On parametrically quasi-elliptic boundary problems

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ On parametrically quasi-elliptic boundary problems © 1980 The Finnish Mathematical Society Published version Purmonen, Veikko Purmonen, V. (1980). On parametrically quasi-elliptic boundary problems. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica, 5(2), 237-262. https://doi.org/10.5186/aasfm.1980.0521 1980 Annales Academire Scientiarum Fennicae Series A. I. Mathematica Volumetr 5, 1980, 237-262 ON PARAMETRICALLY QUASI.ELLIPTIC BOUI\DARY PROBLEMS VEIKKO T. PURMONEN Introduction Let A(z,D) be a partial differential operator with a complex parameter z such that the corresponding polynomial A(z,r) is quasi-elliptic of type x, in which case A(2, D) will be called parametrically quasi-elliptic of type x. Let Br(2, D), ..., B*(2, D) be x partial differential operators and set B(2, D): (Br1z, »1, ..., B,(2, D)). We consider the boundary value problem (P) (A(z,D)u,yoB(z,D)u): U, g). In the elliptic case this problem was investigated by M. S. Agranoviö and M. I. Vi§ik in [2]. In this paper the problem (P) is studied in the above setting with operators of more general types but, on the other hand, in the so-called canonical situation. After the preliminary section we introduce in Section 2 the notion of a parametrically quasi-elliptic operator A(2, D) and consider the equation A(2, D)u:f. Section 3 is devoted to the boundary problem (P) with quasi-homogeneous operators. We prove there an a priori estimate which satisfles certain uniformity requirements, and study the unique solvability of (P). The results are extended for the nonhomogeneous case in Section 4. As we shall flnally (in a.5) note, one obtains as a consequence a known result for boundary problems of the form (,1. (o1 u + ;.u, y o B (D) u) : U, g) under mildly reduced assumptions. A brief remark concerning generalizations is also made. In a forthcoming paper we shall study general initiai-boundary value problems by making essential use of the results of this paper. Acknowledgement. For financial support I am indebted to the Emil Aaltonen Foundation. doi:10.5186/aasfm.1980.0521 238 Vpxro T. PunuoNEN L. Preliminaries 1.1. Let typical points in Rn and in its dual Rn:Rf, be denoted by !:(!',!o):(hr,..tln-t;/,,) and 4:(!'r4n):(4r,...t4n-1> 4r), respectively, and set (y, rt) : (y', rt') + y,rl n : !r4t* ... * ! nr4,- r * ! n4 n. Tf a:(ar,...,an)eNo is a multi-index, an n-tluple of nonnegative integers ocr(N, we write Dn : Di: DT,... D?,", where Do: -i\lDy* with the imaginary wit i€C. Likewise, we set q" : rll' ...q?,". In what follows we shall use also the often more appropriate notations y:(x, t): (xr, ... txn-tt t) and tt:(€,r):(€t, ...,\n-r,x). The inverse 9-r of the Fourier transformatiort fi:fin, (gu)(r» - r, f e-i(t"r) u(y) dy, is denoted by 1, and similarly 4:4-'. Let 9* and fi stand for the partial Fourier transformations, too, (9*u)(€, t) : n,-1f e-i(x'il 116, 71 4*, where z*:( 2n)-t"lz. (%u)(x' t) : q f e-it"u(x' t) dt' Note. It is convenient, sometimes, to let II(w) stand for a function H it variable w. 1.2. Let mp,k:0, l, ...,fr, be positive integers, trt:max{m1,\, Q*:Flffi*, and set 4:(q' , 4n):(qr, ... , Q, -r, 4,) - Consider a (complex-valued, appropriately defined) function H:H(2, ry), z€Z,q:((,0 with C(R"-', (€ft or ((C, where the parameter set z is the sectot Z:Z(ar, ra,r)cC, (Dr3(,)2, defined by Z(ar,cc,r): {z(Cl@r< arg, < ,az}. We shall say that 11 is homogeneous with weight (qo, q) or (qo, q)-homogeneous of degree s€R, and write (qo, q)-degä:s if H()'ao z, lo4) : tr H(2,4) for all ,1.>0, where 7eq : ()ter4r, ..., lq^qo); the homogeneities with other weights are defined and indicated analogously. On parametrically quasi-elliptic boundary problems 239 Next, we define €€-R*L, T€.R, , 4-(€,r)€R", z(.C, and set furthermore, h(2, w): (z)+(w), K(2, w): 1t +(z)'zf (w)')'t', K(w): K(0, w): (1+(w)z;vz. Note that the (.)-functions and their possible sums are homogeneous of degree 1 with respect to the corresponding weights. 1.3. .F1"-spaces. Let Ro*:{y:(x, r)€R'l t=0} and, when convenient, Iet O stand for .R', R! or R'-1. The norms of the Lebesgue space Z2(O) and the (anisotropic) Sobolev space ff'(O) (see L. Hörmander [3], L. N. Slobodeckii [7], L. R. Voleviö-8. P. Panejah [9]), here employed for s>0, are then denoted by II . Jlo and ll . ll",o, respectively; O will be omitted in the case Q:N, and replaced by the symbol + in the case O:Al and by y in the case O:R'-l. We recall the definitions of the I/"(O)-spaces: Let 9'(R') be the space of tempered distributions in R'. Then fI'(.R') is defined by ä" (R) : {u (,9' (R')l K(rD' gu € Lz (R\} and the norm ll'll" bv ,ur": ,K(vil'gurThe definition of ä"(A'-1) is analogous, and llall",r : llK(O' g.ullr. The space H"(Ri) consists of restrictions Å* U of t/€,FI"(Ro), and Il.llo* is given by ll u ll", * : inf {ll ull" lu( H" (R'), R* u : u\; here Ä* is the operator restricting functions (distributions) defined on Ro to R!. clearly Ho(a1:721(2), so that ll. ]lo,o can be replaced by ll'llo. It is well-known that Co-(R) is dense in ä'(.R') and Cfr(R!) in H'(Ri); the symbol C*(ni), for k€N or k:-, is used to denote the space of restrictions to [!:{y:(x, rXR'lt>0} of Cå(R)-functions (Ck-functions in R' having compact support). We recall also that H'(Q)cH'(O) algebraically and topologically for s>r. ((): (:ävoY.) , (t) - lrlltu", Qil - (((),, *(r)u1tru (r) - lzlltuo, if w-€ or 4, VBrco T. PunrvroNEN 240 1.4, Let z€C. The definition llulll, ",, : llull?, a * (z)'" llullk, ue H" (Q), yields another norm on ä'(O), equivalent to ll.11",,2 for any fixed z(C. Let H:(O):(H'(O), ll.ll",",o) be the space fI"(O) with norm ll.ll,,",o. Note that llull|," : llrll2,",^ - [ *e, qf'lnulz dq, u6H" (R"), . llull2,",, : llull|,",n- - I *e, C)2"|g*ulz dC, u(H"(R"-r), and llulll,",*: llzlll,",nl - inf {llull,," I u€FI"(A'), R*(J: u}, uqH"(R"*); here, as well as below, the notation A-B for two expressions I and .B means that crA=B=CzA with two suitable positive constants c, and C, (only with admissible dependences). Note. The symbol c will be used, throughout this paper, to denote a generic positive constant. 1.5. Lemma. If s>r>0, there is a constant C>0 such that, for all z(C, llul,l,,,,a s Cllutl",",o, u€H"(A). Proof. Since and llull,,,,a - llull,.o+Q)'llull" we obtain llull"a= cllull"'o' llull,,,,, = c(ll a ll", o + (1 + (z)") ll ull ") = c llull ",,, n. 1.6. If X and Y arc two (complex) normed spaces, g(X;y) will denote the normed space of all bounded (linear) operators of X into y. we recall that there exists an extension operator .E from r/,(Ri) into ä"(R,), i.e., an operator E<9(H'(R!);ä'(R')) satisfying (l) R*Eu: u, ueH"(Ro+). In fact, let v be a positive integer and defi.ne for u€Cf (ni) (u(x,t) if r=0 (E"u)(x,', : f äi ).1u(x, -jt) ir t < o, where the coeffcients 1r, ..., ).u*, are determined by the system of linear equations v+1 Z ?DoÄ,: t, k:0, ..., v. j:L On parametrically quasi-elliptic boundary problems 241 Then for any s>0 with s<y (note that then Ci(,R')cf1"(,&') and y= max{ke Nlk=slq*}+t) the operator E,: Cr-(Ei) * Cd(R') extends to an operator E: En(g(H"(R+);ä"(R) having the property (1) (see [7]). Moreover, if B:9,..., v and we set fot u(C;"(Ri) lu(x, t) if t > 0 (E..D u)(x,,, : lä (fip ).1u(x, _jt) ir t < o, then the operator B(fr):B@) mapping C;(f,i) into Ci-r(R,) satisfies DfE : E$) Df. 1.7. Lemma. Let s>0 and r>0 be giuen. Then there is a constant C>O such that (r)" llull,,",o = c llull,,"+,,a, u(H"+' (Q), for all zQC. Proof. The statement follows from the inequality ( r)u llullS,o = C llull|, "+,,a, u€ H"+" (e), whose proof in the case d):R" or -R'-1 is straightforward, and which then in the case Q:R\ is obtained by use of the extension operator E:En with v>s*r. 1.8. The trace operator yo: Ci"(R|)*Cå(Å,-t) is defined by (you)(x): u(x,O) for u€Cf,(P*) and, for s>q,f2, extends by continuity to a continuous operator ls i u + you : H" (R*) * Hss"lz (Rn-L) (see, e.g., [8]). 1.9. Lemma. If s>q,12, there exists a constant C>O such that lll oull,,,- q^t2,, = c llull ",,, *, u( H" (R*), for all z€C. Proof. Let u€CtrCN+) and put (J:Eu, where ä:3:, with y>s. Then we have llyoull?,"-q^r2,, - [ K1z, t)2"-s"l(g"yoU)(Ol, d(, 242 Vrrrro T. PunuoNEN where Now so that = 7T?{ (Xtr,()'r**r')lgul'dr { W I : zK(r,11-u [ (X{r, ä)'q"*d)l7Ulz tu. Hence it follows that ll y oulll, " - q"r 2,, = c [ (x {2, O'" + K (2, 4)zs - 2 sn <r>zs^) I r u p a I ar =cIrQ,4)z"lrUlzdq \ = C llull|," = C llull|,,, *. 1.10. Lemma. Let s>0. There is a constant C>0 such that for any aQN' with (u, q)=s we haue llDo ull,, "_ qo,n1, + S C llull,,,, +, u€ H' (R*), for all z€C. Proof.Suppose a(-FI"(Ai) and set U:Er(H'(R'), where E:8, with v=s. If r:(.a,4), we have llD, ull\, " -,, + - I * Q, tl)zs - 2" lgF Dq (J l, dn : I * k, q)'" - " rt'" lfr U l' cht. l(s.To(D(Ol2 - [email protected])(€,O)l' = n?(f tsul dr)' dr llD'ull',,s-r,+ = C { K(r,vil"lquPdry= Cllttli',,s,*. 2. Parametrically quasi-elliptic operators 2.1. We shall consider partial differential operators of the form A(2, D) : ooo*åo)=uakdzkDd (k€N, a(N"), where the coefficients aka are complex constants and the parameter z€C. The principal part Ao(2, D) of A(2, D) is given by (l) ao(2, D) : ooo*årr:oqkdzkDd, so that the corresponding polynomial Ao(z' q) : ono*änr:oa"ozk4o' the principal symbol of A(2, D), is (q0,4)-homogeneous of degree p. On parametrically quasi-elliptic boundary problems 243 2.2. Lemma. If s=p, there is a constant C>0 such that ll Ao (2, D)ull,,"-, 5 C llull,,,, u( H" (R), for all zQC. Proof. There exists a constant Co>O such that lAo(zo,qo)l a Cr, zo(C, qo(N, h(zo,4o) : l. For arbitrary z(C and 4€R' with h(2,rfi>O we find Ao (2, tD : h (2, 4)r Ao (z', qo), where zo:h(r,q)-'oz, qo:h(z,ri-qq (see 1.2 arrd 2.7), and therefore lA'(z,til< Csh(z,q)P, z(C, qQR. Thus we have llAo(2, D)ull1,§-& - I *G, q)%-zplgAo(2, D)ulz d4 = c I xQ,q)z'-zuh(z,41zu19ul2dn = Cllull\,". 2.3. Definition. The operator A(z,D) is said to be parametrically quasielliptic if it satisfies the condition (QE) Ao(2,4) * 0, z(Z : Z(ar,ruor), q(R, h(2,4) = 0, or, equiualently (cf.l31), if lAo(z,q)l > crh(2,4)u, z(2, r1(R, with some constant co>O. Let us now assume that the condition (QE) is fulfilled. Consider Ao(2,(,r) as polynomial in the complex variable z. Then there are functions ro:q(2, O, k:1,...,mo, corrtituous in Z(R'-L, such that for each fixed (2,€)€ZxR"-t they are the roots of the polynomial Ao(2,(,'e), Ao(r, (,r1,(2, O) : 0, k : l, ..., mn. 2.4. Lemma. The roots rx(z,t), k:1,...,mn, are (qs,q')-homogeneous of degree qn. Indeed, we have Ao(r, (, ),-'t"rn(Lqoz, Lc'E)):0 for every )">0, aod the function )v+)-c"xo17aoz,)dt) from R* into C is continuous. 2.5. In what follows we shall mostly consider operators A(2, D) which satisfy a somewhat stronger condition: 244 Vrrrro T. PuRuoNEN condition (QED). The operator a(2, D) is parametrically quasi-eiliptic of determined type 24, l=2't<mo, that is, it satisfies (QE) and, moreouer, the condition (D): (D) For euery z€Z and euery ((N-L with h(2, O>O the polynomial A'(z,t,r) in the complex aariable r has exactly x, roots r!(z,t), j:1,...,x, with positioe imaginary part, Imr!(2,€)=0. Remark. When n>2, (QE) implies (D) and hence (eED). 2.6. Theorem. suppose that condition (eE) is satisfied, and ret s>trt. Gitsen Q=0, there is a constant C>0 such that the a priori estimate (1) llutl,,"= CllAo(z,D)ull,,"_r, ueH"(R), is ualid for euery z€Z with (r)=5. Furthermore,for eaery ze Z\{0}, the operator Ao(2, D)($(n:(n\; H:-t'(R")) is an isomorphism (for the locally conaex structures). Proof. To prove (1), pick u€H'(R\. By virtue of (eE) we then have, if (z)=0, Hence lFul = c;Lh(2,7)-ulgAo(2, D)ul. llull?," = C I X(2, q)r"h(2, r1-zulFAo(2, D)ulz dr1. Since here obviously h(r,ri > CK(z,q) with C:C(e), we obtain llulll,"= C { XQ, ry)%-zrlgAo(2, D)ulz dq = CllAo(2, D)ulll,"_r. It is clear that Ao(z,D) is now an isomorphism of H:(R") onto H'-p(R\ provided that it is surjective. This, however, is easy to see. Indeed, if fEnj-u1p"1, then and u. : FnAo(z,ri-rgf €H:(R) Ao (2, D)u : 9, Ao(2, 4) 9u : f . Boundary value problems 3. The case of principal parts 3.1. Let x be a positive integer, and let Bl(z,D),...,82(r,D) be x operators defined by 4G, D) : Z bil,ozkDo, j : 1, ..., %, kqo+<q,q):il. where the coefficients bi6 zracomplex constants,the parameter z€c, and pr>0. on parametrically quasi-elliptic boundary problems 251 what we now have to prove is therefore the estimate (B); this will be done in the rest of this section. 3.5.2. lf u(H'(R"a) arrd Ao(2, D)u:O, then g : 4Ao(2, D)u : Ao(2, (, D)$*u. Hence, for (almost) all <€R"-1, (4dG,) is an exponential solution of the equation (4) in 3.3, so that it belongs to 9(R*) (cf. [1]) and, furthermore, to fr(z,E). Accordingly, Lemma 3.3.8 implies that (g,u)((, D : 2 co(2, O No(2, (, t), k:t where now (see the proof of Lemma 3.3.6) co(2, O : yoBfl(2, t, D)(\u)((, t) : yofi"Bfl(2, D)u' 3.5.3. Choose an integer v such that y>§ and t>slq,*112, and write again E:En. We have (4) llall?,",+ =llEulll,"- [ *@,4)2"l9Eul2d4. }Jere FEu:4E4u, so that (5) fiEu: 2 ,oe, C)(glENk)(2, (, t))(r). k:1 Therefore it suffces to consider the integral (6) I *@, r)ulc,,(2, O(gt(EN)(2, (, t))(r)12 dn - [ uk, .t)2'1c1,(2, O'P(l l(r,1nNo11z, C, t))(x)|'z d4dC + [ lcr,Q, or([ $)*l(q,1zwo11z, 1, t))('»|'z dr)dc : It*Iz. 3.5.4. Consider next the function (7) No(,,€,o:* [ffirtfi,*re, f< where .l-, is a rectifiable Jordan curve encircling the roots zrt(z,E), .i:1, "',x' There are R>0 and ä>0 such that, for i:1,...,x, lrf(zo, (o)l = A and llmrt Qo, (o)l = ä for all zo<Z and all (0(R'-1 with h(zo, (0):1' Hence, with these z0 and to' 252 Vrxro T. PunvroNEN the curve r, in (7) can always be deformed into the same Jordan curve, say, into .l-*,0 consisting of the paths l(l:lR,Im(>ä and Im C:ö, l[=n From Lemma 2.4 it follows that for arbitrary z(Z and (<R"-, with h(2,()>0 we can take fs:h(2, t)o"l^,0. The function Lo, (8) Lo(r, 1, O: y+t': cn'() , h(2, o = o, (€c, A \t;9t l) is (q0,4)-homogeneous of degree -ilr,-7, (see (l) of 3.3 and Lemma 3.3.9), so that, in particular, Lo(r, t, C): h(2, t)-Po-c"L*(zo, (o, (o) when (2, t,():h(z,11teud(zo,(o,(\. From this and the factthat, for a suitable constant C>0, lLo(r', (o, (o)l = C for all zo,(o with h(zo,1o1:1 and all (06.1-*,u, we see that (9) lLok, 4 gl = Ch(2, t)-r"-r" for all z, ( with h(2, O>0 and all (qf r:1(2,1)o"f ^.u. By virtue of (7), (8) and (9) we thus obtain lNo(r, 1, t)l = Ch(z, O-Pu-q"l(f )maxleit(1, where /(l-,) is the length of i-6. Since and l(r e) =. (2+n) Ah1z, 17e" le"(l = exp (- h (2, 11e" öt), we finally have (10) lnf- (r, €, t)l = Ch(2, O-pr eXp (-h(2, gr" öt) for all z, ( with h(r, O=0, and for all t>0. 3.5.5. We return now to (6).First, in the integral Ir, we have I Wa(EN)(r, (, r))(r)1, a' : f l(Er,{o)(r, (, t)lz dt = [r*(v* ,;ä:]w,'ri rrr- (,, €,t)tzdt (see 1.6), where, bV (10), i tlr{o(r, (. t)1, dt = eh(r, 1-Ztt*-en. Noting that h(r, () - K(2, () when (r)=- g, we therefore obtain (11) hs C I K(r, C)r'lyrg.$k, D)ulrK (2, O-zt k-q^ d( = C llyoBf,(r, D)ull\,s_ t x_ ,ntz,T . On parametrically quasi-elliptic boundary problems 253 3.5.6. To treat the integral 1, let us rewrite it in the form Ir: I lc1,Q, Ol2 Ho!2, t) d(, where HoQ, E) : I (r) l(s,(EN)(2, (, t))(r)1'zdr. 3.5.7. First of all, we shall verify the continuity of He(2, O when h(z,O>0. To do this, let e>0 be given, and let zo€Z and (o€R'-' such that h(zo, $)>0. Restrict the consideration to a neighbourhood of (zo, (o), and notice that for all (2, () in this neighbourhood the curve .l-g in (7) can be deformed into the same rectifiable Jordan curve, say, into I with the property r : min {Im (l(e f} = O (cf. 3.5.a). If Zo denotes again the function given by (8), then there is ä">0 such that, for afl (€f , lLo(2, t, O - Loko, h, 0l = e whenever lz-zol*l(-(ol=ä". fn view of 1.6, if O=fi=v, we thus have for ,>0 (cf. 3.5.4) l(Df No)Q, €, t)-(Dl N)(ro, €o, t)l = * { lLo(2, €, O-L*(zo, €o, OllUllettelfi(l = C(f,t)ee-'t and then for / <0 l(Df EN)(2,4, t)-(Df ENk)@o, h, t)l : l(E@) Dl N)(2, (, t)-(E@ Df No)Qo, g, t)l y*1 = z le j)pl lljl l(Df N)(2, €, - jt) - (Df N)(zo, h, - jt)l i:1 = C(f,v)es-'ltl' hence, for all r€R, l(Df EN)(2, C, t)-(Df ENk)(ro, 4r, t)l < Ct,e-'ttt if lz-zol*l(-€ol=ö". This yields (12) lra((fi,(ENp)(2, €, t))k)-(F,(EN*)(zo, (0, r)(")l = n, I le - t" ((Df E Nk) Q, t, t) - (Dl E Np) (z o, $, t))l d t = Ce, provided that lz - zrl*l( - (ol =ä,. Next, a small computation shows that (13) lHoQ, O-HoQo, *)l : l/{rX"{tt F,(ENo)e, (, t))(r)|,-l(e,(ENy)(zo, g, t))(r)l)dtl = 2ab+b2, where o : ([ k)"1(r,1ntto11zo, i,o, t))(r)12 dx)rtz, t : ([ (")*l(q,1rNo11z, (, t))(i-(4(EN)(zo, 1o, t1)141', a)'t'. With O=B<y we have (see 1.6) Fu (s,(z x ) (Z o, Eo, r) (d I : l(qt@@, D f Nk) e o, ( o, t)) (r)l if v+l o I : "rl{ e-tt"(Df N)(zo,to,t)dt+ Z?ilPli _fe-t'"(Df Np)ko.h,-iDdtl, and since it follows from (9) that l(Df N)(zo, h, t)l = Ch(zo, (o)- Puqne-", 2s4 Vrrco T. PunruoNEN we obtain (cf. 3.5.4) lro (,q,(E lrt) (ro, (0, t)) (r) I = C h (ro, h)- FuQn . Hence we get az= c {# [tr+lrl,) (,E@r,{o)(ro, h,t))(r)lzdr =C(zr,Cr)1ffiu, = C(zr, €o), since v>sf qn+112. From (12) we derive in the same way that bz = Ce. Therefore, it flnally follows from (13) that whenever lz - zol *lC - (ol = ä, . 3.5.8. There exists a constant C>0 such that (14) Ho(zo, C\ = C, z0€2, (o€.R"-r, h(ro, (o) : 1. Let then ze Z and €<R"-' with h(r, €)=0, let 1>0, and consider Ho(Aooz, )a' () : [ (r)r'l@r(EN)(Äuoz, ),q' (, r))(r)1, dr. On parametrically quasi-elliptic boundary problems 255 In the case ,>0 we obtain (see 3.5.4) No(),aoz, )ts' t, t) : ),- PuNt (2, (, [e"t), so that, for all r€X, (EN1)(Lqoz, ).s' (, t): l-pu(ENr)(2, (, )'q"t). Consequently, (qt@N)Q,soz, ),q't, r))1r; : 1-a-e"(fi(EN)(2, E, t))(1,-e"r}. Hence we see that Ho(),aoz,Ls',O: A-zpu-zq" r (r),'l(41nNo)Q, { t))().-s"x)12 dr: )2s-z1ru-rnHo(2, €), from which it follows, by (14), that Ho@, €) = Ch(2, ()2s-2tlk-s^, z€2, €€R-t, h(2, O > 0. Now we return to the integral I, and reach the conclusion that Ir: I lc1,Q, Ol2 Ho(z, C) dC = C I h(2, 02"-zuu-a"lg*(2, Ol2 d( = c I xQ, Ozs-z,k-s^lyo7,$Q, D)ulz d(, whence (15) Iz= Cllyofl,Q, D)ullf;,"-ru-qsz,y. 3.5.9. By combining (4), (5), (6), (ll), and (15) we flnally obtain llull|,",* = C i I K(2, O2"lc1,(2, ()(g,(ENu)(2, (, t))(,4lz d4 = C 2 fiy, B?,(2, D) ullf, , " - uu-q^12.y. &:1 This completes the proof of the estimate (B). 3.6. Proof of the latter part of Theorem 3.4. To show that Po(2, D) is an isomorphism of H)(R'*) onto /{)(Ri*R"-'), for every z(Z\{0}, we shall, in fact, construct its inverse operator. In what follows, let E:En,v>s. 3.6.1. Fix z(Z\{0} and define Krf : R* firAo(2, ni-'gEf, I< H;- P (R+)' Then Ko is an operator mapping H:-t'(R"+) into II,"(R!). Indeed, Ef€H:-t"(R') and llfinAo(z,rD-'gEfll\,"= C I K(z,q)'"lAo(z,q)-r9Ef l2 d4 = C I Xe, r1r" h(2, t»-zp lg Ef lz d4 = cllEfllT,"_p, so that llKo fll\. ". + = c ll Ef ll2, "-, 5 c ll f ll2, " -,, * ; 2s6 Vrxro T. PunrvloNEN this implies, furthermore, that Ko € s (H :- t (n!) ; rri(n!)). 3.6.2. For i:1,..., ?4 we set Ki g : %a(Nifr*g), g(Hi-u1-t"t2(r?4-1). First, one easily checks (see 3.5.5) that Krg(Lr(ni), at least. Since now and so EKig - 4(@U)g,s) we obtain fiEKig: (%EN)fi*s' (1) I *Q, r)2"l3EKislL d4 - I *@, 0^ ([ lF,ENip dr)ls"sl, d(+ [ ([ k)" l.fi,EN1l, dr)ls.sl, d€ : It*Iz. Here we have (see 3.5.5) tr= c I xQ, c)'"h(r, €)-2pt-q"l4sl, it( = cllSll?,"-p:-q^iz,t and (see 3.5.8) Ir: t H,k, Olg-sl'dE = c I lr@, ozs-zt'i-snlgF sl'd€ = cllsll1,,-pt-q^rz,t. It therefore follows from (1) that and hence EKlg€H)(R)' Ki8: R+EKjg(H:(R"+); moreover, 11 xt sll,, ", * = C ll slll, " - u - ent 2, y. Consequently, K i e e (n ;- u r - c"tz (Rn - L)' H : (R^*)). 3.6.3. Let us now set KoF : K"f + ; Ki(s,-yo4Q, D)Kof), F : (f , gt, ..., g,)(tr:(R+, Ro-r). j:1 Then K0 is an operator on af,)(Rl,R'-t), valued in Hi(R"), byvirtue of 3.6.1 and 3.6.2, and furthermore llKoFll,,", * = cllfll","-p,+*c å-(lr,ll",,-ri-q,rz,y*llTo4k, D)Ksfll.,s-p1-q^12,y), j:r where (see Lemmas 1.9 and 3.2) llt o fi @, D) Ko f ll,, " - u: - hntz, y = c ll I ll,, " - u. * . Thus we have Ko < g (/f : (R\, R - L) ; H ) (R"*)). On parametrically quasi-elliptic boundary problems 257 3.6.4. The operator K0 has the property Po(2, D)KoF : F, F : (f, Sr, ..., g*)€.#)(Ro*,.R'-1). To see this, let F:(f,gr, ...,g)Caf;gf*,R'-') and set u:KoF(Hi(Ri). Then we have Ao(2, D)u: Ao(2, D)Kof + 2 Oo@, D)K1(si-tofiQ, D)Kof), t:t where (note that Ao(z,D) and R* commute) Ao (2, D) Knf : Ao (2, D) R+ g4Ao (2, q)-L I Ef : R + fr,rAo Q, q) Ao (2, rD-' g Ef :f and (see 3.3.7) Ao(2, D)K/Ei-To4Q, D)Kof): Ao(2, D)g<(N jg,(s1-loB](2, o)Kofl) : ge((Ao(2, 4, D)N)g"(ci-To4Q, D)Krfi) -0, so that Ao(z,D)u:f. By (12) of 3.3, we obtain further I o 8fl.(2, O) K lg 1 - t o Bl (2, D) K o f) : 9e(0 o 4(2, (, D) N ) 3,(g i - y o4Q, D) Kofi) : ö ;r,(g i - T o B] (2, D) Ko f), whence yr*t Q, D)u: yoBlk, D)Krf + åa,o(s1-to4Q, D)Kof) : 8k, as required. If thus follows that the operator P0(2, D) is a continuous bijective linear map and therefore an isomorphism from H;(R"+) onto af,)(R"*, R'-t), with inverse K0. The continuity of Ko was, in fact, found also directly in 3.6.3. 4. The general case In this section we shall gereralize the results of Theorem 3.4 to cover the case of nonhomogeneous operators. 4.1. We shall need the following two lemmas. 2s8 Vu«o T. Punrr,roNEN 4.1.1. Lemma. Suppose s1>5r>5rr3Q. Giuen any e>O there is a constant C(e)>0 such that llull",*,* = e llall,,",, a * C(e) llrll,,*, *, u€H\(R\), rttr all z(C. Proof. From the well-known inequality (equal to the above with z:0) llall*,* = ellall",,*+C(e)llull",,+, u(H\(R\), we obtain llull",,,,*-llull*,++(z)"llzll*=ellull,,,a*c(e)llall",,**(z)"llull*. The assertion follows therefore from the inequality (r)", = e (z)t + c1e; (z;t, which in turn is a consequence of the elementary inequality oA=!orrr!60', a,b>0, p=1, p,: pl@-l); pp as a matter of fact, choose p : +, a : (pe)rto(z)"t\, b : (pe)-tto (z)ssro" ' sr-sa 4.1.2. Lemma. Let s>O and r>0. To euery e>O there is C(t)>O such that Z llDull",",* = e llull,,"*,, a *C(e)llull*, u(H"-"(R'+), (a,q)=r for all z€C. Proof. By Lemma l.l0 we have llD" ull ". ", * = C llull,, " * (n, q), + . If we now put k: max(a, a)la(N", (a, q) - r\ = r. it follows from Lemma 1.5 that ,,,?, =,llull " " * (n' q)' + 1 c llull " " * o' *' where, by virtue of Lemma 4.1.1, llull,,"*0, * = ellall,,"*,, + * C(e)llzll * . 4.2. ln the general case we have to consider operators of the form P (2, D) : Po (2, D) + Poo (2, D), where P0(2, D), the principal part of P(2, D), is defined as in 3.1, and Poo(2, D) On parametrically quasi-elliptic boundary problems 259 is given 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)) with troo(z,D)- Z akdzkDa, kqo*(a'q)=P 4o (', D) : oro*l*-o,b ivozk D; here the coefficients a1,n and bi*a arc complex constants, of course. 4.3. Theorem. Suppose that Po(z,D) satisfies Conditions (/ED) and (CC), and let s€5. Then there is Q-0 such that, for some constant C>0, llull,, ", + = C lllP (2, D)ulll,, ", u ( H" (R*), for all zQZ,(z)>-7. Proof.In what follows let u be any function in ä"(R|) and let z€Z\{0}. 4.3.1. According to Theorem 3.4 we first have llull,.".+ = ClllPo(2, D)ulll,,", which yields (1) llull,,,, + = C r(lllP (2, D) ulll,, " * llAoo (2, D)ull ", "- p, + + 2 Wrry'(2, D)ull,,"-u,-q.tz,y). j=L 4.3.2. Given any e>0 we obtain, by virtue of Lemma 4.1.2, ll Aoo (2, D) ull,, " _ r, * = c, o?^l rlo r,, r, i _ r, oll D u 11,,, _,,, + = crt oZo(z)keollull,,"-rqo, * *G(e) o?^"(r)o,,llul*. Since ) (z)k e" llull * = c (l + (zlu - ao) (z) -" ll u 11,,,, *, k<mn we have, by Lemma 1.7, (2) llAoo(2, D)ull","-r,* =(coe+cu(e)(1+(z;"-n)(z)-)llull,,",*. Fix now e>0 satisfying Crr={, and choose go>0 such that c,c)t#_ = ä. 260 Vrmro T. PunuoNEN Then it follows from (2) that (3) llAoo(z,D)ull,,"-r,. =)rlull",",* for all z€2, (z)=po. 4.3.3. Next, in view of Lemma 1.9, we find that lly o 4o (r, D) ull,,, - u, - on t z, t 1 " u oFo,l rlo r,, o, f, _ oo oll D' ull,, " - u,, * . Hence, given any e>0, we obtain again from Lemma 4.1.2 lll o S @, D) ull,, " - u - q,rz, t = Cee ) (z)kullull,,"-r,qo,a-fC7@) ) (z)ka,l1u11*. kqo=pl kqo-pJ Since Z <z>ks,llull+ = c(l *(z)ut11r7-"llull,,",* kqo<ili (obviously, this inequation could be strengthened), we have, as above, (4) lltrfio (r, D)ull,,"_ u,_ n^tz,t s (C, e + Cs (e) (t + (zyu1qr1-") llull,,,, * . If we now take e>0 so small that Crr= rfi; and choose Q;>0 such that c^G\ t+atjt = I _u \_/ Ai _ gxCr, then it follows from (4) that (5) llyoBlo(2, D)ull,,"-r,-r^,r,, = irrLlull,,", * for all z(2, (r)=pi. 4.3.4. To completetheproof,itisnow enoughtodefine q:max{q,, Qr, ..., Q,}. Indeed, combining (1), (3), and (5), we have Ilull,,", * = CllllP(2, D)ulll,,"++llull,,", * * ,JrLO)lull,,", * and thus llull,,".* = ClllP(2, D)ulll,," for all z(2, (z)> p. 4.4. Theorem. If Po(z,D) satisfies Conditions (QED) and (CC), and if s(,S, then there is p>0 such that, for euery z€Z with (r)=_0, the operator P(2, D) is an isomorphism of H;(R"+) onto Jf,i(R"*, R,-1). Proof. We shall show that P(2, D) can be represented as the product of po(2, D) and a tri(R"+,,R'-1)-automorphism.