On parametrically quasi-elliptic boundary problems

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY 4.0 h ps://c ea i ecommons.o g/licenses/by/4.0/ On pa ame ically quasi-ellip ic bounda y p oblems © 1980 The Finnish Ma hema ical Socie y Published e sion Pu monen, Veikko Pu monen, V. (1980). On pa ame ically quasi-ellip ic bounda y p oblems. Annales Academiae Scien ia um Fennicae. Se ies A I. Ma hema ica, 5(2), 237-262. h ps://doi.o g/10.5186/aas m.1980.0521 1980 Annales Academi e Scien ia um Fennicae Se ies A. I. Ma hema ica Volume 5, 1980, 237-262 ON PARAMETRICALLY QUASI.ELLIPTIC BOUI DARY PROBLEMS VEIKKO T. PURMONEN In oduc ion Le A(z,D) be a pa ial di e en ial ope a o wi h a complex pa ame e z such ha he co esponding polynomial A(z, ) is quasi-ellip ic o ype x, in which case A(2, D) will be called pa ame ically quasi-ellip ic o ype x. Le B (2, D), ..., B*(2, D) be x pa ial di e en ial ope a o s and se B(2, D): (B 1z, »1, ..., B,(2, D)). We conside he bounda y alue p oblem (P) (A(z,D)u,yoB(z,D)u): U, g). In he ellip ic case his p oblem was in es iga ed by M. S. Ag ano iö and M. I. Vi§ik in [2]. In his pape he p oblem (P) is s udied in he abo e se ing wi h ope a o s o mo e gene al ypes bu , on he o he hand, in he so-called canoni- cal si ua ion. A e he p elimina y sec ion we in oduce in Sec ion 2 he no ion o a pa a- me ically quasi-ellip ic ope a o A(2, D) and conside he equa ion A(2, D)u: . Sec ion 3 is de o ed o he bounda y p oblem (P) wi h quasi-homogeneous ope - a o s. We p o e he e an a p io i es ima e which sa is les ce ain uni o mi y equi e- men s, and s udy he unique sol abili y o (P). The esul s a e ex ended o he nonhomogeneous case in Sec ion 4. As we shall lnally (in a.5) no e, one ob ains as a consequence a known esul o bounda y p oblems o he o m (,1. (o1 u + ;.u, y o B (D) u) : U, g) unde mildly educed assump ions. A b ie ema k conce ning gene aliza ions is also made. In a o hcoming pape we shall s udy gene al ini iai-bounda y alue p oblems by making essen ial use o he esul s o his pape . Acknowledgemen . Fo inancial suppo I am indeb ed o he Emil Aal onen Founda ion. doi:10.5186/aas m.1980.0521 238 Vpx o T. PunuoNEN L. P elimina ies 1.1. Le ypical poin s in Rn and in i s dual Rn:R , be deno ed by !:(!',!o):(h ,.. ln- ;/,,) and 4:(!' 4n):(4 ,... 4n-1> 4 ), espec i ely, and se (y, ) : (y', ') + y, l n : ! 4 * ... * ! n- 4,- * ! n4 n. T a:(a ,...,an)eNo is a mul i-index, an n- luple o nonnega i e in ege s oc (N, we w i e Dn : Di: DT,... D?,", whe e Do: -i lDy* wi h he imagina y wi i€C. Likewise, we se q" : ll' ...q?,". In wha ollows we shall use also he o en mo e app op ia e no a ions y:(x, ): (x , ... xn- ) and :(€, ):(€ , ..., n- ,x). The in e se 9- o he Fou ie ans o ma io i: in, (gu)( » - , e-i( " ) u(y) dy, is deno ed by 1, and simila ly 4:4-'. Le 9* and i s and o he pa ial Fou ie ans o ma ions, oo, (9*u)(€, ) : n,-1 e-i(x'il 116, 71 4*, whe e z*:( 2n)- "lz. (%u)(x' ) : q e-i "u(x' ) d ' No e. I is con enien , some imes, o le II(w) s and o a unc ion H i a iable w. 1.2. Le mp,k:0, l, ..., , be posi i e in ege s, :max{m1, , Q*:Fl i*, and se 4:(q' , 4n):(q , ... , Q, - , 4,) - Conside a (complex- alued, app op ia ely de ined) unc ion H:H(2, y), z€Z,q:((,0 wi h C(R"-', (€ o ((C, whe e he pa ame e se z is he sec o Z:Z(a , a, )cC, (D 3(,)2, de ined by Z(a ,cc, ): {z(Cl@ < a g, < ,az}. We shall say ha 11 is homogeneous wi h weigh (qo, q) o (qo, q)-homogeneous o deg ee s€R, and w i e (qo, q)-degä:s i H()'ao z, lo4) : H(2,4) o all ,1.>0, whe e 7eq : () e 4 , ..., lq^qo); he homogenei ies wi h o he weigh s a e de ined and indica ed analogously. On pa ame ically quasi-ellip ic bounda y p oblems 239 Nex , we de ine €€-R*L, T€.R, , 4-(€, )€R", z(.C, and se u he mo e, h(2, w): (z)+(w), K(2, w): 1 +(z)'z (w)')' ', K(w): K(0, w): (1+(w)z; z. No e ha he (.)- unc ions and hei possible sums a e homogeneous o deg ee 1 wi h espec o he co esponding weigh s. 1.3. .F1"-spaces. Le Ro*:{y:(x, )€R'l =0} and, when con enien , Ie O s and o .R', R! o R'-1. The no ms o he Lebesgue space Z2(O) and he (aniso opic) Sobole space '(O) (see L. Hö mande [3], L. N. Slobodeckii [7], L. R. Vole iö-8. P. Panejah [9]), he e employed o s>0, a e hen deno ed by II . Jlo and ll . ll",o, espec i ely; O will be omi ed in he case Q:N, and eplaced by he symbol + in he case O:Al and by y in he case O:R'-l. We ecall he de ini ions o he I/"(O)-spaces: Le 9'(R') be he space o empe ed dis ibu- ions in R'. Then I'(.R') is de ined by ä" (R) : {u (,9' (R')l K( D' gu € Lz (R } and he no m ll'll" b ,u ": ,K( il'gu - The de ini ion o ä"(A'-1) is analogous, and llall", : llK(O' g.ull . The space H"(Ri) consis s o es ic ions Å* U o /€,FI"(Ro), and Il.llo* is gi en by ll u ll", * : in {ll ull" lu( H" (R'), R* u : u ; he e Ä* is he ope a o es ic ing unc ions (dis ibu ions) de ined on Ro o R!. clea ly Ho(a1:721(2), so ha ll. ]lo,o can be eplaced by ll'llo. I is well-known ha Co-(R) is dense in ä'(.R') and C (R!) in H'(Ri); he symbol C*(ni), o k€N o k:-, is used o deno e he space o es ic- ions o [!:{y:(x, XR'l >0} o Cå(R)- unc ions (Ck- unc ions in R' ha ing compac suppo ). We ecall also ha H'(Q)cH'(O) algeb aically and opologically o s> . ((): (:ä oY.) , ( ) - l ll u", Qil - (((),, *( )u1 u ( ) - lzll uo, i w-€ o 4, VB co T. Pun oNEN 240 1.4, Le z€C. The de ini ion llulll, ",, : llull?, a * (z)'" llullk, ue H" (Q), yields ano he no m on ä'(O), equi alen o ll.11",,2 o any ixed z(C. Le H:(O):(H'(O), ll.ll",",o) be he space I"(O) wi h no m ll.ll,,",o. No e ha llull|," : ll ll2,",^ - [ *e, q 'lnulz dq, u6H" (R"), . llull2,",, : llull|,",n- - I *e, C)2"|g*ulz dC, u(H"(R"- ), and llulll,",*: llzlll,",nl - in {llull,," I u€FI"(A'), R*(J: u}, uqH"(R"*); he e, as well as below, he no a ion A-B o wo exp essions I and .B means ha c A=B=CzA wi h wo sui able posi i e cons an s c, and C, (only wi h admissible dependences). No e. The symbol c will be used, h oughou his pape , o deno e a gene ic posi i e cons an . 1.5. Lemma. I s> >0, he e is a cons an C>0 such ha , o all z(C, llul,l,,,,a s Cllu l",",o, u€H"(A). P oo . Since and llull,,,,a - llull,.o+Q)'llull" we ob ain llull"a= cllull"'o' llull,,,,, = c(ll a ll", o + (1 + (z)") ll ull ") = c llull ",,, n. 1.6. I X and Y a c wo (complex) no med spaces, g(X;y) will deno e he no med space o all bounded (linea ) ope a o s o X in o y. we ecall ha he e exis s an ex ension ope a o .E om /,(Ri) in o ä"(R,), i.e., an ope a o E<9(H'(R!);ä'(R')) sa is ying (l) R*Eu: u, ueH"(Ro+). In ac , le be a posi i e in ege and de i.ne o u€C (ni) (u(x, ) i =0 (E"u)(x,', : äi ).1u(x, -j ) i < o, whe e he coe cien s 1 , ..., ).u*, a e de e mined by he sys em o linea equa ions +1 Z ?DoÄ,: , k:0, ..., . j:L On pa ame ically quasi-ellip ic bounda y p oblems 241 Then o any s>0 wi h s<y (no e ha hen Ci(,R')c 1"(,&') and y= max{ke Nlk=slq*}+ ) he ope a o E,: C -(Ei) * Cd(R') ex ends o an ope a o E: En(g(H"(R+);ä"(R) ha ing he p ope y (1) (see [7]). Mo eo e , i B:9,..., and we se o u(C;"(Ri) lu(x, ) i > 0 (E..D u)(x,,, : lä ( ip ).1u(x, _j ) i < o, hen he ope a o B( ):B@) mapping C;( ,i) in o Ci- (R,) sa is ies D E : E$) D . 1.7. Lemma. Le s>0 and >0 be giuen. Then he e is a cons an C>O such ha ( )" llull,,",o = c llull,,"+,,a, u(H"+' (Q), o all zQC. P oo . The s a emen ollows om he inequali y ( )u llullS,o = C llull|, "+,,a, u€ H"+" (e), whose p oo in he case d):R" o -R'-1 is s aigh o wa d, and which hen in he case Q:R is ob ained by use o he ex ension ope a o E:En wi h >s* . 1.8. The ace ope a o yo: Ci"(R|)*Cå(Å,- ) is de ined by (you)(x): u(x,O) o u€C ,(P*) and, o s>q, 2, ex ends by con inui y o a con inuous ope a o ls i u + you : H" (R*) * Hs- s"lz (Rn-L) (see, e.g., [8]). 1.9. Lemma. I s>q,12, he e exis s a cons an C>O such ha lll oull,,,- q^ 2,, = c llull ",,, *, u( H" (R*), o all z€C. P oo . Le u€C CN+) and pu (J:Eu, whe e ä:3:, wi h y>s. Then we ha e llyoull?,"-q^ 2,, - [ K1z, )2"-s"l(g"yoU)(Ol, d(, 242 V o T. PunuoNEN whe e Now so ha = 7T?{ (X ,()' ** ')lgul'd { W I : zK( ,11-u [ (X{ , ä)'q"*d)l7Ulz u. Hence i ollows ha ll y oulll, " - q" 2,, = c [ (x {2, O'" + K (2, 4)zs - 2 sn < >zs^) I u p a I a =cI Q,4)z"l Ulzdq = C llull|," = C llull|,,, *. 1.10. Lemma. Le s>0. The e is a cons an C>0 such ha o any aQN' wi h (u, q)=s we haue llDo ull,, "_ qo,n1, + S C llull,,,, +, u€ H' (R*), o all z€C. P oo .Suppose a(-FI"(Ai) and se U:E (H'(R'), whe e E:8, wi h =s. I :(.a,4), we ha e llD, ull , " -,, + - I * Q, l)zs - 2" lgF Dq (J l, dn : I * k, q)'" - " '" l U l' ch . l(s.To(D(Ol2 - [email p o ec ed])(€,O)l' = n?( sul d )' d llD'ull',,s- ,+ = C { K( , il"lquPd y= Cll li',,s,*. 2. Pa ame ically quasi-ellip ic ope a o s 2.1. We shall conside pa ial di e en ial ope a o s o he o m A(2, D) : ooo*åo)=uakdzkDd (k€N, a(N"), whe e he coe icien s aka a e complex cons an s and he pa ame e z€C. The p incipal pa Ao(2, D) o A(2, D) is gi en by (l) ao(2, D) : ooo*å :oqkdzkDd, so ha he co esponding polynomial Ao(z' q) : ono*än :oa"ozk4o' he p incipal symbol o A(2, D), is (q0,4)-homogeneous o deg ee p. On pa ame ically quasi-ellip ic bounda y p oblems 243 2.2. Lemma. I s=p, he e is a cons an C>0 such ha ll Ao (2, D)ull,,"-, 5 C llull,,,, u( H" (R), o all zQC. P oo . The e exis s a cons an Co>O such ha lAo(zo,qo)l a C , zo(C, qo(N, h(zo,4o) : l. Fo a bi a y z(C and 4€R' wi h h(2, i>O we ind Ao (2, D : h (2, 4) Ao (z', qo), whe e zo:h( ,q)-'oz, qo:h(z, i-qq (see 1.2 a d 2.7), and he e o e lA'(z, il< Csh(z,q)P, z(C, qQR. Thus we ha e llAo(2, D)ull1,§-& - I *G, q)%-zplgAo(2, D)ulz d4 = c I xQ,q)z'-zuh(z,41zu19ul2dn = Cllull ,". 2.3. De ini ion. The ope a o A(z,D) is said o be pa ame ically quasi- ellip ic i i sa is ies he condi ion (QE) Ao(2,4) * 0, z(Z : Z(a , uo ), q(R, h(2,4) = 0, o , equiualen ly (c .l31), i lAo(z,q)l > c h(2,4)u, z(2, 1(R, wi h some cons an co>O. Le us now assume ha he condi ion (QE) is ul illed. Conside Ao(2,(, ) as polynomial in he complex a iable z. Then he e a e unc ions o:q(2, O, k:1,...,mo, co i uous in Z(R'-L, such ha o each ixed (2,€)€ZxR"- hey a e he oo s o he polynomial Ao(2,(,'e), Ao( , (, 1,(2, O) : 0, k : l, ..., mn. 2.4. Lemma. The oo s x(z, ), k:1,...,mn, a e (qs,q')-homogeneous o deg ee qn. Indeed, we ha e Ao( , (, ),-' " n(Lqoz, Lc'E)):0 o e e y )">0, aod he unc ion ) +)-c"xo17aoz,)d ) om R* in o C is con inuous. 2.5. In wha ollows we shall mos ly conside ope a o s A(2, D) which sa is y a somewha s onge condi ion: 244 V o T. PuRuoNEN condi ion (QED). The ope a o a(2, D) is pa ame ically quasi-eilip ic o de e mined ype 24, l=2' <mo, ha is, i sa is ies (QE) and, mo eoue , he con- di ion (D): (D) Fo eue y z€Z and eue y ((N-L wi h h(2, O>O he polynomial A'(z, , ) in he complex aa iable has exac ly x, oo s !(z, ), j:1,...,x, wi h posi ioe imagina y pa , Im !(2,€)=0. Rema k. When n>2, (QE) implies (D) and hence (eED). 2.6. Theo em. suppose ha condi ion (eE) is sa is ied, and e s> . Gi sen Q=0, he e is a cons an C>0 such ha he a p io i es ima e (1) llu l,,"= CllAo(z,D)ull,,"_ , ueH"(R), is ualid o eue y z€Z wi h ( )=5. Fu he mo e, o eae y ze Z {0}, he ope a o Ao(2, D)($(n:(n ; H:- '(R")) is an isomo phism ( o he locally conaex s uc u es). P oo . To p o e (1), pick u€H'(R . By i ue o (eE) we hen ha e, i (z)=0, Hence lFul = c;Lh(2,7)-ulgAo(2, D)ul. llull?," = C I X(2, q) "h(2, 1-zulFAo(2, D)ulz d 1. Since he e ob iously h( , i > CK(z,q) wi h C:C(e), we ob ain llulll,"= C { XQ, y)%-z lgAo(2, D)ulz dq = CllAo(2, D)ulll,"_ . I is clea ha Ao(z,D) is now an isomo phism o H:(R") on o H'-p(R p o ided ha i is su jec i e. This, howe e , is easy o see. Indeed, i Enj-u1p"1, hen and u. : FnAo(z, i- g €H:(R) Ao (2, D)u : 9, Ao(2, 4) 9u : . Bounda y alue p oblems 3. The case o p incipal pa s 3.1. Le x be a posi i e in ege , and le Bl(z,D),...,82( ,D) be x ope - a o s de ined by 4G, D) : Z bil,ozkDo, j : 1, ..., %, kqo+<q,q):il. whe e he coe icien s bi6 z acomplex cons an s, he pa ame e z€c, and p >0. on pa ame ically quasi-ellip ic bounda y p oblems 251 wha we now ha e o p o e is he e o e he es ima e (B); his will be done in he es o his sec ion. 3.5.2. l u(H'(R"a) a d Ao(2, D)u:O, hen g : 4Ao(2, D)u : Ao(2, (, D)$*u. Hence, o (almos ) all <€R"-1, (4dG,) is an exponen ial solu ion o he equa ion (4) in 3.3, so ha i belongs o 9(R*) (c . [1]) and, u he mo e, o (z,E). Acco dingly, Lemma 3.3.8 implies ha (g,u)((, D : 2 co(2, O No(2, (, ), k: whe e now (see he p oo o Lemma 3.3.6) co(2, O : yoB l(2, , D)( u)((, ) : yo i"B l(2, D)u' 3.5.3. Choose an in ege such ha y>§ and >slq,*112, and w i e again E:En. We ha e (4) llall?,",+ =llEulll,"- [ *@,4)2"l9Eul2d4. }Je e FEu:4E4u, so ha (5) iEu: 2 ,oe, C)(glENk)(2, (, ))( ). k:1 The e o e i su ces o conside he in eg al (6) I *@, )ulc,,(2, O(g (EN)(2, (, ))( )12 dn - [ uk, . )2'1c1,(2, O'P(l l( ,1nNo11z, C, ))(x)|'z d4dC + [ lc ,Q, o ([ $)*l(q,1zwo11z, 1, ))('»|'z d )dc : I *Iz. 3.5.4. Conside nex he unc ion (7) No(,,€,o:* [ i i,* e, < whe e .l-, is a ec i iable Jo dan cu e enci cling he oo s z (z,E), .i:1, "',x' The e a e R>0 and ä>0 such ha , o i:1,...,x, l (zo, (o)l = A and llm Qo, (o)l = ä o all zo<Z and all (0(R'-1 wi h h(zo, (0):1' Hence, wi h hese z0 and o' 252 V x o T. Pun oNEN he cu e , in (7) can always be de o med in o he same Jo dan cu e, say, in o .l-*,0 consis ing o he pa hs l(l:lR,Im(>ä and Im C:ö, l[=n F om Lemma 2.4 i ollows ha o a bi a y z(Z and (<R"-, wi h h(2,()>0 we can ake s:h(2, )o"l^,0. The unc ion Lo, (8) Lo( , 1, O: y+ ': cn'() , h(2, o = o, (€c, A ;9 l) is (q0,4)-homogeneous o deg ee -il ,-7, (see (l) o 3.3 and Lemma 3.3.9), so ha , in pa icula , Lo( , , C): h(2, )-Po-c"L*(zo, (o, (o) when (2, ,():h(z,11 eud(zo,(o,( . F om his and he ac ha , o a sui able cons an C>0, lLo( ', (o, (o)l = C o all zo,(o wi h h(zo,1o1:1 and all (06.1-*,u, we see ha (9) lLok, 4 gl = Ch(2, )- "- " o all z, ( wi h h(2, O>0 and all (q :1(2,1)o" ^.u. By i ue o (7), (8) and (9) we hus ob ain lNo( , 1, )l = Ch(z, O-Pu-q"l( )maxlei (1, whe e /(l-,) is he leng h o i-6. Since and l( e) =. (2+n) Ah1z, 17e" le"(l = exp (- h (2, 11e" ö ), we inally ha e (10) ln - ( , €, )l = Ch(2, O-p eXp (-h(2, g " ö ) o all z, ( wi h h( , O=0, and o all >0. 3.5.5. We e u n now o (6).Fi s , in he in eg al I , we ha e I Wa(EN)( , (, ))( )1, a' : l(E ,{o)( , (, )lz d = [ *( * ,;ä:]w,' i - (,, €, ) zd (see 1.6), whe e, bV (10), i l {o( , (. )1, d = eh( , 1-Z *-en. No ing ha h( , () - K(2, () when ( )=- g, we he e o e ob ain (11) hs C I K( , C) 'ly g.$k, D)ul K (2, O-z k-q^ d( = C llyoB ,( , D)ull ,s_ x_ ,n z,T . On pa ame ically quasi-ellip ic bounda y p oblems 253 3.5.6. To ea he in eg al 1, le us ew i e i in he o m I : I lc1,Q, Ol2 Ho!2, ) d(, whe e HoQ, E) : I ( ) l(s,(EN)(2, (, ))( )1'zd . 3.5.7. Fi s o all, we shall e i y he con inui y o He(2, O when h(z,O>0. To do his, le e>0 be gi en, and le zo€Z and (o€R'-' such ha h(zo, $)>0. Res ic he conside a ion o a neighbou hood o (zo, (o), and no ice ha o all (2, () in his neighbou hood he cu e .l-g in (7) can be de o med in o he same ec i iable Jo dan cu e, say, in o I wi h he p ope y : min {Im (l(e } = O (c . 3.5.a). I Zo deno es again he unc ion gi en by (8), hen he e is ä">0 such ha , o a l (€ , lLo(2, , O - Loko, h, 0l = e whene e lz-zol*l(-(ol=ä". n iew o 1.6, i O= i= , we hus ha e o ,>0 (c . 3.5.4) l(D No)Q, €, )-(Dl N)( o, €o, )l = * { lLo(2, €, O-L*(zo, €o, OllUlle el i(l = C( , )ee-' and hen o / <0 l(D EN)(2,4, )-(D ENk)@o, h, )l : l(E@) Dl N)(2, (, )-(E@ D No)Qo, g, )l y*1 = z le j)pl lljl l(D N)(2, €, - j ) - (D N)(zo, h, - j )l i:1 = C( , )es-'l l' hence, o all €R, l(D EN)(2, C, )-(D ENk)( o, 4 , )l < C ,e-' i lz-zol*l(-€ol=ö". This yields (12) l a(( i,(ENp)(2, €, ))k)-(F,(EN*)(zo, (0, )(")l = n, I le - " ((D E Nk) Q, , ) - (Dl E Np) (z o, $, ))l d = Ce, p o ided ha lz - z l*l( - (ol =ä,. Nex , a small compu a ion shows ha (13) lHoQ, O-HoQo, *)l : l/{ X"{ F,(ENo)e, (, ))( )|,-l(e,(ENy)(zo, g, ))( )l)d l = 2ab+b2, whe e o : ([ k)"1( ,1n o11zo, i,o, ))( )12 dx) z, : ([ (")*l(q,1 No11z, (, ))(i-(4(EN)(zo, 1o, 1)141', a)' '. Wi h O=B<y we ha e (see 1.6) Fu (s,(z x ) (Z o, Eo, ) (d I : l(q @@, D Nk) e o, ( o, )) ( )l i +l o I : " l{ e- "(D N)(zo, o, )d + Z?ilPli _ e- '"(D Np)ko.h,-iDd l, and since i ollows om (9) ha l(D N)(zo, h, )l = Ch(zo, (o)- Pu- qne-", 2s4 V co T. Pun uoNEN we ob ain (c . 3.5.4) l o (,q,(E l ) ( o, (0, )) ( ) I = C h ( o, h)- Fu- Qn . Hence we ge az= c {# [ +l l,) (,E@ ,{o)( o, h, ))( )lzd =C(z ,C )1 iu, = C(z , €o), since >s qn+112. F om (12) we de i e in he same way ha bz = Ce. The e o e, i lnally ollows om (13) ha whene e lz - zol *lC - (ol = ä, . 3.5.8. The e exis s a cons an C>0 such ha (14) Ho(zo, C = C, z0€2, (o€.R"- , h( o, (o) : 1. Le hen ze Z and €<R"-' wi h h( , €)=0, le 1>0, and conside Ho(Aooz, )a' () : [ ( ) 'l@ (EN)(Äuoz, ),q' (, ))( )1, d . On pa ame ically quasi-ellip ic bounda y p oblems 255 In he case ,>0 we ob ain (see 3.5.4) No(),aoz, ) s' , ) : ),- PuN (2, (, [e" ), so ha , o all €X, (EN1)(Lqoz, ).s' (, ): l-pu(EN )(2, (, )'q" ). Consequen ly, (q @N)Q,soz, ),q' , ))1 ; : 1-a-e"( i(EN)(2, E, ))(1,-e" }. Hence we see ha Ho(),aoz,Ls',O: A-zpu-zq" ( ),'l(41nNo)Q, { ))().-s"x)12 d : )2s-z1 u- nHo(2, €), om which i ollows, by (14), ha Ho@, €) = Ch(2, ()2s-2 lk-s^, z€2, €€R- , h(2, O > 0. Now we e u n o he in eg al I, and each he conclusion ha I : I lc1,Q, Ol2 Ho(z, C) dC = C I h(2, 02"-zuu-a"lg*(2, Ol2 d( = c I xQ, Ozs-z,k-s^lyo7,$Q, D)ulz d(, whence (15) Iz= Cllyo l,Q, D)ull ;,"- u-qsz,y. 3.5.9. By combining (4), (5), (6), (ll), and (15) we lnally ob ain llull|,",* = C i I K(2, O2"lc1,(2, ()(g,(ENu)(2, (, ))(,4lz d4 = C 2 iy, B?,(2, D) ull , , " - uu-q^12.y. &:1 This comple es he p oo o he es ima e (B). 3.6. P oo o he la e pa o Theo em 3.4. To show ha Po(2, D) is an iso- mo phism o H)(R'*) on o /{)(Ri*R"-'), o e e y z(Z {0}, we shall, in ac , cons uc i s in e se ope a o . In wha ollows, le E:En, >s. 3.6.1. Fix z(Z {0} and de ine K : R* i Ao(2, ni-'gE , I< H;- P (R+)' Then Ko is an ope a o mapping H:- '(R"+) in o II,"(R!). Indeed, E €H:- "(R') and ll inAo(z, D-'gE ll ,"= C I K(z,q)'"lAo(z,q)- 9E l2 d4 = C I Xe, 1 " h(2, »-zp lg E lz d4 = cllE llT,"_p, so ha llKo ll . ". + = c ll E ll2, "-, 5 c ll ll2, " -,, * ; 2s6 V x o T. Pun loNEN his implies, u he mo e, ha Ko € s (H :- (n!) ; i(n!)). 3.6.2. Fo i:1,..., ?4 we se Ki g : %a(Ni *g), g(Hi-u1- " 2( ?4-1). Fi s , one easily checks (see 3.5.5) ha K g(L (ni), a leas . Since now and so EKig - 4(@U)g,s) we ob ain iEKig: (%EN) i*s' (1) I *Q, )2"l3EKislL d4 - I *@, 0^ ([ lF,ENip d )ls"sl, d(+ [ ([ k)" l. i,EN1l, d )ls.sl, d€ : I *Iz. He e we ha e (see 3.5.5) = c I xQ, c)'"h( , €)-2p -q"l4sl, i ( = cllSll?,"-p:-q^iz, and (see 3.5.8) I : H,k, Olg-sl'dE = c I l @, ozs-z 'i-snlgF sl'd€ = cllsll1,,-p -q^ z, . I he e o e ollows om (1) ha and hence EKlg€H)(R)' Ki8: R+EKjg(H:(R"+); mo eo e , 11 x sll,, ", * = C ll slll, " - u - en 2, y. Consequen ly, K i e e (n ;- u - c" z (Rn - L)' H : (R^*)). 3.6.3. Le us now se KoF : K" + ; Ki(s,-yo4Q, D)Ko ), F : ( , g , ..., g,)( :(R+, Ro- ). j:1 Then K0 is an ope a o on a ,)(Rl,R'- ), alued in Hi(R"), by i ue o 3.6.1 and 3.6.2, and u he mo e llKoFll,,", * = cll ll","-p,+*c å-(l ,ll",,- i-q, z,y*llTo4k, D)Ks ll.,s-p1-q^12,y), j: whe e (see Lemmas 1.9 and 3.2) ll o i @, D) Ko ll,, " - u: - hn z, y = c ll I ll,, " - u. * . Thus we ha e Ko < g (/ : (R , R - L) ; H ) (R"*)). On pa ame ically quasi-ellip ic bounda y p oblems 257 3.6.4. The ope a o K0 has he p ope y Po(2, D)KoF : F, F : ( , S , ..., g*)€.#)(Ro*,.R'-1). To see his, le F:( ,g , ...,g)Ca ;g *,R'-') and se u:KoF(Hi(Ri). Then we ha e Ao(2, D)u: Ao(2, D)Ko + 2 Oo@, D)K1(si- o iQ, D)Ko ), : whe e (no e ha Ao(z,D) and R* commu e) Ao (2, D) Kn : Ao (2, D) R+ g4Ao (2, q)-L I E : R + , Ao Q, q) Ao (2, D-' g E : and (see 3.3.7) Ao(2, D)K/Ei-To4Q, D)Ko ): Ao(2, D)g<(N jg,(s1-loB](2, o)Ko l) : ge((Ao(2, 4, D)N)g"(ci-To4Q, D)K i) -0, so ha Ao(z,D)u: . By (12) o 3.3, we ob ain u he I o 8 l.(2, O) K lg 1 - o Bl (2, D) K o ) : 9e(0 o 4(2, (, D) N ) 3,(g i - y o4Q, D) Ko i) : ö ; ,(g i - T o B] (2, D) Ko ), whence y * Q, D)u: yoBlk, D)K + åa,o(s1- o4Q, D)Ko ) : 8k, as equi ed. I hus ollows ha he ope a o P0(2, D) is a con inuous bijec i e linea map and he e o e an isomo phism om H;(R"+) on o a ,)(R"*, R'- ), wi h in- e se K0. The con inui y o Ko was, in ac , ound also di ec ly in 3.6.3. 4. The gene al case In his sec ion we shall ge e alize he esul s o Theo em 3.4 o co e he case o nonhomogeneous ope a o s. 4.1. We shall need he ollowing wo lemmas. 2s8 Vu«o T. Pun , oNEN 4.1.1. Lemma. Suppose s1>5 >5 3Q. Giuen any e>O he e is a cons an C(e)>0 such ha llull",*,* = e llall,,",, a * C(e) ll ll,,*, *, u€H (R ), all z(C. P oo . F om he well-known inequali y (equal o he abo e wi h z:0) llall*,* = ellall",,*+C(e)llull",,+, u(H (R ), we ob ain llull",,,,*-llull*,++(z)"llzll*=ellull,,,a*c(e)llall",,**(z)"llull*. The asse ion ollows he e o e om he inequali y ( )", = e (z) + c1e; (z; , which in u n is a consequence o he elemen a y inequali y oA=!o !60', a,b>0, p=1, p,: pl@-l); pp as a ma e o ac , choose p : +, a : (pe) o(z)" , b : (pe)- o (z)ss o" ' s -sa 4.1.2. Lemma. Le s>O and >0. To eue y e>O he e is C( )>O such ha Z llDull",",* = e llull,,"*,, a *C(e)llull*, u(H"-"(R'+), (a,q)= o all z€C. P oo . By Lemma l.l0 we ha e llD" ull ". ", * = C llull,, " * (n, q), + . I we now pu k: max(a, a)la(N", (a, q) - = . i ollows om Lemma 1.5 ha ,,,?, =,llull " " * (n' q)' + 1 c llull " " * o' *' whe e, by i ue o Lemma 4.1.1, llull,,"*0, * = ellall,,"*,, + * C(e)llzll * . 4.2. ln he gene al case we ha e o conside ope a o s o he o m P (2, D) : Po (2, D) + Poo (2, D), whe e P0(2, D), he p incipal pa o P(2, D), is de ined as in 3.1, and Poo(2, D) On pa ame ically quasi-ellip ic bounda y p oblems 259 is gi en by P oo (2, D) : (Aoo (2, D), y o Boo (2, D)) : (Aoo (2, D), y o Bl0 (2, D), .. ., y o 4 @, D)) wi h oo(z,D)- Z akdzkDa, kqo*(a'q)=P 4o (', D) : o o*l*-o,b i ozk D; he e he coe icien s a1,n and bi*a a c complex cons an s, o cou se. 4.3. Theo em. Suppose ha Po(z,D) sa is ies Condi ions (/ED) and (CC), and le s€5. Then he e is Q-0 such ha , o some cons an C>0, llull,, ", + = C lllP (2, D)ulll,, ", u ( H" (R*), o all zQZ,(z)>-7. P oo .In wha ollows le u be any unc ion in ä"(R|) and le z€Z {0}. 4.3.1. Acco ding o Theo em 3.4 we i s ha e llull,.".+ = ClllPo(2, D)ulll,,", which yields (1) llull,,,, + = C (lllP (2, D) ulll,, " * llAoo (2, D)ull ", "- p, + + 2 W y'(2, D)ull,,"-u,-q. z,y). j=L 4.3.2. Gi en any e>0 we ob ain, by i ue o Lemma 4.1.2, ll Aoo (2, D) ull,, " _ , * = c, o?^l lo ,, , i _ , oll D u 11,,, _,,, + = c oZo(z)keollull,,"- qo, * *G(e) o?^"( )o,,llul*. Since ) (z)k e" llull * = c (l + (zlu - ao) (z) -" ll u 11,,,, *, k<mn we ha e, by Lemma 1.7, (2) llAoo(2, D)ull","- ,* =(coe+cu(e)(1+(z;"-n)(z)-)llull,,",*. Fix now e>0 sa is ying C ={, and choose go>0 such ha c,c) #_ = ä. 260 V m o T. PunuoNEN Then i ollows om (2) ha (3) llAoo(z,D)ull,,"- ,. =) lull",",* o all z€2, (z)=po. 4.3.3. Nex , in iew o Lemma 1.9, we ind ha lly o 4o ( , D) ull,,, - u, - on z, 1 " u oF- o,l lo ,, o, , _ oo oll D' ull,, " - u,, * . Hence, gi en any e>0, we ob ain again om Lemma 4.1.2 lll o S @, D) ull,, " - u - q, z, = Cee ) (z)kullull,,"- ,qo,a- C7@) ) (z)ka,l1u11*. kqo=pl kqo-pJ Since Z <z>ks,llull+ = c(l *(z)u 11 7-"llull,,",* kqo<ili (ob iously, his inequa ion could be s eng hened), we ha e, as abo e, (4) ll io ( , D)ull,,"_ u,_ n^ z, s (C, e + Cs (e) ( + (zyu1q 1-") llull,,,, * . I we now ake e>0 so small ha C = i; and choose Q;>0 such ha c^G +a j = I _u _/ Ai _ gxC , hen i ollows om (4) ha (5) llyoBlo(2, D)ull,,"- ,- ^, ,, = i Llull,,", * o all z(2, ( )=pi. 4.3.4. To comple e hep oo ,i isnow enough ode ine q:max{q,, Q , ..., Q,}. Indeed, combining (1), (3), and (5), we ha e Ilull,,", * = CllllP(2, D)ulll,,"++llull,,", * * ,J LO)lull,,", * and hus llull,,".* = ClllP(2, D)ulll,," o all z(2, (z)> p. 4.4. Theo em. I Po(z,D) sa is ies Condi ions (QED) and (CC), and i s(,S, hen he e is p>0 such ha , o eue y z€Z wi h ( )=_0, he ope a o P(2, D) is an isomo phism o H;(R"+) on o J ,i(R"*, R,-1). P oo . We shall show ha P(2, D) can be ep esen ed as he p oduc o po(2, D) and a i(R"+,,R'-1)-au omo phism.