Pseudo-differential operators of multiple symbol and the Calderon-Vaillancourt theorem. (Received Nov. 12, 1973)

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1 J. Math. Soc. Japan Vol. 27, No. 1, 1975 Pseudo-differential operators of multiple symbol and the Calderon-Vaillancourt theorem By Hitoshi KUMANO-GO (Received Nov. 12, 1973) N x', Introduction. In the present paper we shall define a class S ~v; my ~,p,s of multiple symbol as an extension of the class SP,s ' of double symbol in our previous paper [6], where my = (ml,.., mv) and m == (ms, ml,.., mv) are real vectors and 1=0, 1, v. The multiple symbol has the form p(x, v, zv) = p(x, e1,, v, xv) and the associated pseudo-differential operator P= p(x, D 3v, X v) is defined as the map P : B -->2 by using oscillatory integrals developed in Kumano-go [7] and Kumano-go-Taniguchi [8], where - denotes the set of C -functions with bounded derivatives of any order in Rn. Then, the (single) symbol 6(P)(x, ) is given by a(p)(x, ) = We shall give a theorem which represents a(p)(x, ) by the oscillatory integral of the multiple symbol p(x,v, xv) and the asymptotic expansion formula for a(p)(x, ) will be given. As an application we shall prove the Calderon-Vaillancourt theorem in [3] (see also [11]) for the L2-continuity of pseudo-differential operators of class Sa,s,s (0 <_ o <1) only by symbol calculus. Another application is found in Tsutsumi [10], where our theorem is used to construct the fundamental solution U(t) for a degenerate parabolic pseudodifferential operator in the class S with a parameter t. We believe that our theorem will be useful when we try to solve operatorvalued integral equations with pseudo-differential operators as their kernels. 1. Oscillatory integrals. DEFINITION 1.1. We say that a C -f unction p(r~, y) in Rij belongs to a class LA z for -0 < m < 00, o < 1 and a sequence z ; 0 <_ ~1 < v2 <... < ~~ when for any multi-index a, $ we have (1.1) I P? (2, y) I Ca,p<7>mCy> ~~I for a constant Ca,p and set A = U U where p= a~ D~ p, Dye = ia/ay;, =1,, n, <y>=/1+j[, <>=/1+I2 ~(cf. [7], [8])

2 N 114 H. KUMANO-GO DEFINITION 1.2. For a p(~, y) A T we define the oscillatory integral by (1.2) OS[ep(j, y)] = OS- ep(ij, y)dydry = lim $$e- e-0 y)p('1, y)dydij, where th2 = (2,r)-ndi, y i = yi 'i+... +yn~n and X E(i, y) = X(, ey), 0 < < 1, for a X E s (the class of rapidly decreasing functions) in R such that x(0) =1 (cf. [4] for the general form). Then, the following propositions are found in [7], [8]. PROPOSITION 1.1. For a p Es, T we choose positive integers 1, 1' such that -21(1-a)+m < -n, -21'+v21 < -n. Then, we can write OS[e'2p(7j, y)] as (1.3) OS[ep] = $$ei?<y> _y -21'<D~>21' {<~>-21<Dy>21p(~, y)} dyd~, and for l = 2(l+l') and a constant C we have OS[e-2p] I <_ C I p I io', where p I is a semi-norm defined by I p I = max inf {Ca,~ of (1.1)}. PROPOSITION 1.2. Let {p}0<<1 be a bounded set of A z in the sense: sup { I pc I } < M10, l = 0, 1, 2,.., for constants M10. - Suppose that there exists a p0 E s, such that p~(7, y) -~ p0(aj, y) as E - O E uniformly on any compact set of Rrny. Then we have liro3[e-z~''2p6] = OS[e2po]. PROPOSITION 1.3. For p E~ we have OS[e-iy"p] = OS[e-vY'l2D p] and OS[e2r1~p] = OS[e-iy 2Dy p] 2. Multiple symbols and theorems. Let AO is a C~ -f unction in R such that for constants Ao and Aa (2.1) 1 _<- A() AO< > and I A() I A0A(e)i-'"~ (cf. [5]). Let (x,.)=(x, xi,, xv) and v =(',, v) be a (v+1)-tuple of points x, x1,, xv E Rx and a v-tuple of ei,, v E R respectively. By (j3o, $1) = (Qo, 431,..., Qv) and &=(ai,, av) we denote a (v+1)-tuple of multi-indices 430, j3i,.., w in Rn and a v-tuple of ai,, av of Rn respectively. DEFINITION 2.1. i) For 0 o p < 1, o <1, real vectors my = ml,.., mv) and m = (ms,.., m') (m; >_ 0, j =0,..,) we say that a C -functions p(x, #v, xv) 0, 1, x, 1 v x v) in R(2v+i)n is a multiple symbol of class my;'mv when =p(x for any & and (j3, j) we have for a constant C=C(&, 430, Qv) i (2.2) I pv) I < CA( i)mo+81 r~ I it 2( ')mipi +o Ii where (ao v cao,..l N N av' v = aav D~8L)~v (~v+1 = 0) p = a i... a Dg... Dxvp. When m = 0 we

3 N Pseudo-differential operators of multiple symbol 115 N N denote Sa simply by Sa,p,s (cf. S~; of [1], [2]). ii) The associated pseudo-differential operator P= p(x, Dxy, Xv) = p(x, Dx1i X1, -, Dxv, X v) with symbol p(x,v, xv) is defined by - Y 1,...,v, x+yy)a(x+yv)dyv " d ~ v (2.3) fo r Pu(x) u =0s- E~ ep(x,1, x+y where dyed v = dy1de1... dyed v, yv v = y1. E yv. Ev, y1= y1,..., yu =yl+... +yv. In case v =1 we often write P= p(x, Dx, X') as in [6]. REMARK 1. For p S,p;s v we define semi-norms p i v 'n, 1, l'=0, 1, -, by (2.4) p I i,l'u' v' = j max inf {C = C(&, j3, j ) of (2.2)}. is I_1, 11e.1 i _l mu;'~v Then makes a Frechet space. 2. For Pj = pj(x, Dx, X') =1,..., v, set p(x,v, xv) =p1(x,1, x1) pv(xv-1,v, xv). Then p(x,v, xv) Sa,p,a for my = (m1i --, mv) and we have N (2.5) P1... Pvu(x) = p(x, D xv, X v)u(x). In fact, take X E( v, yv) = X(~ 1, ~y1), X(sr, ~yv) for X(, x) E S in R,x such that X(0,0)=1, and set pj,e(x, e, x')=pj(x,, x')x(e, E(x'-x)), =1,..., v. Then by the change of variables x+91= z1,, x+yv = zy in (2.3) we have (2.6) p(x, Dxv, xv)u(x)=lim $... H (ej-l-z'j pj,e(z~-1, ~~ zj))u(zv)dz"d " and have (z =x), $$ei_1_pj,(zi_1, (zjzj) ~ s1, z')v(zj)dz~ci -->(pj(x, Dx, X')v)(z'-1) in S for v E S. So we have (2.5). 3. For P= p(x, Dx) E S,s we have (2.7) p(x, ) = ep(e), since e-ix P(eix ) = e-ix.e0 [e-ip(x, )e'] =01[e-iycp(x, ~1)]=0.,_$e-ip(x, +~)dyd~=p(x, S) The following lemma is fundamental in the present paper. LEMMA A, For a symbol p(x, r, xv) E Sr, define a single symbol q0(x,, x'), I O I1, by

4 116 H. KUMANO-GO ~ -iyv-.iw 1 1 (2.8) qe(x, E, x') =Os- e l -1 P(x, E+0, x+y,, -e~v-1' x+yy-1, e, x'), dyv-1 yv-1 Then there exists a constant C > 0, depending on M= independent of v, such that v-1 ~=1 (I m; + I m; I )+mo but (2.9) qe(x,, x') c Cv+1 P iiv,mv )~()mv+mu (101<1), v... +my, my = mo+m-r i... +my, an m where = m1+ ' ' (2.10) l = 2[n/2+1], l' = 2[(n+M)/(2(1-o))+1]. Proof will be given in 3. THEOREM 2.1. For P= p(x, D Tv, Xv) E Sa,p;s v, set.11) q(x,, x') (=q1(x, e, x') of (2.8)) (2 =O - e-iy"v-ll.p(x +;rl x+y e+1, v-x+9v-1 xl d v-1dv-1 Then we have (2.12) P=q(X, Dx, X') and q(x, S, x') E S,p dmj Furthermore, for any 1, l' there exists a constant C such that (mv;~v) 2.13 ~ C'v+1 lpl where 10=1+2[n/2+11, (2.14) v-1 1(=1'+2[(n+ (I m' + I in~ )+mo+,ol+o1')/(2(1-o))+1]. 7=1 PROOF. As we got (2.6), by the change of variables x'=x+yv=x+yv-1+yv, =v, ~'=e-e,,..., v-1, we have Pu(x) = lim $$e1_ C xx')'s~~~e-tidy-l.w-lx~( x+y',..., x+yv-1', f-x-yv_1)p(x,..., x+yv_1,, x,)dyv_1,s v-llu(x')dx'de, and using Proposition 1.2 we get P=q(X, Dx, X'), q`p'p)(x, c, E, x') Since = 0 - e-2yy-i.;w-1 as D~D~ P(x +~1.., x+yv-1 x')dyv-1 ' we have by Lemma A (2.13 and q E S s Q. E. D. THEOREM 2.2. Let P; = p;(x, Dx. X') e j =1, "', v. Then, Q = Pi -' P E S~,p,s and for any 1, l' there exists a constant C=C(1, l') such that

5 Pseudo-differential operators of multiple symbol 117 v (2.15) 16(Q) I i i~' _< Cv+1 I p I io off', where 10, to is defined by (2.14). Proof is clear from Theorem 2.1. THEOREM 2.3 (Calderon-Vaillanc.ourt [3]). Let P= p(x, Dx, X') E S,p,o (0 < o < 1). Then there exists a constant C such that (2.16) IIPuIIL2~CIPll1,l211uIIL2 for ue8, where l1= 2[n/2+1], 12 = 2[n/(2(1-o))+1]. PROOF. I) For PO = p0(x, Dx, X') E- S,p,s we first assume that p0(x,, x') = 0 for I x I + I I + I x' I >_ R (>0). Then setting K0(x, x') = $e1')p0(x, cx-x'~, x')d we have P0u(x) _ $K0(x, x') u(x') dx', and I K0 (x, x') I CR I P I o;o for CR = the volume of { I I R}. So, noting K0(x, x') = 0 for I x I + I x' I >_ R, we have (2.17) IIPOuIIL2 CRI P ~,ollulil2 for u E= s. II) Consider Qv = P*P. P*P for v =2', 1=1, 2, Then we have II P I v II Qv. Note that a(qv)(x, e, x') = 0 for I x I + I I + I x' I? R if p(x,, x') = 0 for I xl + I + I x' I >-_ R, and that Q(P*)(x,, x') = p(x', e, x). Then we get by (2.17) and Theorem 2.2 IIPIIv~ Qvl SC o(q) CC1(ipI12) and by letting v-->oo we get (2.16). For the general P= p(x, Dx, X') E S3 we have Pu = lira P~u in L2 for u E S, if we set a(pe)(x,, x') = X (x, s, ax') p(x, e, x') for X (x,, x') E Co in { I x l + I S I + I x' < l} such that X(0) =1. Hence, noting a(pe) = 0 for x + I + I x' I >_ ~-1 we get (2.16) for the general case. Q. E. D. N wr 1 THEOREM 2.4. For P= p(x, D;, X1) E= Sa,p;s v set (2.18) pay-1(x, x')-p11':0)a1++10) (x, e,..., x,, x'), v 1 1 which belongs to Sa,p+s v'-gyp-a' t"v-1', where (2.19) av-1 = (a1,..., av-1), Then for any N there exists rn(x,, x') v-j I av-1 I = I a1 I I av-' such that (2.20) a(p)(x,, x') _ -1 1 pa~-1(x,, x') a"~-1 <N IT (ai I... ay-j!) 3-1 +rn(x,, x') (cf, the expansion form in [9]).

6 ., 118 H. KUMANO-10 PROOF. We take Taylor's expansion for p(x, E+'i1,.., x+yv-1, E, x') with respect to ~. Then by Proposition 1.3 and Lemma A we get easily (2.20). 3. Proof of Lemma A. For n = [n/2+1] we first write by integration by parts (1+(_ f~j)n ('( ~8'lj)2npv) q0(x, e, x')=os_ (3.1) e-iyu-1.rv-1v-1 X II (1+2(+O ~j)2n o yj 2n )-1p(x, e+o ',..., x+yv-1,, x')dy~-ld~v-1 Then by the change of variables: (y',, y~-1)--,(9',., yv-1) such that y1= y',, yv-1=y1+ +yv-1, and by integration by parts we have for 0_<kj=k, (,2i, ~;+1) l'/2 (3.2) q8(x,, x') = i3-j+1(2kj) where v-1 x (-d y j)k'r0(x, e, x'; v-1, yv-1)a1w-l ~v-1 (~v = 0), (3.3) re(x,, x'; v-1, vv-1) ^ (1+(_drj)n (~( Le~j)2n ~~.) Noting that we have for a constant ~=1 v-1 _ x (1+2(e+8~i)2n I yj_yj-1 2foy1p(x, ±O',..., x+yv-1, e, x') (9 =0). $(1+A(e+oi)2foo y~yj-yj-1 12n )-1dyj C12(E+e~j) no, C=C(1, l') (3.4) I qe(x,, x') _< Cy+1 I P 1 p;'j~~()my+my' v-1 x (I ~j-,~j+1j-2kj~(~b~j)rnj-no((+8~1)+~(+e~~+1))mj~+2kji))v-1 (i=0). We have from (2.1) (3.5) A(e +eij'+1)/2 2(e+~9~j) < 2A(e+e1) and for ~1j-,i'+11 c c 2(~ f-6i ') (3.6) A(E+e~j) C2I for I I

7 H Pseudo-differential operators of multiple symbol 119 for a small constant c0 > 0 and a large constant C2 > 0. Set Q,,1 = {~; j. I7 2j-~p 7 j+1 I C c02(+6,j+1)s} ~I s Qj,2= {~j ;. c02(+oij)+1)s I j- j+1 I co2(e+6~j+1)} and Q;,3 = {~j ; I vi- j+1 > c02(e+6ij+1)}, and set k; =0 for ij j E Q,,l and = l'/2 for ry j E Q;,2 U 9j,3 Then we can prove by induction (3.7) Aj0= S ~( +Th?')m0H {I 1+1I-2kjA(+6~j)mj x (A(+67)+~( +6~1+1))mj'+2/ } ~~j < C3p+1~(+6,~j0+1)mj~+mjQ (1o =1, 2,..., y-1) for a large constant C3 independent to prove that $ { I jo_ 3o+1 I -2kj0~( +6~1 o)m1 o+;m ja_1-ns of j0 and v. To get (3.7) we have only X(( +6~jo)+A(e+6~'o+l))m'j0+2kjoS}t~jo<C4~\+6o+l)mjo+m'j0, which can be done by dividing the integrand into three parts Q,1, Q,0,2 and Q0,3 and using (3.5) and (3.6). Here we use the condition (2.10) to obtain A(+6j1 o)mio+m 1o_1-nS < C42(+6'! jo+l)(mjo)++m' j0-1, -l'(l-o)+n+(m; 0)+ < m j0, jo =1,... v-1, (m;0)+ = max (0, m20). Finally setting j0 = v-1 in (3.7) we get (2.9) from (3.4). Q. E. D. References [1 ] R. Beals and C. Fefferman, Spatially inhomogeneous pseudodifferential operators, I, Comm. Pure Appl. Math., 27 (1974), [2] R. Beals, Spatially inhomogeneous pseudodifferential operators, II, Comm. Pure Appl. Math., 27 (1974), [3] A. P. Calderon and R. Vaillancourt, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A., 69 (1972), [4] L. Hormander, Fourier integral operators, I, Acta Math., 127 (1971), [5] H. Kumano-go, Pseudo-differential operators and the uniqueness of the Cauchy problem, Comm. Pure Appl. Math.,, 22 (1969), [6] H. Kumano-go, Algebras of pseudo-differential operators, J. Fac. Sci. Univ. Tokyo Sect. IA, 17 (1970), [7] H. Kumano-go, Oscillatory integrals of symbols of pseudo-differential operators and the local solvability theorem of Nirenberg and Treves, Katata Simposium on Partial Differential Equation, 1972, [8] H. Kumano-go and K. Taniguchi, Oscillatory integrals of symbols of pseudo. differential operators on Rn and operators of Fredholm type, Proc. Japan Acad., 49 (1973), [9] M. Nagase and K. Shinkai, Complex powers of non-elliptic operators, Proc. Japan Acad., (1970),

8 120 [10] [11] H. KUMANO-GO C. Tsutsumi, The fundamental solution for a degenerate parabolic pseudo. differential operator, Proc. Japan Acad., 49 (1974), K. Watanabe, On the boundedness of pseudo-differential operators of type p, b with 0_<p=8<l, Tohoku Math. J., 25 (1973), Hitoshi KUMANO-GO Department of Mathematics Faculty of Science Osaka University Toyonaka, Osaka Japan