Aug 2, 2013 - Massimo CICOGNANI and Luisa ZANGHIRATI. I. Introduction. In this paper we cosider a class of analytic operators with multiple non-involutive ...
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J. Math. Kyoto Univ. (JMKYAZ) 33-3 (1993) 633-646
Propagation of analytic and Gevrey singularities for operators with non-involutive characteristics By Massimo CICOGNANI and Luisa ZANGHIRATI
I. Introduction
In this paper we cosider a class of analytic operators with multiple non-involutive characteristics and study the propagation of analytic and Gevrey singularities. To state precisely our result, we begin by recalling that f E C (X), X an open set in Rn, is said to be of Gevrey class G', 1 1 . If l< d then we have Go Y(X)OE d
-
d
d
d
(d )
d
d
( d )
(d )
(d )
(d
(d Y
Go i "(X). (d
)
The d-wave front set WFd(f) of f E Go "(X ), d >1, is defined as follows: for a fixed (xo, eo)E T*(X )\0, we say that (xo, eo)qWFd(f) if there exist Go (X) with 0(x)=1 in a neighborhood of xo, and positive constants C, E such that the Fourier transform (çbf)^ of Of satisfies: ( d )
d
(1.2)
1(sbf)"( ) 1< c exp(—Elel("fl
for all e in a conic neighborhood of eo. We also define the 1-wave front set (analytic wave front set) WFI(f) of f EGo d r(X ), d >1: let us consider a sequence {q5,1c Go (X ) , q5,(x)=1 in a neighborhood of xo, such that there exists a constant C and for every 6 >0 a consotant Cc satisfying (
Communicated by Prof. N. Iwasaki, November 5, 1991
d
634
Massimo Cicognani and Luisa Zanghirati ID a+flo , ( x )1‹ c e( colai e.ifil(g!)d
for every xE X, vEZ+, a, ,GEZ-En, v. Then we say that (xo, $0)qWF1(i) if there exists a constant A such that: J( f)A(E) 1 A (A)))z)(1 +1$1)
(1.3)
- 1
)
for a ll E in a conic neighborhood of $ 0 , vEZ+. If f G o "(X), d >1, the projection on X of WFdi(f ), for 1 d 1 d, is the di-singular support of f , th a t is the complement of the largest open set in X where f is of G ' class (see for example HOrmander [6] for this and other properties of d-wave front sets). We have also W Fd(f )cW Fdi(f )cW FI(f Let I ' be a conic open set in T * (X )\ 0 . We introduce the equivalence relation for f ,g Go "(X ) ( d )
d
( d )
f gWFd(f — g)n F=
and define for d>1 the space M d(F ) of the d-microdistributions in F to be the factor space Go Y(X)/ — . We also define mi(r) to be D ' ( X ) I . For d 1, the d-wave front set W Fd(u) o f a microdistribution u E M (T ) is a well defined conic closed subset of F. Let Ic R , X c R n be open sets, 0E/, and put Y=/ x X . Coordinates in T *( Y )\0 w ill b e d e n o t e d b y (t, x ; r, E). W e c o n s id e r a n a ly tic pseudodifferential operators P in Y of the form ( d
d
(1.4)
P (t, x, D t, D .)= (tD t)m + (2,(t, x, D x)(1D 1)m ' 3=1 -
and assume Q, is (1.5)
an analytic pseudodifferential operator of order 8j, 0< 81, P : md(r) md(r) is a well defined operator. Note that (1.5) means only that the principal symbol of P is O thus P is a model of an operator with non involutive characteristics. W e rem ark that near a point zE T *( Y )\O, away from the set f(0, x; 0, E)} the propagation of C , a n a ly tic and Gevrey singularities for an operator of this type is well known (see for example Duistermaat-Htirmander[4], Sato-Kawai-Kashiwara [12], Rodino-Zanghirati[11]). W e study P in a conic neighborhood ro in T *( Y )\0 of a point 2o=(0, xo; 0, $0) and define the four half -bicharacteristics at zo to be the sets: ,
(
(1.6) (1.7)
h
= f (t
XO;
0, $ 0 )E T*( Y )\O; (-1 )" t >01 ,
h=1, 2
Sk•={(0, Xe; r, eo)E T *( Y )\O; (-1) r > 0 } , k=1, 2 . h
m ,
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A nalytic an d Gevrey singularities
Our main result is: Theorem 1 . 1 . L et P satisfy (1.4), (1.5) and let u ellI d (r o ),1
d < là , with
z a W F d (P u ). A ssume that there exist h {1, 2) and k (1, 2} such that (LhU S h)n w F , ( 0 = . T h e n zoqW Fd(u).
The propagation of singularities for an operator (1.8)
(tDt)m + ± , x , D t , D x ) ( t D t ) ' 7z
m
with (1.9)
Order Q.1-0
assuming u regular in Lh U Sh for some h and k, has been studied by several authors in the D ' framework. We refer to Hanges [5], Ivrii [9] and Melrose [10] for the case m = 1, and to Bove -Lewis-Parenti [2] fo r m > 1. Their results are similar to Theorem 1.1 with WFd replaced by the C wave front set. We note that -
(1.10
(tat
-1 )(to f (x ))=1 )=(tat+1 )(a(t)o f (x ))
for all f E Go( d r (X ), hence we can not replace Lh U Sk by LIU L2 or by S1U S2 in Theorem 1.1. The problem of the propagation of the singularities for the operator (1.8) with (1.9), assuming u regular in Li U L2 or Si U S2, has been studied in [5] and [2] in the D ' framework. Recently Uchikoshi [13] has considered the same problem in the analytic framework for an operator of the form (1.8), with order Q., < j - 1 , in the case (Li U L2)C1 WFi(u)= 0 . In [13] the result of Theorem 1.1 with d =1 is not given. We will consider the extension of Theorem 1.1 to operators of the more general form (1.8) in a future work. The plan of the proof of Theorem 1.1 is as follows: in Section 2 suitable ultradistribution kernels are defined, and their singularities estimated, by using the theory of symbols of infinite order of Zanghirati [14] and CattabrigaZ anghirati[3]. W e p ro ve in Section 3 the existence of m icrolocal parametrices for the operator P with kernels of the type studied in the previous section. The estimate of the wave front sets of the kernels yields Theorem 1.1. 2. Symbols of infinite order and ultradistribution kernels
First step of the proof of Theorem 1.1 is the introduction of a class of symbols.
636
Massimo Cicognani and L uisa Z anghirati
Defintion 2.1. Let d be a real number, d > 1 . We denote by S ' (Rn) the space of all functions p E C ( R ) such that for every compact K c R n there exists a constant A and for every c >0 a constant Cs satisfying d
2
( 2 .1
)
ID,apx/3p(x, t9 )1‹
for every a,
l
c c A
e and x E K ,
la+fli a ! R ! du + 1
inleXP(Eloilid)
E R ', 101>A.
S (Rn) is one of the classes of symbols of infinite order studied in [14] and [3]. W e refer to these papers for the proofs not given here and more details. The space of all analytic sym bols of o rd er m is a subspace of S ' (R n ) for every m and every d >1. The pseudodifferential o p e ra to r P (x , D x ) w ith sy m b o l p E S ' ( R n ) is defined for uE Go ( Rn) by: - 4
d
-
d
d
(2.2)
p(x , D x )u(x )=(27) nf e
P(x, 0)11'(9)&9 .
-
The operator P(x, Dx) maps Go (R ) into G (Rn) continuously and can be extended as an operator from G r ( R ) to Go Y(Rn). According to ultradistribution kernel Theorem (see K om atsu [8]) there e x ists an unique KE Go ( R n) s u c h th a t =, u,v E d
n
d
(d
2
(d Y
Go (Rn); formally: d
(2.3)
K (x , y )= (27) nf e' < " P(x, t9)dt9 -
)
2 . -
T o g iv e a m eaning to (2.3) w e c a n look a t the in teg ral in the above equality as a particular oscillatory integral of the type studied in [3]. Let gr,} be a partiton of unity in Co ( Rn) such that for constants L >0, A >0 -
(2.4)
supp groC{1011 .
with
;
The composition of two elements in FS way: (2.11)
- 4
(Rn) is defined in the following
Ep,(x, t9). E q ,(x , 0 )= Er,(x, r9)
1,0
1,0
;,0
where (2.12)
r1 (x ,0 )=
E (a!)
-1
lal+h+k=i
a9aPh(x ,0)Dx aqh(x ,0)
We introduce now an equivalence relation in F S " ( R ) : -
Definition 2.6.
Ep,(x, 3) and q , ( x , 3 ) from FS - 4 (Rn) are said to be 1
equivalent (Ep,(x, 3)— E q,(x , 0)) if for every compact K c .R n there exists a constant A and for every E >0 a constant Ce such that: (2.13)
E (p,(x , t9) —
i0 a constant Ce satisfying: i
2
d
2
(2.14)
ID2WDoaD,fla(z, w, x, 3)10 , t >0, a holomorof the variables ( t + ip, s + p) E C+ with values in G 0 (R 2 ) -
d
phic function Ea
2
'
by: (2.15)
Ea(t+ ip, x , s+ ip, y )= Os — f x m> a(t + ip, s+ip, x, 0)cla (
-
As a consequence of (2.14) we have that for every bounded set L c C+ and every OE Go (r i ) there exists a constant C satisfying: d
(2.16)
1 0 an d a corresponding operator EA : G0 11 1, ml >M ( ro, m) such that 2
(d )
-
-
d
d
TEA = I w here I is the natural projection f rom Go 'ITI, m] onto M ( To, m) and EA has the kernel d
(d
H(t - s)EA' = H (t - s)(0 s- f e -
i
< X -" )
s +
i O
,
P ro o f W e look for A with asymptotic expansion A -
x
,
0)C1C9).
E A , in FS
- 4
7 (R Z,
m ) . W e have
T (H(t— s)EA +)=÷ta(t— s)EA ++H(t— s)T E A+ therefore in view of Theorem 2.9 and properties (2.24), (2.25), we shall prove Proposition 3.1 if the A , can b e ch o sen to satisfy the following transport equations for (z, w ) C + 2 (zDz- B(z, x, z9))A0 =0
{
(3.3)
(3.4)
A 0 .=
—
I( /
is
the identity matrix)
{ (zDz- B(z, x, t9))A,=G, Avl-w-0
where:
G ,=
EE
k =1 lal=k
(a !)
- l
a d a B D x a A ,-k
.
For L={.zEC+; z= pe , pl< p< p2, 71< y< 721 a bounded set, 0< pi < p2, 0< i7
< 72.< 7r, we define
0) 1= sup t1B(z, x, t9)I; zELI , No(z, w, x, 0 )=
l
w
.
exP(1113(x,
We represent the solution Ao of (3.3) as the limit of the successive approximations 21(i defined by
X0=7 .1 , 7
)
1 x w, x, t9)=- i(—I + f
W
A
c 9')
w, x, t9)d0 ,
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Massimo C icogn a n i and L uisa Z anghirati
-where A is the path A = {"(t) --= inductively the estimate
(IIBII ilogz logwp —
1
t
tzt;
1}. It is then easy to prove
h
h!
ludh=o
for every z , w E L , thus we obtain 14/101 N 0
for every z, w E L . By taking derivatives in eqality (3.3) we can prove inductively, that for every compact K C IV there exists a constant C such that: (3.5)
1Dzi.DzokDoaDx8A0(2, w, x, 9)1 . 1 \10C l a + f i l + " kj!k!a!,3!( 1 +101) - l a i lz1- 1w1 - k • j
1 ( 1 + 9 1 )ar k 1 4 E (1 +
r= 0 r!
s=0.3!
1,9 ,8s 1)
E
05h5la+/31 rt
logz — log w ih(1 +10 r
for every j, k , a, f i and every z, w E L , x E K , c9E R ',191> C. The solution A , of (3.4) can be represented by (3.6)
A ,(z , w )=LA 0(z,
w )cl .
By taking into account the inequality (3.5), the representation (3.6) and the following property of No: N o(z, -.)/N/0( ", w )= -6 N o (z , w ) for e v e ry E A ,
then we can prove by induction on j+ k + y that for every compact K c R n there exists a constant C such that: (3.7)
1DiDw'DoaDxfill,(z, w, x, 9)1
• r=o i i (1+101) 8 s ± ( 1r!+ 1 9 1 ) 8 r s=os! •
E
1 —,,Ilogz—logw 1h(l+O r
05h5lcr+51+2v
for every j, k , d , g, I ) and every z, w E L , x E K , t9 E R ',1 9 1 > C . From the above inequality (3.7) and the assumption 1 < d < —16 w e have for '
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Analytic and Gevrey singularities
each fixed A> 0 and all A, that for every compact K c R n and every bounded set L c C+ , there exists a constant B and for every e > 0 a constant Ce satisfying: (3.8)
ID2AvhDoaDiA,(z, w, x, c9)i
c‘,13 1.+,91+.7+k+v! k !a! g ( 1 + 101)-lal-v !2 )!
XeXP(Elt 14z1 1W1 9
-
-k
eXP(B1 2 1- À +BIWI - '1 )
for every j, k , a, fi and every z , w E L , x E K , c9E R n,101> B. If follows from (3.8) that A E o'(C+ ; S ' (Rn , m ) ) . This completes the proof. 2
d
Remark 3 .2 . In inequality (3.8) we have 8! instead of firi; therefore A satisfies a condition stronger than (2.14) that permits us to obtain also for the analytic wave front set WFI(EA ) the same estimates given for WFd(EA+) in Proposition 2.11 and in the following part of Section 2 (See also Boutet de Monvel [1]). +
We can construct other right param etries for T , b y u sin g the same methods. More preciselly, if we introduce operators of the type E = liM 0 S f e' (x "› A (t + ip, s+ i,u, x , 0)dt9 -
we have four right parametrices with kernels of the form H(t —s)EA or H(s —t)EA . Also four left parametrices with kernels of the same form exist, because the transposed operator T * has the same properties of T . A ctu ally we consider T as an operator from Go [Fi, m] to Go "[Fi, m] and, taking a left parametrix E A , we have E A T = I . We can prove Theorem 1.1 by using these left parametrices. ±
±
(d )
(d r
Proof of Theorem 1 . 1 . Let 1 d < E Go
(d Y
(3.9)
[ii, m]
a and U E M
d
(Fo, m ), U = IV
with V
satisfying
(LIU
n wFd( V )= 0 ,
zo
W F d(TV ) .
Considering the left param etrix EA with kernel H (t —s)EA+, we have U = E A T V . Thus zolWF0( U ) follows from the assumption (3.9), the remark 3.2 in the case d=1, and the estimate (2.23). The other cases of Theorem 1.1 can be proved in a similar way by choosing the appropriate left parametrix. Dipartimento di Matematica, Università di Ferrara Via Machiavelli 35, 1-44100, Ferrara, Italy
646
Massimo Cicognani and Luisa Zanghirati References
L. Boutet De Monvel, Opérateurs pseudo-différentiels analytiques et opérateurs d' ordre infini, Ann. Inst. Fourier, 22 (1972), 229-268. [ 2 1 A. Bove, J. E. Lewis and C. Parenti, Propagation of singularities for fuchsian operators, Lecture Notes in Math, 984 (1983), 1-160. [ 3 1 L. Cattabriga and L. Zanghirati, Fourier integral operators of infinite order on Gevrey spaces. Applications to the Cauchy problem for certain hyperbolic operators, J. Math. Kyoto, 30 [il
(1990), 149-192. [ 4 ] J. J. Duistermaat and L. Htirmander, Fourier integral operators II, Acta Math., 128 (1972), 183-269. [ 5 ] N. Hanges, Parametrices and propagation of singularities for operators with non-involutive characteristics, Indiana Univ. Math. J., 28 (1979), 86-97. [ 6 ] L. Htirmander, The analysis of linear partial differential operators I, Springer Verlag, 1985. [ ] H. Komatsu, Ultradistributions I. Structure theorems and a characterisation, J. Fac. Sci. Univ. Tokyo, 20 (1973), 25-105. [ 8 1 H. Komatsu, Ultradistribution II. T h e kernel theorem and ultradistributions with support in a submanifold, J. Fac. Sci. Univ. Tokyo, 24 (1977), 607-628. [ 9 1 V . Y. Ivrii, Wave fronts of solutions of certain pseudodifferential equations, Soviet Math. Dockl., 17 (1976), 233-236.
[10] R. Melrose, Normal self-intersections of the characteristics variety, Bull. Amer. Math. Soc., 81 (1975), 939-940.
[11] L. Rodino and L. Zanghirati, Pseudodifferential operators with multiple characteristics and Gevrey singularities, Comm. Partial Differential Equations, 11 (1986), 673-711. [12] M . Sato, T. K aw ai and M . Kashiwara, Microfunctions and pseudodifferential equations, Lecture Notes in Math., 287 (1973), 265-529. [13] K. Uchikoshi, Microdifferential equations with non involutory characteristics, Comm. Partial Differential Equations, 15 (1990), 1-58. [1 4 ] L. Zanghirati, Pseudodifferential operators of infinite order and Gevrey classes, Ann. Univ. Ferrara, Sez. VII, Sc. Mat. 31 (1985), 197-219.
