfies(ii)-(v). Concerning embeddings of Schwartz distributions, we have the following result, where r W c :={[xe ]∈[W ]|∃K ⋐W ∀0e : xe ∈K} is called the set of compactly supportedpointsinW ⊆Rn. f Theorem4.UndertheassumptionsofLemma3,letW ⊆Rnbeanopensetandlet (y eb)bethenetdefinedinLemma3.Thenthemapping iWb :T ∈E′(W )7→ T∗y eb (−) ∈r GC¥ (r W c,r R) h(cid:16) (cid:17) i f e uniquelyextendstoasheafmorphismofrealvectorspaces ib:D′−→r GC¥ (r (−) ,r R), c g e andsatisfiesthefollowingproperties: (i) Ifb≥ dr −aforsomea∈R>0,thenib|C¥ (−):C¥ (−)−→r GC¥ (r (−)c,r R) isasheafmorphismofalgebras. g e (ii) IfT ∈E′(W )thensupp(T)=supp(ib(T)). W (iii) lime→0+ W iWb(T)e ·j =hT,j iforallj ∈D(W )andallT ∈D′(W ). (iv) ib comm´utes with partial derivatives, i.e. ¶ a ib(T) =ib (¶ a T) for each W W T ∈D′(W )anda ∈N. (cid:0) (cid:1) ConcerningtheembeddingofColombeaugeneralizedfunctions,werecallthatthe specialColombeaualgebraonW isdefinedasthequotientGs(W ):=E (W )/N s(W ) M ofmoderatenetsovernegligiblenets,wheretheformeris EM(W ):={(ue )∈C¥ (W )I|∀K⋐W ∀a ∈Nn∃N∈N:sup|¶ a ue (x)|=O(e −N)} x∈K andthelatteris N s(W ):={(ue)∈C¥ (W )I|∀K⋐W ∀a ∈Nn∀m∈N:sup|¶ a ue(x)|=O(e m)}. x∈K Usingr =(e ),wehavethefollowingcompatibilityresult: Theorem5.AColombeaugeneralizedfunctionu=(ue )+N s(W )d ∈Gs(W )d de- fines a generalized smooth map u:[xe ]∈r W c −→[ue (xe )]∈Rd which is locally LipschitzonthesameneighborhoodoftheFermattopologyforallderivatives.This f e InverseFunctionTheoremsforGeneralizedSmoothFunctions 9 assignment provides a bijection of Gs(W )d onto r GC¥ (r W ,r Rd) for every open c setW ⊆Rn. f e For GSF, suitable generalizations of many classical theorems of differential and integral calculus hold: intermediate value theorem, mean value theorems, Taylor formulasin differentforms,a sheaf propertyfor the Fermattopology,and the ex- tremevaluetheoremoninternalsharplyboundedsets(see[18]).Thelatterarecalled functionallycompact subsets of r Rn and serve as compactsets for GSF. A theory of compactlysupportedGSF hasbeen developedin [17], andit closely resembles e the classical theory of LF-spaces of compactly supportedsmooth functions. It re- sultsthatforsuitablefunctionallycompactsubsets,thecorrespondingspaceofcom- pactlysupportedGSFcontainsextensionsofallColombeaugeneralizedfunctions, andhencealsoofallSchwartzdistributions.Finally,inthesespacesitispossibleto provetheBanachfixedpointtheoremandacorrespondingPicard-Lindelo¨ftheorem, see[37]. 2 Local inverse function theorems As in the case of classical smoothfunctions,any infinitesimalcriterionfor the in- vertibilityofgeneralizedsmoothfunctionswillrelyonthe invertibilityofthe cor- responding differential. We therefore note the following analogue of [24, Lemma 1.2.41](whoseprooftransfersliterallytothepresentsituation): Lemma4.LetA∈r Rn×nbeasquarematrix.Thefollowingareequivalent: (i) Aisnondegeneerate,i.e.,x ∈r Rn,x tAh =0∀h ∈r Rnimpliesx =0. (ii) A:r Rn→r Rn isinjective. e e (iii) A:r Rn→r Rn issurjective. e e (iv) det(A)isinvertible. e e Theorem6.Let X ⊆r Rn, let f ∈r GC¥ (X,r Rn) andsupposethatfor some x in 0 the sharp interior of X, Df(x ) is invertible in L(r Rn,r Rn). Then there exists a e 0 e sharp neighborhoodU ⊆X ofx anda sharp neighborhoodV of f(x ) such that 0 0 f :U →V isinvertibleand f−1∈r GC¥ (V,U). e e Proof. Thm. 2.(vi) entails that f can be defined by a globally defined net fe ∈ C¥ (Rn,Rn). Hadamard’s inequality (cf. [11, Prop. 3.43]) implies |Df(x )−1| ≥ 0 n 1|det(Df(x )−1)|,whereC∈R isa universalconstantthatonlydependson C 0 >0 q the dimension n. Thus, by Lemma 4 and Lemma 1, detDf(x ) and consequently 0 alsoa:=|Df(x )−1|isinvertible.Next,pickpositiveinvertiblenumbersb,r∈r R 0 suchthatab<1,B (x )⊆X and 2r 0 e |Df(x )−Df(x)|<b 0 10 PaoloGiordanoandMichaelKunzinger for all x ∈B (x ). Such a choice of r is possible since every derivative of f is 2r 0 continuous with respect to the sharp topology (see Thm. 2.(iv) and Thm. 3.(iii)). Pickrepresentatives(ae ),(be )and(re )ofa,bandrsuchthatforalle ∈I wehave be >0,ae be <1,andre >0.Let(x0e )bearepresentativeofx0.Since[Bre (x0e )]⊆ B2r(x0), by Lemma 2 we can also assume that Bre (x0e ) ⊆ W e , and |Dfe (x0e )− Dfe (x)|<be forall x∈Ue :=Bre (x0e ). Nowlet ce := 1−aaeebe . Thenc:=[ce ]>0 andby[13,Th.6.4]weobtainforeache ∈I: (a) Forallx∈Ue :=Bre (x0e ),Dfe (x)isinvertibleand|Dfe (x)−1|≤ce . (b) Ve := fe (Bre (x0e ))isopeninRn. (c) fe |Ue :Ue −→Ve isadiffeomorphism,and (d) settingy0e := fe (x0e ),wehaveBre/ce (y0e )⊆ fe (Bre (x0e )). ThesetsU :=hUe i=Br(x0)⊆X andV :=hVe iaresharpneighborhoodsofx0 and f(x0),respectively,by(d),andsoitremainstoprovethat[fe |U−e1(−)]∈r GC¥ (V,U). Wefirstnotethatby(a),|Dfe (x)−1|≤ce forallx∈Bre (x0e ),whichbyHadamard’s inequalityimplies 1 |det(Dfe (x))|≥C·cne (x∈Bre (x0e )). (1) Nowfor[ye ]∈V and1≤i,j≤nwehave(seee.g.[11,(3.15)]) 1 ¶ j(fe−1)i(ye )= det(Dfe (fe−1(ye )))·Pij((¶ sfer(fe−1(ye )))r,s), (2) where P is a polynomial in the entries of the matrix in its argument. Since ij [fe−1(ye )]∈U ⊆X,itfollowsfrom(1)andthefactthat f|U ∈r GC¥ (U,r Rn)that (¶ j(fe−1)i(ye ))∈Rnr . e Higherorderderivativescanbetreatedanalogously,therebyestablishingthatevery derivativeof ge := fe |−1 is moderate.To provethe claim, it remainsto show that Ue [ge (ye )]∈U=hUe iforall[ye ]∈V =hVe i.Sincege :Ve −→Ue ,weonlyprovethat if(xe )∼r (ge (ye )),thenalsoxe ∈Ue fore small.Wecansetye′ := fe (xe )because fe is defined on the entire Rn. By the mean value theorem applied to fe and the moderatenessof f′,weget |ye′ −ye|=|fe (xe )−fe(ge (ye ))|≤r eN·|xe −ge(ye )|. Therefore(y′e )∼r (ye )andhencey′e ∈Ve andge(ye′)=xe ∈Ue fore small. ⊓⊔ FromThm.2.(iv),weknowthatanygeneralizedsmoothfunctionissharplycon- tinuous.Thusweobtain: Corollary1.Let X ⊆r Rn be a sharply open set, and let f ∈r GC¥ (X,r Rn) be suchthatDf(x)isinvertibleforeachx∈X.Then f isalocalhomeomorphismwith e e respecttothesharptopology.Inparticular,itisanopenmap.