; TeX output 1998.02.19:1936 |f7fKzDt G G cmr17H9older7tRegularitqyofSubsellipticPseudo-dierentialOpseratorsMc卍aꍍ KK`y 3 cmr10P!engfeifGuan Departmen!tfofMathematics PrincetonfUniv!ersityiJQ Curren!tfAddress: Departmen!tfofMathematics McMasterfUniv!ersity /Hamilton,fOn!tarioL8S4K1 \Canada *|f7fte #+"V 3 cmbx10Abstract:9Wee7considerHfolderregularit!yofsubMellipticpseudo-dierentialopMerators.xOurmainresultsarethe :9follo!wing.:91.MIn!general,?Hfolderregularit!ydoMesnotholdforsubellipticoperators.MWeeconstructaclass:9ofsecondordersubMellipticdieren!tialoperatorsinb> 3 cmmi10R2 cmmi8nF(n !", 3 cmsy103)forwhic!hHfolderregularitybreaks:9do!wn.غMoreover,foranys >0,ifnlarge,thereexistsf8c2 s'͍lKoc (Rn),an!ydistributionsolutionuis:9notfev!enloMcallybounded.:92.yOnzStheotherhand,Ww!eprovetheHfolderregularityforsubMellipticpseudo-dierentialopMerators:9indiagonalform.WeealsoestablishtheHfolderregularit!yforanysubMellipticpseudo-dierential:9opMeratorsfint!wofvdDariableswhic!hsatisfyNirenbMerg-Terevescondition(P).:93.0As8^anapplication,N`w!eobtaintheoptimalHfolderregularityresultsforthesubMellipticoblique:9derivdDativ!efproblem. i !|f17f:9InttroYduction. :9SinceQHansLewy'scelebratedexampleofanon-solvdDableopMerator,|wm!uchQimportantQworkhas:9bMeenSdoneonthestudyofsolvdDabilit!yandhypMoellipticitySofpseudo-dierentialopMerators.(e.g.[H1],:9[NT1],[NT2],[T],[BF],[E1],[H3],etc.)'ThepimpMortanceofstudyofthepseudo-dieren!tialoperators:9isfalsoillustratedb!yitsconnectiontootherproblems([H2],[FK]).:9The|follo!wingcondition( )andcondition(PV)introMducedbyNirenbMergandTerevesarefunda-:9men!tal.:9Conditionf( )::9Letp(x;1 )bMetheprincipalsym!bolofPV,giv!enanypMoint(xz|{Y cmr80;1 0@)0A2 }fRn;nfOMgg,andsuch:9thatfthereexistssomez2 C7mnnfOMg;) Bp(xz0;1 z0@) =0;dȮuRe(z{Ip)(xz0; z0@)6=0:9theKfunctionI m(z{Ip)(x;1 );restrictedtothebic!haracteristicscurveofRe(z{Ip)(x;1 )through(xz0; 0@) :9canfnev!erchangesignfrom-to+whenonemovesinthepMositivedirection.:9Conditionf(PV)::9ThegfunctionI m(z{Ip)(x;1 )inabMo!vegdoesnotc!hangesigninthebicharacteristiccurveof:9rMe(z{Ip)(x;1 )fthrough(xz0; 0@).:9Conditionn(PV)isnecessaryandsucien!tforloMcalsolvdDabilityofdierentialopMerators.v([NT2],[BF]).:9Condition^ ( )isnecessaryforloMcalsolvdDabilit!yofpseudo-dierentialopMeratorP`andforhypMoellip-:9ticit!yof;j4ĖP:([M]).FeorthesubMellipticitye,caverystrongresultofEgorovcharacterizesallsubMelliptic:9opMerators.A5pseudo-dieren!tial6operatorPXissubellipticifandonlyif;jTĖPsatisescondition( ):9andthefunctionI mz{IpvdDanishesonlyniteorderalongthebic!haracteristiccurvesofRez{Ip.Ifkis:9thesmallestn!umbMersuchthattheorderofvdDanishingislessthanorequaltokX?,thenwecantake:9,A=K b1] fe PAk6+1,0 thisissharp.(1Egoro!v'soriginalproMof([E1])ofthesuciencydoesnotseemtobev!ery:9rigorous.جTherstcompleteproMoffordieren!tialoperatorsw!asgivenbyTereves([T]).Hformander:9([H3])flatersuppliedthemissingpartintheproMofof[E1].(seealso[F]).:9In.thispapMer,w!estudyHfolderregularityofsubMellipticpseudo-dierentialopMerators. wItis:9w!ell-knownfthatifP+isanellipticpseudo-dieren!tialopMeratoroforderm,then) սu 2DMޟzK cmsy80( );1PVu2z:jcompZ( )! 5u2e+m%lKocӹ( );:9whereA 7XisHfolderspacewithindex> 0.Itw!asexpMectedthatsubellipticpseudo-dieren!tial :9opMeratorsnmw!ouldhavethesimilarpropMertye. 5Surprisingly,nthisnmisnottrueingeneral,ev!enfor:9subMellipticdieren!tialoperators.Inthefollo!wingtheoremwegiveaclassofdierentialopMerators:9infRn(n 3)whic!hdoMesnotbeha!vewellinHfolderspaces.$:9Theorem21.1eeLetPO=K@x-:Aa cmr62|, fe /^@xt2G @(K@x-:233 fe ş@xx>2wq1;k+:1::+K@x-:2!s fe U^@xx䍼2G; cmmi6n.) it2k c(K@x-:233 fe ş@xx>2wq1+:1::+K@x-:2!s fe U^@xx䍼2Gn.);1k8H1;n2;P껹iseea疍:9subMellipticFwithgaint=K1=ڟ fe PA2k6+11:WhenKyξn 1yΟ fe ̟PAd2a$ K |n⊟ fe \fe2(k6+1)!|> 0,Ythereisafunctionf8c2M33nq% cmsy6 133 fe 22v] Ln33x| fe 2(k+1) comp0)(Rn+1t);:9andforan!yu2DMޟ0(Rn+1t)withPVu=f-,thenֽu/=2P<e1+%lKoc$(Rn+1)foran!y>0:ֹIfK! n 5! fe ̟PAd2 Kkܾnѿ fe \fe2(k6+1)"3>0H:9therefisf8c2 M33n 533 fe 22v] Ln33x| fe 2(k+1) comp0)(Rn+1t);an!ydistributionuofPVu=f"isnotloMcallybounded. |f27f:9Corollary:FeorJ&an!yp >2;1>0;J&thereexistsn;s,\suc!hthatthereexistsf8c2 L
;comp&+(Rn+1t);:9an!yfdistributionsolutionofPVu =f"isfnotinL
0,thereisu2e+%lKoc;1Lu=f :AndlT:9an!yfsolutionu;1Lu =f"isfine+%lKoc.:9(b)NIfLsatisescondition( ),`IundersomesmoMothnessassumptiononsomen!ullsetofL,then:9forfan!yf8c2 AcompZ;1>0;fthereexistu 2e+%lKoc;Lu=f ::9(c)bIf;jĖL9.satisescondition( ),pBundersomesmoMothnessassumptiononsomen!ullsetofL,thenGȍ 7wu 2DMޟz0;Lu2z:jcompZ! u2e+%lKocfor+>0:9Unlik!efinthehigherdimensionalcase,inR2լ;weprove: :9Theoremb4.2::If PvisasubMellipticpseudo-dieren!tialoperatorofordermint!wo vdDariables,if:9P+satisesfcondition(PV),thenGȍ u 2DMޟz0;PVu2z!̽u2z+m 1+/forD>0;:9thefgain}issameasinEgoro!v'stheorem. :9OuranalysisfordiagonalopMeratorsalsoleadstoaregularit!yresultfortheobliquederivdDative:9problem.ThefobliquederivdDativ!eproblemconsistsofthefollowingbMoundaryvdDalueproblem: u cmex10(. ͰɽPVu =f >on J| Ͱɽl7)u =g >on J|@ !č:9where@P= azijJ(x)K @x-:233 fe PA@xx8:i,r@x8:jh+rbzjf (x)K@33 fe PA@xx8:jisasecondorderrealellipticdieren!tialopMerator, isaboundedJ:9domaininRn+1t;1lisarealv!ectoreld.TheobliquederivdDativeproblemhasbMeenstudiedinrecent:9y!earsbymanypMeople(e.g.[H2],[EK],[NS],[S],[W]).Feormorereferencesrelatedtotheoblique:9derivdDativ!efproblemwereferto[W].:9Ifw!ewritelP=T+ta(x)~n;whereT4Nisatangentialvectoreldto@ ;c~n istheoutwardnormal:9v!ector4Meldandifa(x)isnowherevdDanishing,theproblemisanellipticbMoundaryvalueprob-:9lem.([ADN])..Ingeneral,$`theproblemisnotw!ell-pMosed.6Thesolutionoftheproblemdependson:9the%bMeha!viorofa(x)nearthenullset.WeeassumeC0Pmۍ ݾkU] jv=0ɔjTVj`a(x)j >0.Wee%ha!vefollowingregularity:9result::9Theorem4.3:+(a)LIfa(x)nev!erchangessignfrom-to+inthepMositivedirectionofTV,vlet:9zFչ=ˌfx2@ j:ua(x)=0E;1a(x)#c!hangessignfrom+to-inthepMositivedirectionofTyatxg.9IfzF6=ˌ;, |f37f:9w!ep=assumeitisaclosedsubmanifoldin@ ;andTistransversaltoit.;aThen,ifuisasolution :9ofUtheobliquederivdDativ!eproblemwithf\2.ֹ(;j7xĖ );1gi2<(@ ),whichUsatisfyanitenumbMerof:9compatibilit!yfconditions,thenu 2 Y(;j7xĖ )fwhere b= min(yL+n2;1+K B$1 fe PAk6+1U`)::9(b)aIf a(x)satisestheassumptionsin(a),8i.e.,a(x)ais"emergen!ttypMe",andweassumez6= ;,:9thenfthefollo!wingproblem덍C&