,v, we have AI$I-(m-k)000$
r I'\l,9l-,n+kO
b
31ibj
j> N
j>N <
2
<
j>k+l
'--
1
because in this sum one has Ja +,31 < j and A > f on supp b3. We thus get a - E3
I
As a particular example of this construction, we take functions a. with com-
pact support in x, (positively) homogeneous of degree m. - j in and Cx outside = 0. As shown above, we can associate symbols b., E S'"-.' to such functions, and the construction of Lemma 2.2 will give us a new sym-
bol a E S', for which we will still write a - >j a3 (a is clearly uniquely determined by the functions aj modulo S-O°). Such symbols are called polyhomogeneous (some authors say "classic"). These polyhomogeneous symbols form a subclass of S°° containing the differential symbols (polynomials in ), the symbols Am(1;) = + 1)-/2 (indeed, the second factor can be written as a power series in Ifs-2 convergent for 1) and closed under differentiation and multiplication (and also, as we will see, under adjunction, composition, and inversion).
2.2
Oscillatory integrals
In this section, we just want to sketch a theory for some integrals of functions that are not absolutely integrable. These integrals will be used in the next section to define operations on the symbols (symbols of the adjoint and of the compound). Thus, we do not try to give here any idea of what a general theory could be, but
only to establish the fundamental properties of these integrals in the particular case we will use most. The functions of which we want to define the integrals are products of an oscillatory term and an amplitude with a controlled growth at infinity, exactly as in the classic integral f 00 (sin x/x) dx. More precisely,
33
Oscillatory integrals
the oscillatory term will be a trigonometric function of a quadratic form, while the amplitudes will be taken in the following spaces: for m >_ 0, a E A"` (the space of amplitudes of order m) if a is a Cx complex-valued function defined on R" such that the functions (1 + IxI2)-+"/2Oaa(x) are bounded on R' for all
a E Z. It is clear that polynomials are such amplitudes, and that derivatives or products of amplitudes are amplitudes. On the space A', we will use the norms IIIaIIIk = maxlaI
Let q be a nondegenerate real quadratic form on R, a E A' and p E S such that cp(0) = 1. Then the limit lim
J
e`Q(x)a(x)cp(ex) dx
exists, is independent of 'p (as long as 'p(0) = 1), and is equal to f e'q(z)a(x) dx when a E V. When a L' we continue to denote this limit by f e'Q(x)a(x) dx, and one has an estimate
if
e=q(x)a(x)dx 1 G Cq,mIIIaIIIm+n+I
where the constant Cq,,,, depends only on the quadratic form q and the order m.
PROOF When a E L' the result simply comes from dominated convergence. In the general case, let us choose a function 0 E Co (R" ), such that b = 1 in B, and V) = 0 outside B2, and set I. = f e`9(x)a(x)V)(2-)x) dx. We are going to show that limi-,,,, I, exists and is equal to limf,o f e'q(')a(x)'p(ex) dx, and this will prove both the existence of the limit for any 'p E S and its independence with respect to V. Furthermore, since for any fixed c > 0,
J
e`Q(=)a(x)V(ex) dx = lim
J
e'q(x)a(x)'(ex)0(2-3x) dx
by dominated convergence, we will just set
li(e) = f e+q(z)a(x)(I
-
A(Ex))V,(2-3x)dx
and show that lim3.o,, Ii exists (resp. that limi_m IJ(e) = 0(e)). To get these results, we use the change of variables y = 2-3x:
li - ii_, =
f
e'Q(x)a(x)(,(2-ix)
- V,(2'-Jx)) dx
= r ei2''9(y)a(21 y)(,0(y) - 0(2y))23' dy
(resp. li(e)-I,_,(e) =1 et22jq(y)a(2iy)(I-'P(e23y))(il'(y)-rl'(2y))2'"dy
.
34
Pseudodiferential Symbols
The function X(y) = i(y) - '(2y) satisfies X E Co and supp X C {y; 1/2 < IyI < 2}. Moreover, for y E supp x, I(&a)(23y)I S
IIIaiIIIQi(1+22jlY12)m/2 <
IIaI I Ilal2m('+2) (resp. I(d°b()(2'y)I <- EC2(*"+i )2 where bE (x) =
and the constant C depends on everything but one and j, for I1 - cp(e2'y)I < I E2iyI sup I p'I < eC23 ). Thus, one gets the following estimates, which imply the theorem:
IIj - h-11 <_
-
(resp. II,(E)
q,m2-'IIIaIII,1+,.+1
- I,-i(E)I <- (C2-J)
by using the next lemma with µ = 2J and N = m + n + 1 (resp. N = m+n+2). 1
LEMMA 2.4
Let q be a nondegenerate real quadratic form on Rn and X E Co with x = 0 near 0; then for all N E Z+,
l
eill 24(11)b(py)X(y) dyl
CNA-N
sup
yEsuPPX,IaI
I (0'b)(uy)I
where the constants CN do not depend on µ > I nor on b E C. PROOF One can perform a linear change of variables so that q(y) = Iy' I2 - ly" I2
with y = (y',y"). Then the operator L = (1/2Iy12)((y',c') - (y",8")) is well defined on supp X (with CO° coefficients) and satisfies Lq = 1. Integrations by parts will involve the transpose of L : tL = (8", y"/2Iy12) - (8', y'/2IyI2), which is also a first-order differential operator with C°° coefficients, and N such integrations by parts thus give
f etvZ((y)b(µy)X(y)dy =
(aµ2)-N f (LNetµ2q(y))b(µy)X(y)dy
_ (iµ 2)-N
J
dy
_ (ii, 2)-N where cµ,N is a linear combination with C°° coefficients of terms of the form µ1 a1((8ab)(py))(81 X(y)) for Ia +,31 < N. The result follows since supp X is compact.
I
The main result of this section is that oscillatory integrals behave essentially as absolutely convergent integrals.
35
Oscillatory integrals
THEOREM 2S
The integrals defined in Theorem 2.3 satisfy the following properties: (i)
Change of variaables. if A is an invertible real matrix. J eiq(AV)a(Ay)I det Al dy = J eiq(x)a(x) dx.
(ii) Integrationn by parts: if a E A", b E A', and a E r+,
(iii)
f
ei9(x)a(x)8'b(x) dx = f b(x)(-a)"(e`q(x)a(x)) dx.
Differentiation under f : if a E A' (1R" x RP), then f eiq(x)a(x, y) dx E Am(1RP)/ and
a° (iv)
J
for all a E Z.
eiq(x) a(x, y) dx = Je'Oa(xY)dx
Interversion of the f : if a E At(1R" x RP) as in (iii) and if r is a nondegenerate real quadratic form on RP,
f
eir(y)
(f elq(x)a(x, y) dx) dy =
fe*()+)a(x,y)dxdy.
PROOF The proof of this theorem consists essentially in checking that all the integrals written in the statement are oscillatory integrals, then in taking the limit 0 as in the definition from Theorem 2.3. Thus, property (i) follows for f dx, from the change of variables x = Ay in the integral f since '(y) = p(Ay) E S and satisfies &(O) = cp(0) = 1 and since b(y) = I det AIa(Ay) is an amplitude of order m.. Similarly, integrations by parts in the
right-hand side of (ii) where we add a factor cp(Ex) give a factor &'(so(Ex)b(x))
),Ell" (013 )(Ex)e-'b(x),
and for 0 # 0, the EIAl gives 0, when taking the limit c -. 0, while for = 0 we get the left-hand side of (ii). (The details are left to the reader.) For (iii) and (iv), the proof is not as easy. Coming back to the proof of Theorem 2.3, one considers the integrals
Ij (y) = Je"a(xY)IJ(2'x)dx which satisfy a&I3(y) = f eiq(x)a&a(x, y)i,t(2-ix) dx thanks to the absolute convergence given by the factor ''(2-l x). Since I a_0a& a(pz, y) I < CQ3 (1 + Ipz12 + Iy(2)m/2 _< CCp5m/2µ'"(l + Iy12)m/2 for Izi < 2, we get from Lemma 2.4 estimates
Ia& I2(y) - 8&Il_1(y)I <_ CC2-1(l +
IYI2)m/2
Pseudodiferential Symbols
36
which imply uniform convergence on every compact set for the sequence 80 Ij (y).
It follows that the limit I(y) of the sequence Ij (y) is in Am(RP) and satisfies 0I(y) = limj-. c' I) (y), which is formula (iii). Moreover, this estimate also shows that
Ir(I(y) - Ij(y))I : C°2-1 (1 +
Iy12)m/2
so that the functions b3(y) = 0(2-3y)(I(y) - Ij(y)) satisfy bj E Am(RP) with lllbjlllm+p+i <_ Co2'j. Then one can write
fei'(y) (feiQa(x,y)dx) dy = Jim00 fe1t1)I(Y)1,(2_iy)dY and
f eir(v)I(y)V)(2-Jy)dy= f eir(y)Ij(y)VG(2-2y)dy+
Je
ib(y
thus get property (iv) since
jx J lim
eir(y) Ij
(y)'(2-' y) dy =
r ei(e(x)+r(y))a(x, y) dx dy
f
and
if
eir(y)b,
(y) dy < Cr,rIllb)Illm+p+l < Cr,mCo2-'
I
Example 2.6 (i) If a E A' (RI), then (27r)-n
J e-i(y,n)a(y) dy dr) = (2ir)-"
J
e-`(y.n)a(7l) dy d77 = a(0).
(ii) If a and 0 E Z+,
y° p dydri _
(27r)-n
J
a! Q!
=
0 if a 0
)3;
a
if a
PROOF The quadratic form on R2n E) (y, r)) F--i (y, 7l) is obviously nondegenerate since the identity (y, q) = (1/4)(Iy+qI2- Iy-7i12) shows its n positive and n negative eigenvalues. On the other hand, the polynomial y°rla is in Al°+131 so that all the integrals written in the statement are oscillatory integrals, as defined in Theorem 2.3. (i) The first equality follows from exchanging y and 77. Then take a p E S such that p(0) = 1. By definition, r e-'(y.n)a(71) dy drl = li m
r J
dy drr.
37
Operations on symbols
Through the change of variables ey = z, e(= rl then integration in z we get
f
f
dz d(
For f < 1,
d(. I(I2)m12lwlo, which
IIIatlIo(1 +
is inte-
grable, so that by dominated convergence we get (21r)-"
f e-;(b,») a(q) dy d77 = (2ir)-" f P(()a(0) d( = V(O)a(O) = a(0).
(ii) For a and 0 E Z+, yae-'(b,'7) = (-D,,)ae-'(Y,')), from which e-i(b,n) yQ
(2.r)-"
a dyd77 =
(27r)-"
e-;ly,n)
Dn (
1
1 dydi7.
The function
a() = satisfies a(O) = 0 if Q follows from (i). I
2.3
(iii)
(Z)k
a
()7s-Q
a and a(0) = (-i)1°I /a! if 3 = a, so that the result
Operations on symbols
We can now define the symbols a* and a#b, which will be the symbols of the adjoint and the compound. THEOREM 2.7
Let a E S' and b E St; then the oscillatory integrals - y,( - 77) dy d77
a* (x, () = (27r) -"
a#b(x, () = (27r)-" r e-'(",v)a(x, - tl)b(x - y, C) dy dij define symbols a* E S' and a#b E S'"+t with the following asymptotic expansions: Ot'Daa
aa
and
± &t aDz b.
a#b a
PROOF The quadratic form (y, 77) is nondegenerate (see the proof of Example 2.6). To see that the function bx, (y, 77) = a(x - y, - r7) is an amplitude,
Pseudodiferenrial Symbols
38
we simply remark that Peetre's inequality (Lemma 1.18) gives the estimates 18" 8» a(x - y, - 0 1 S
C°3\, (C - n)
< CR,321"'lAm(S)(l + Iy12 +
I71I2)Im1/2
for all a,$ E Z', from which bx, E AI-I(lY2") With Illbx,{IIIImj+2,'+I CoV (l;). Thanks to the estimate given in Theorem 2.3, it follows that a-"`a' S"'-1131, we get and 8x 3 a E is bounded, and since 82 8 (a`) the boundedness of A I0I -'8x 8. a' for any a, 0 E Z. by the same argument, so that a' E S'. The proof of a#b E S`+P is similar, since the function cx,{ (y, rl) = a(x,e - rl)b(x - y,{) satisfies c,.,f E AI'nl (R2') With and since
IIICX,
8°(a#b) _
I (a`ja)#(a°-fib)
for all aEZ+. To get the asymptotic expansions, we use Taylor's formula (cf. Theorem 1.1):
(a) ()F
Ox - Y'!; - n) _
a(x, ) + rk (x, , y, rl)
I°+131 <2k
with
rk (x, C y, rl) = 1 °+RI =2k
2k(-y)°
(-rl)a
a!
,0!
r°Q(x, , y, rl )
and 1I
r.0 (x, , y,'7) =
Jo
(1 -
t)2k-18y 8{ a(x
- ty, - trl) dt.
The terms for ka + of < 2k give after integration the terms of the expansion in view of Example 2.6(ii). For rk, which is clearly in AIm1+2k in (y, 77), we integrate by parts as in Example 2.6(ii): J
e
a!
Q!
r°,5(x, , y, n) dy drt
p
1
(e-`(b,")) dy dq
y, rl) Dn J(a fe_'")(-D,')((_Dt)°-r°y(x,,y,))dydn
&
Or
7
(a)1-Y I
/\
0)
/e-;(b,+r)(_)A-7(-Dn)°-7r°p(x,,y,rl)dydrl-
39
Operations on symbols
A second integration by parts then gives (-i)171y! 7
y \a/
a!/3!
Y
V e-
Recalling the definition of ra,q, one has S, y, rl)
f(l 1
=
- ty, - tq) dt.
Since y < a and -y < 0, one also has Iry(< k and Ia + Q - -yl > k, then
+v-_Yaf +f1-'a E Sm-k. Thus, the calculations given above can be rephrased
f e`(y.n)rk(x,C y,rl)dyd?1 =
f
where now the amplitude sk satisfies sk E AI"'-kl with IIISkIIIIm-kl+2n+1 < CkA' -k(t ). We thus get the boundedness of
Ak_m(S) f e-1(y.^'rk(x,C y, z) dydrl, and using the same argument as above, we get
f
f, y, rl) dydrl E Sm-k
just by remarking that i3Ot rk is the rest of index 2k in Taylor's expansion Sm-101. Finally, the of c,'8 a(x - y, - r)), for which one has E asymptotic expansion of a#b can be obtained along the same lines, but its technical verification is left to the student. I
REMARK AND EXAMPLE 2.8
If a is a differential symbol (that is, a polynomial
in t: with coefficients in HO0), we can follow the previous proof with k = m + 1, which leads to the terms which are identically 0 since J a + 3 - y I _> k > m. Thus the asymptotic formula for a' is exact and contains only the terms with Ial < m. For the same reason, we also have
a#b = E{al<,n(I/a!)O I) b for any b E St. It would be interesting to give the same remark if a is a polynomial in x with coefficients depending on l;, but such functions are symbols only if the degree of the polynomial is 0, that is, if a depends only on . Yet, the argument still works
and gives a" = a and b#a = ba for any b E St. In particular, this is the case for the symbols A' which thus satisfy (A'")' = A"' and At#Am = .1t+` (and more generally, b#Am = ba'n for any b E St). As for higher order polynomials in x, which are not symbols, we will see (in Example 3.5(ii)) a formula quite similar to the finite development of a#b we could expect. I
Pseudodifferential Symbols
40
PROPOSITION 2.9
The operations defined in Theorem 2.7 satisfy the properties (i) (a*)* = a. (ii) a#1 = 1#a = a. (iii)
a#(b#c) = (a#b)#c.
(iv)
(a#b)' = b'#a'.
PROOF One has
(a*)+(x,() _
(2Tr)-2n
(27r)
f
(fe-;(Y'h1)a(x_z_Y_(_77)dYd1)) dzd(
-2n f e=((b,n)-((z,())a(x
- z - y, - ( - 77) dy dij dz d(.
Using, then, the change of variables Y = -y, H = 77 + (, Z = z + y, Z = (, for which (y, 77) - (z, () _ -(Y, H) - (Z, Z) and dy d&J dz d(= dY dH dZ dZ, we get
(a*)* (x, O = (27r)-2nJ
e-i((YH)+(z.Z))a(x
- Z, ( - H) dY dH dZ dZ
(27r)-2n re-i(z.z) (fe_"1a(x_Z,
_
= (27r)-n J e-'(z,z)a(x
H) dY dH) dZ dZ
- Z, )dZdZ =
where the last two equalities come from Example 2.6(i). Formulas a#1 = 1#a = a simply follow from Remark 2.8 above. To prove (iii), one writes
a#(b#c) (x, () _
(27r)-zn
f e-(1,I) a(x, ( - rf)
(Jeb(x _ (2r)-2nJ
y,
- ()c(x - y - z, () dz d() dy di
e''((v,n)+(z,S>)a(x,
- r!)b(x - y,( - () c(x - y - z, () dy dri dz d(,
Operations on symbols
41
then
(27r)
-2n f e-+(Z.Z)
2 - H)b(x - Y,
CJ
e-i((YH)+(Z,Z))a(x,
_ (27r) -2n
f
.2) dY dH) c(x - Z, ) dZ dZ
- 2 - H)
b(x - Y, - Z)c(x - Z, ) dY dH dZ dZ,
and these two quantities are equal through the change of variables y = Y,
77=H+Z,z=Z-Y,c=Z.
Finally, for the last formula we have b* #a*
(27r) -3n J e-+(t'T)
e-=(z,S)b(x
\J
- z,
rr - () dz do/
(Je_"a(x - t - y,
= (27r)
-3n Je-
- 7]) dy d7)) dt d7-
t - y, - 7l) c)dyd7/dzdOdtdr
= (27r) -3n r e-'(-(Y,H)+(Z,Z)+(x,=))a(x
b(x-Z-Y,e-Z)dYdHdZdZdXdr
J _ (27r)-2n
J
- Z, - 2 - H)
e-'(Z,Z)
(fetc'.h1)a(x - Z, - Z - H)b(x - Z - Y, - Z) dY dH)
dZ dZ
after a change of variables (Y = z - t - y, H = 77--r-C, Z = t + y, 2 = 7-+(, X = z - t, E = 77 - r) then integration in (X, E-) (cf. Example 2.6). This ends the proof of Proposition 2.9, since
(27r) -n
J
e-'(YH)a(x - Z, - 2 - H)b(x - Z - Y, - Z) dY dH.
I
42
Pseudod{ferential Symbols
To close Chapter 2, we examine the problem of inverting a symbol for the operation #, that is, to solve - at least approximately - the equation a#b = 1
(or b#a = 1) for a given a E S'". Since b E St implies a#b E S-+' and 1 E So, it seems natural to seek in S-' the inverse b. The result can then be stated as follows. THEOREM 2.10 ELLIPTIC SYMBOLS
If a E S', the following four statements are equivalent:
There exists a b E S-' such that a#b - 1 E S. (ii) There exists a b E S-' such that b#a - I E S. (i)
(iii)
(iv)
There exists a b o E S such that abo - 1 E S-1. There exists an e > 0 such that Ia(x, C)I > eA"'(l;) for ICI >
/e.
Moreover, when these conditions are fulfilled, a is said to be elliptic, and there
exists an ae E S-' such that b solves (i) PROOF
a
b solves (ii)
a
b - a0 E S-°°.
If a E S"' and b E S-'n, then a#b = b#a = ab modulo S-1 in
view of the asymptotic expansion given in Theorem 2.7. Thus each of the statements (i) and (ii) implies (iii); then (iii) implies the existence of an e > 0 such that 11 < 1/2 for ICI > 1/e, so that we have for such : 1/2, then since bo E S-'n Ambo bounded, and this is (iv). Conversely, if (iv) is satisfied, then c = A-'"a E So and satisfies Ic(x, ) I > e for Il;') > 1/e. If F(z) is a C°° function defined on C and equal to 1/z for
Izi > e, F(c) E So thanks to Lemma 2.1, and then bo = A-mF(c) E S-' satisfies abo(x,l;) = 1 for ICI > 1/e, which implies (iii). Now, if (iii) is satisfied, then a#bo = I - rl and bo#a = 1 - sl with rl and sl E S-1. Let
us set rj = rl#rj_l E S-j, sj = S,_l#sl E S-J, b, = bo#rj E S-"'-I, cj =sj#b0ES-'"-j, and finally bNF_12!ob, ES-' and c,,bo+>,2,lcj E S-' by using the construction given in Lemma 2.2. One then has for any fixed k E Z the following equalities modulo S-k:
a#b=a#l:bj=(1-rl)#(1+ > rj I =1-rk=1 j
c#a= (b0+
0<j
I
E Cj #a= I+ E s, #(1-s1)=I-sk=1 0<j
0<j
since rk and Sk E S-k. We thus get a#b - 1 E S-O° and c#a - 1 E S-O°, for k is arbitrary.
43
Exercises
Finally, if b solves (i) and c satisfies c#a - 1 E S-°°, one has modulo S-1
b = (c#a)#b = c#(a#b) = c which implies the uniqueness of the inverse modulo S- O°. Conversely, if b-a",
a#a# - 1, and a"#a - I are in S', this is also true for
aft-1 = (a#aM-1)+(a#(b-a#))
and
b#a-l = (aa#a-1)+((b-ae)#a)
since this is true for the four terms appearing in the right-hand sides.
I
The symbols A"' are all elliptic since they satisfy A"#A-' = 1. One can remark that they even have exact inverses. For a differential symbol
a(x,f) = Ejai<ma"(x)e (and more generally for a polyhomogeneous symbol), the principal symbol is defined as the homogeneous part of degree m, that is, the function p(x, t;) = a,, (x)t;". By using the characterization (iv), it is easy to show that such a symbol is elliptic if and only if Ip(x, e>0
forxEW' and ICI= 1.
Exercises 2.1
Let e > 0, f? _ {S E C"; Jim (1 < elRe(I} and a(() a holomorphic function on f1 satisfying an estimate Ia(()I < C(1 + ICI2)'"/Z for large (in 1. By using Cauchy's integral formula, show that (1 - :p(t;))a(i;') E S°' for some function
p EC. with cp=l near l;=0. 2.2
The goal of this exercise is to show that a C°` function a(x,t;) is a symbol of order m as soon as the functions A-'&.',a and ak-'".9E1a are bounded for all j < n and k E Z+. (Notice that there is no assumption on the mixed derivatives.) Let cp and
be two C°° functions of one real variable t satisfying
V(t)=l fort<1,
yp(t)=0fort> f,
ra(t) = I fort E (f/2, 11,
(t) = 0 for t (1/2, 03). Then, for a E C°°(RZ") satisfying the assumptions given above, x E R", p > 1,
and(y,i)ER" xR",set a..,. (y, 17) = v(IyI)'(InI)i °'a(x+y,p71)
if µ>2,
if p < 2. = v(IyI)v(IiiI)p ma(x + y, p i) Show that for (y, rj) E supp a,,,µ one has (1 /2)A(pi1) < p < 2A(µ7j). Show that there exists a sequence Ck independent of x E R" and u > I such that Iay,ax.pl° < Ck and IO,k,,ax.µIo < ck for all j < n and k E 4+. Show that there exists a sequence Ck independent of x E R" and p > I such that ax,, E H°°(R" x ax.,.(y, )7)
ar
R") with Ilax,,,Ilk < Ck for all k E 4. Show that a R", the function a(x) =
f) _
that E satisfies a E A°. If A is a real symmetric nonsingular n x n matrix, show that the oscillatory integral
Pseudodiferential Symbols
2.4
e'(^=._)l= UW = f e;('t= =)/2a(x) dx is in P° and satisfies (U, gyp) = f p(x) dx for all V E S. Then compute U(.) by using Exercise 1.6(b). More general oscillatory integrals. Let tP be a real-valued C°° function defined on 9t" \ {0} such that Op never vanishes on It" \ {O}.
(a)
Show that there exists a first-order linear partial differential operator L with
C"° coefficients in R" \ {0} such that Lv = 1. Let k E JR and X E Co be such that X = 0 near 0. Show that
J
elakw(v)b(py)X(y) dyl
CNI.tN('-k)
sup
I (Ob)(µy)I
LEsuppx.jnjGN
for all N E Z+ where the constants CN do not depend on u j> 1 nor on b E C°°. (b)
Let k and i E lR be such that k + i > 1, and assume that p is also homogeneous of degree k. Let a be any amplitude taken from the space
AL = {a E C°`; At* -m(x)O"a(x) is bounded on 1R" for all a E Z. }. Show that for any t' E S such that t;,(0) = 1. the limit
J
lim I e"0(=)a(x)y)(ex) dx o
exists, is independent of io (as long as t!)(0) = 1) and is equal to
J e"°t )a(x) dx when a E L) (this limit will also be denoted by f e"')=)a(x) dx when (c)
Give an estimate of this integral, as in Theorem 2.3. If 96 0, show that the function W4 (x) (x, ) satisfies the assumptions of question (b).
If a E A' for some m and i > 0, show that the function f e" °f (=)a(x) dx satisfies A E P. Show that if t' E S satisfies ib = 0 near 2.5
= 0. then (A,ty)
conclusion? Let a E S'" and b E St. Rewrite the asymptotic expansions of Theorem 2.7 to get
simple expressions of a' and a#b modulo S` and Sm+r-2, respectively. Then, relate the symbol a#b - b#a to the Poisson bracket of a and b, which is defined as (a, b} = (efa, 8=b) - (8=a, O (b). 2.6
Let a and b be two elliptic symbols. Prove that a' and a#b are elliptic and express
their inverses (a')' and (a#b)' in terms of the inverses a' and V.
2.7
Conversely, assume that a and b E S°° are such that a#b is elliptic. Show that a and b are elliptic. What conclusion could you give when you assume that a#b#c is elliptic? Let a E S' satisfying eX"'({) for all I/e and some e > 0,
and let k E Z.+ \ {0}. Show how to construct a symbol b E Sm/k such that
a = b#b#...#b (k terms) modulo S. 2.8
This problem is made up of two exercises: in questions (a) and (b), one proves that if a symbol a of unknown order satisfies a single estimate Ial < Cat, it is automatically in St+ = n>IS,; in questions (c) and (d), one uses this property in the study of nilpotent and idempotent symbols.
45
Exercises
(a)
Let k and q be two positive integers. Using Holder inequality, show that IIaIIk <- IIaIIo-(1/°)IIaIIk/° for all a E Hoc.
Let K be a compact set. Show that there exist constants C depending on K, k, and q but not on a such that IIaIIo 5 CIabo and 1/q
Sup Ia°alo IIaIIk < CIa11-(I/9) (Ial!5kq
(b)
for all a E C' satisfying supp a C K. Let f < m < p and assume that a E S' is such that A-ta is also bounded. For x E R" and µ > 1, define the same functions a..,, as in Exercise 2.2. Show that there exist constants CO independent of x E R" and p > I and loO a=,,,Io < C°Qµp-". such that Ia:,,Io 5 Cooµt Use question (a) to show that there exists a sequence Ck independent of
x E R" and µ > I such that IIa,,,,Ilk 5 Ck for all k E Z+. As in Exercise 2.2, conclude that a E S'". (c) A symbol a E S°° is said to be nilpotent if there exists a k E Z+ such that a#a# ... #a = 0 (k terms). Show that if this relation holds for an a E S', then a E S"'-('/2k), and conclude that nilpotent symbols belong to S-°`. (d) A symbol a E S°` is said to be idempotent if a#a = a. Show that a is idempotent if and only if I - a is idempotent. Show that if a is idempotent, then a E S' for some m > 0 implies that a E S2m/3 U S"'-(1/3), and a E Sm for some m < 0 implies that a E S2m. Show that an idempotent symbol a satisfies a E Sol = fl,">OSm and that a2 - a = b for some b E S-213. Then, show that there exists an e > 0 such 1/4,Re(1 +4b(x,e)) > 0 (so that that for ICI > 1/e one has you can take the usual definition of (1 + 4b)' /2), 11 - (1 + 4b(x, ) )' /2I < Finally, conclude that 4Ib(x,t)i and I1+(1+4b(x,C))'/2I >
a E S-'13 or 1- a E S-'/3 (one assumes n> 1). 2.9
Prove that an idempotent symbol a satisfies a E S-°° or 1- a E S-O°. This long exercise will be continued in Exercise 3.6. (a)
Quasi-elliptic operators. For any p E Z+ such that µ, > 0 for all j < n, one sets Ia : µI = (a,/µ1)+...+(a"lµ") and Ii;(N)I = (Ej 2"' )'l2. The differential operator a(x, D) is said to be quasi-elliptic at xo if there exists a p as above such that a(x, D) = >I°:,AI
Let a(x, D) be quasi-elliptic at as and p(x,l:) = FI° Fl_i be its quasi-principal symbol. Show that there exists a constant C and a neighborhood Il of xo such that CIC(,.)I
for x E S2.
Then, prove that the functions (Ama)'' and a -' AP1 I gs ea are bounded
hand ICI > 1/e} for some mE R,e>0,andp>0,and
on
all a,l3EZ+. (b)
The S,7,0 calculus. Let p > 0 and m E R, then define the class S,7,,
of symbols as the set of smooth functions a(x, l;) such that the functions APIQI--O."
a are bounded on R" x R" for all a and,3 E Z.
Pseudodifer+ential Symbols
46
" for j E Z+ , one can construct an a E Show that if a, E oS" that a - E2
SP o such
Show that if a E SP0 and b E SP, 0, one can define symbols a' E S'"110
and a#b E So o by the same formulas and with the same asymptotic expansions as in Theorem 2.7. (c)
Inversion of quasi-elliptic symbols. Let p > 0 be fixed, then let a E So
be such that the functions (,\'a)-' and a-' )PlB1OOa are bounded on ICI > 1/e for some m E R and c > 0, and all a, 8 E Z. (cf. question (a)). Show that there exists a E Co (R") such that the function bo(x, = co(x,l;) _ (1 -,(l;))/a(x,satisfies bo = co E SP0 and .1P10+61(0,0 a) (&'Obo) bounded for all a, )3, y, b E Z. Show that the symbols b, and c, defined by induction as
(a)(Dzbk)
bj = -bo Inl+k=j k<j and
(Mt ck)(Dx a)
c' = -co
1a1+k= <j
satisfy b, and c, E Sp o P2 and and
AP(J+Ip+61)(ooc,)(8
a)
b E Z+ . bounded for all a, Show how to construct symbols b and c E S" no such that a#b - I and c#a - 1 are in S_0°1
3 Pseudodifferential Operators
3.1
Action in S and S'
For a E SO° and V E S, the integral defining a(x, D)V(x) in the introduction to Chapters 2 and 3 (see also the statement of Theorem 3.1 below) is absolutely convergent, and we are going to show that it even defines a function in S. THEOREM 3.1
If a E SO` and cp E S, the formula a(x, D),p(x) = (27r)
n
r el(=.f)a(x,
)(P(C) d
defines a function a(x, D)So E S. and there exist constants N E Z+ and Ck for k E Z+ depending on a such that Ia(x, D)wlk < CkIwik+N (continuity property). PROOF
Since St C S'n for £ < m, one can assume that a E S2,n for some
m E Z+. Then wp E S implies cp E S and one can write l a(x, D)w(x)I <_
(27r)-n r
which shows (cf. Lemma 1.3) that a(x, D)cp is bounded with (a(x, D)cplo < then la(x, D)cwlo < Colcpl N with N = 2m + 4n in view of Theorem 1.8. Moreover, one gets O) (a(x,
a(x, D)(a,w)(x) + (0 ,a)(x, D)Ax)
by differentiating under f, and also
xi(a(x, D)V(x)) = a(x, D)(xicp)(x) + i(Ot,a)(x, D)cp(x)
47
Pseudodiferential Operators
48
by integrating by parts. Thus, x"Oa(a(x, D)cp(x)) can be written as a linear combination of terms (bid{a)(x, D)(xa-6OQ-7cp)(x) so that a(x, D)cp E S I with Ia(x,D)plk !5 The next step consists in proving that the operations of adjunction and composition of operators a(x, D) correspond to the operations introduced in Section 2.3. THEOREM 3.2
For any aand bES°°and 'pand 1P ESone has (a*(x, D)p,
(i) (ii)
t) = ( a(x, D)V,) ,
(a#b(x, D)cp, bb) = (a(x, D)b(x, D)cp, ).
PROOF The quantity Io = (a' (x, D)cp, ) is equal to the oscillatory integral Io = (21r) -2n
=
f
y,
(f
/ e,((=,e>-(=-=.e-<>)a(z,
(2T)-2n
- rl) dy d7l ) 0(e)0(x) dx dl;
dx Adz d(.
Similarly, Io = (a#b(x, D)cp, 0) is equal to the oscillatory integral 1o
=
(27r)-2n
e=(t=
o)b(z, )c3() {x) dx d dz do.
On the other hand, the quantities 1' = (
I' =
(22r)-2n
f Ab)
(Jei> (Je(z() (fe'(x)dx) doJ dz) d I" = (27r)-2n
f
(Je"a(x,) (fe>
(fei'
We thus have to prove IS = I' and 100 = it.
b(z,
4) dz) do) f(x) dx.
Action in S and S'
49
Here we will provide the proof only for * since it is a good exercise for the student to write the details for #. First, we write that Io is the limit for e 0 of the integral IE _ (227r)-2n
t J where x E S can be chosen so that x = I in B1. Then 1' - I. = IE + IE + IE with
IE =
(21r)-n
fe
(l
- X(e)X(Ez))a(z, D)O(z) d. dz,
x(E())0(()d4 dzd(,
1, = (27r) -2n IE = (21r) -2n
ei((=,()+(_,()-(z,())g()a(z,
J
()x(e )X(Ez)x(E()
(1 - x(ex))(x) Adz d(dx. The integral IE tends to 0 with a by dominated convergence. The integrals IE and IE also tend to 0 with e, thanks to the following result. I
LEMMA 3.3
Let a(x, y) E A'n(Rn x RP), cp a real-valued function, and x, -0, and w E S with x = 1 in B1. Then dx dy = 0.
y)w(Ex)(1 -
lim I E---o
PROOF The change of variables z = ex gives
f
ei'P(=/(,Y)a(z/E,
y)w(z)(1 - X(Ey)) (y)e-n dz dy,
then a(z/e, y) is estimated by IIIaIIIo(1 + Iz/EI2 + Iy12)m/2 <
IIIaIIIoe-m(1 +
IZI2)m/2(1
+
IyI2)m/2.
For y E supp (1 - x(ey)), one has IyI > 1/c, and this gives 11 + IyI2 1
1 +p
m-n-p
)
<
IyI2)-
Pseudodifferential Operators
SO
for Y E supp (1 - X(ey)), so that finally Y)a(z/E,y)w(z)(1 - X(Ey))'+G(y)C' I C EplllailIoC,0(1 + jz12)m12lw(z)I(1 + IyI2)-
which gives the result after integration (cf. Lemma 1.3).
1
Theorems 3.1 and 3.2 allow us to extend the operator a(x, D) : S - S as an operator from S' into S' as follows. DEFINITION 3.4 Given an a E S°°, we call pseudodifferential operator of
symbol a the operator a(x, D) : S' - S' defined by (a(x, D)u, cp) = (u, a*(x, D)p)
for u E S', ,p E S.
If a E S", a(x, D) is said to have order m. The set of pseudodifferential operators of order m will be denoted by W'. The set of all pseudodifferential
operators is q" = U,,,%P m and the elements of-°° = fl,,,4m are called smoothing operators (because of the result in Corollary 3.8).
for is that of A' (D), which was defined in Section 1.3 by the formula a"'(D)u = At u. Indeed, ProposiA simple example of a pseudodifferential
tion 1.20 shows that this operator is the same as the pseudodifferential operator
of symbol A'° as defined in Definition 3.4; of course, Am(D) has order m. Another example is that of differential operators with coefficients a,, E H°O: the student will benefit by showing that in this case the definition of a(x, D)u through Definition 3.4 coincides with the usual definition (through differentiations and multiplication by coefficients as defined in S'). Even if these pseudodifferential operators are very similar to the differential
operators, one must take care that they do not share all their properties. In particular, this is the case for the local property (control of supports): indeed, in general, pseudodifferential operators do not satisfy
supp (a(x, D)u) C supp u
for all u E S'.
(One can even prove that the only operators a(x, D) E W°° possessing this property are the differential operators; cf. Exercise 3.1.) However, it is clear that if p E C°° satisfies p = I in a neighborhood of supp u, then one has a(x, D)u = a(x, D) (cpu) = a#W(x, D)u, and it follows from the asymptotic formula for the operation # (Theorem 2.7) and from Lemma 2.2 that a#cp = b+r with supp b(x, D)u C supp xb C supp cp and r E S-O°. Thus one will keep some control on the supports provided that the computations are achieved modulo smoothing operators.
Si
Action in S and S'
Instead of the local property, we will see in Corollary 3.8 that pseudodifferential operators satisfy the so-called pseudolocal property (control of singular supports), i.e., they do not increase singular supports:
sing supp (a(x, D)u) c sing supp u
for all u E S.
Actually, we will eventually show (in Section 4.2) a more precise result, namely the microlocal property (control of wave front sets), which implies the pseudolocal property. Before beginning the next section, let us show that the rather implicit Definition 3.4 allows us to carry out explicit computations: formula (i) below will show that one can recover the symbol of a pseudodifferential operator from the
operator itself (thus the correspondence between S°° and 'P°° is one to one), while formula (ii) extends Remark 2.8 to the case of polynomials in x. Indeed, if E(aI
b(x, D)u = E xa(ba(D)u)
for u E S'
Ial
since multiplication by polynomials is well defined on S'. Formula (ii) below then means that for such an operator and any a E SOD, a(x, D)b(x, D) = c(x, D) where c is formally given by the expansion of a#b obtained in Theorem 2.7 with only a finite number of nonzero terms.
Example 3.5 (i) Let a E S°°. Then, for any fixed
E R, one has
a(x,D)e'(x,f) =
a(x,S)ei(x,f>.
(ii) For any polynomial p(x), a E S°°, and u E S', one has
Dzp(x)(O a)(x,D)u.
a(x, D)(p(x)u) PROOF
(
a
(i) We first compute a* (x, D)cp for cp E S as follows. i, a* (x, D)cp)
=
a* (x, D)cp) = (a(x, D)'i', cp)
for ' E S,
and since the Fourier transform of r/' is (2rr)nm , this is also equal to
r (ir -n J et(x,')a(x,
e)(2ir)'
de) ;P(x) dx
=
f(e)
(fea(x)(x)dx)
dC
52
Pseudodiferential Operators
f e'(x,{>a(x, by Fubini's theorem, so that a (x, orem 1.6(ii)). Then one has for every W E S
(a(x,D)e'(=,f>,v) =
dx (cf. The-
(x, D)(x) dx
Je'a(x)(x)dx
= a* (x,
which proves (i).
(ii) Using the linearity of the operator a(x, D) it is sufficient to prove the formula for p(x) = xR. First, for p E S,
xpa'(x,
(27r)-n
f
(21r)-"fe(_D(a*(x,())d
( l ((-DF)"a')(x,
= (27r)-" J e'(2,E>
\
a
J
a! (Dt a)"(x, since
(aaJ xR /
=
CEO xR-
Then we get
(a(x, D)(x'u), cc) = (u,?Ra*(x, D)c) _
a! (u, (DE a)"(x, D)((ol x' )c))
_ Ea a! ((a°xI)(DF a)(x, D)u,'p) which gives the result, since the factor (-i)I°I can be transferred from the D{
to the 8.
3.2
1
Action in Sobolev spaces
For the definition and elementary properties of Sobolev spaces, we refer to Section 1.3. In particular, we showed in Corollary 1.19 that differential operators
of order m with coefficients in HOD map continuously H' into H'-' for any s E R. This property can be extended to any pseudodifferential operator of any
order m E R
Action in Sobolev spaces
53
THEOREM 3.6
Let a E S"`; then for every s E R there exists a constant C, such that a(x, D)u E H'-m for all u E H', with IIa(x, D)ulla-m < Ca Ilufl8. PROOF Assume first that we have proved that for any b E So there exists a constant Cb such that for all r' 1E S.
IIb(x, D) 'IIo <- CbIIV IIo
Then, if a E S' and s E R, we can set b = A-'#a'#A'-' E So, and using Proposition 1.20, one gets for W E S Ila* (x,
IlA-`(D)a'(x,
IIb(x,
< CbIlAm-'(D)WIlo = CbllCppll
o
-s
so that for all u E H' and still p E S I(a(x,D)u,')I = I(u,a'(x,D)W)I <- Ilullslla*(x,D)VII
Thanks to Proposition 1.15, this implies that a(x, D)u E H'-m with IIa(x, D)ulla-m S Cbllulls as required. Thus, all we have to prove is the estimate IIb(x, D)0110 S CbIIV'IIo, t' E S, for any b E So. This will be done in three steps: first when b E S-n-', then when b E S' for some m < 0, and finally in the general case b E So. (i) Operators of order -n - 1. Let b E S-"-' and t' 1E S. One has (21r)-n J e"x,l) b(x, f)tG(e) df
b(x, D)tP(x) =
= (21r)-" J e=(x-v,N(x, )V)(y) dy 4
=
f
K(x, y)',(y) dy
where
K(x, y) = (27r) -' f
e'(x-u,f)b(x,C) dC
since all these integrals are absolutely convergent thanks to the condition b E
S-"-' (cf. Lemma 1.3). This kernel K(x, y) satisfies for any a E ?+
I(x - y)QK(x,y)I = (27r)-" J(x -
(2a)-" t ei(x-v,f)
y)°e`(x-v,E)b(x,
0A
b(x,t)dal
Pseudodiferential Operators
54
after integration by parts since A"+1+I'IOb is bounded. Therefore, one has (I + Ix - yi2)" I K(x, y)I < C/ir" for some constant C, so that
f IK(x,y)Idx
and
The result then follows from the next lemma, known as Schur's lemma. LEMMA 3.7 SCHUR'S LEMMA
Let K(x, y) be a measurable function defined on R" x lR" and satisfying
f
I K(x, y)I dx < C
and
f
for some uniform bound C. Then for any IP E L2, one has K(x, y)yP(y) E L' (dy) for almost every x, and the function p(x) = f K(x, y)zb(y) dy is square integrable over R' with C110110 PROOF The function K(x, y)Vi(y)K(x, z) ,b(z) is integrable over R" x R' x R1 since 1,0(y)V(z)I < 1(It'(y)I2 + 1,0(Z) 12) and
Jk(y)I2 (JIK(xY)I (JIK(xz)Idz) dx)
2
00.
dy <
Thus, by Fubini's theorem, K(x, y)z'(y) E L'(dy) for almost every x and
f Ip(x)I2 dx =
1(1
K(x, y)V(y)
dy)
U
K(x, z)''(z)
dz) dx
< C111,0112
0*
(ii) Operators of negative order. Since U,
S-1/2k,
for all k E Z, and this is done by induction on k. Indeed, this is true for some
negative k, say for k = -n, according to step (i). Then, if b E
S-'/2k+,,
b*#b E S-;12k, and one can write IIb(x, D),O112 = (b(x, D),O, b(x, D),O) = (b*#b(x, D)O,,O)
< IIb*#b(x,D)'IIo11'IIo 5 by using the recurrence hypothesis, so that IIb(x, D),0IIo <- CbIIiPIIo with Cb = 1/2
(iii) Operators of order zero. Now if b E So, b is bounded and IbIo - Ib12 E S°
is nonnegative. We can choose a function F E C°°(C) such that F(z) _
-
(1 + z)1/2 for z E R+, and it follows from Lemma 2.1 that a = (1 + Ib1o IbI2)'/2 = F'(Ib1o - Ib12) E So. One has a* = a and a*#a = Ia12 modulo S'1, then
a*#a + b*#b = 1+ Iblo + c
for some c E S-'.
55
Action in Sobokv spaces
Thus one can write Iib(x, D)iIIo < IIa(x, D),PIIo + IIb(x, D)'IIo = ((a'#a + b'#b)(x, D)?P, 0)
((I +
(c(x, D)1, VG) <- (1 + Iblo + CC)Ik1'IIo
since c E S-' satisfies IIc(x, D)-+'I10 <- CeII'IIo, thanks to step (ii).
I
As a consequence of Theorem 3.6, we can show that operators of order -oo are smoothing (this explains the terminology used in Definition 3.4), and that pseudodifferential operators do not increase singular supports (the so-called pseudolocal property). Since elliptic operators (i.e., operators with elliptic symbols) are invertible modulo smoothing operators, they do not decrease either singular supports. This latter property is called hypoellipticity. COROLLARY 3.8
If a E S-°°, then a(x, D) maps E' into S and S' into P; similarly, if a E S°°, then a(x, D) maps S into S and P into P. Finally, any pseudodifferential operator a(x, D) E 41°° has the pseudolocal property sing supp (a(x, D)u) C sing supp u
for all u E S',
and one even has sing supp (a(x, D)u) = sing supp u if a is elliptic. PROOF For a E S-°° and u E E' one has u = i4u for some & E Co (provided
that * = 1 near supp u) from which u E H-N for some N E Z+ thanks to Lemma 1.16. From Theorem 3.6 we then get a(x, D)u E H" which contains only bounded continuous functions (cf. Proposition 1.14(ii)). Using the same argument as in the proof of Theorem 3.1, we can now write x'BA(a(x, D)u) as a linear combination of terms (8T8{a)(x, D)(x°-b81-yu) which are bounded continuous functions for the same reason, and thus we get a(x, D)u E S. If now u E S', one can write
D' (a(x, D)u) = ba (x, D)u
where ba = r°#a E S-O°.
Thanks to Lemma 1.16, there exists an N E Z+ such that v = (1 + IxI2)-Nu E
H-N. Thus, ba(x, D)u = ba(x, D) ((I + IxI2)`vv)
(DO (I + IXI2)N)(e, b,,,) (x, D)v
(cf. Example 3.5(ii)), and one still has 8{ b(k E S-°° for all a and,3. It follows that (&ba)(x, D)v is a bounded continuous function, since it is in H" thanks to Theorem 3.6, from which Da(a(x, D)u) = b,, (x, D)u E P°.
Then, if a E S', Theorem 3.1 shows that a(x, D) maps S into S. If u E P, we first remark that \2k(D)u E P° for any k E Z+, since a2k(D) is then a differential operator with constant coefficients. Therefore there exists
56
Pseudodifferential Operators
an N E Z+ depending on k such that v = ( 1 + I x t 2) - N A2k (D) u E L2 (cf. Lemma 1.3). Thus, for any fixed a E Z+, we take k such that 2k > m + Ica I + n,
which implies that ba = fa#a#A -2k ES-". We then write D"(a(x, D)u) = ba(x, D) A 2k(D)u = b,, (x, D)((1 + Ix12)NV) (DO(1 + Ix12)N)(O ba)(x, D) v.
Here, we have 4ba E S-' and v E L2 = H°, so that (Obn)(x, D)v E H", thanks to Theorem 3.6, and we can conclude as above.
For the pseudolocal property, let a E S°°, U E S', and set St = R" \ sing supp u. Then 'Ou E Co for all ' E Co (1l), and for any can find a V) E Co (S2) with = I near supp ;p and write
E Ca (SZ) one
cpa(x, D)u = cpa(x, D)('u) + cpa(x, D)((1 - iP)u).
The first term is in S since 7Pu E Co C S, and the second term can be rewritten b(x, D)u with b = cpa#(1 - P) E S-'°, thanks to the asymptotic expansion for the operation # (Theorem 2.7) since the supports of cpa and 1 - iP
do not meet. We thus get cpa(x, D)u E C'° for all for all ICS > 1/E and some E > 0 (cf. Theorem 2.10(iv)). Since on ICI < I /f a is bounded (from above) and A2ni' 1 is bounded from below, this assumption clearly implies Re a + Coa'-'"-1 _> EVrit everywhere for some large constant Co, and conversely if this latter estimate holds, Re a can be proved to satisfy an elliptic type estimate with a smaller f > 0. In our statement we will therefore use this more convenient form of the assumption. THEOREM 3.9 GARDING'S INEQUALITY
Let a E S2m and assume that for some Co and c > 0 one has Re E)I2m (assumption satisfied in particular when Re a(x, C) > I /E). Then for any N > 0 there exists a constant CN such that 2Re (a(x, D)V, gyp) > EIIV1122 - CNIIc I12 _N
a+CoA2m- i >
for 1 I >
for all cp E S.
Action in Sobolev spaces
57
Let us set b = \-m#a#a-m E So. Since b = A-2ma modulo S-1, the assumption on a implies that Re b + (Co + CI )A- I > for some constant C1, so that b itself satisfies the assumption in the theorem with m = 0. If we assume momentarily that the result is proved when m = 0, we can write for ,p E S PROOF
2Re (a(x, D),p, ,p) = 2Re (b(x, D)Am (D),,, A' (D),p) ?EIIA"`(D)WIIO-CNIIA'(D)wII? N=EIIwIIm-CNIIwII,,,-N (cf. Proposition 1.20). Therefore, it is sufficient to prove the theorem in the case
m = 0. Thus assume a E So with Re a + CoA-' > E. We can choose a function F E C°°(C) such that F(z) = z)1/2 for z E lll, and since 2(Rea + CoA-' - ) E So is nonnegative, it follows from Lemma 2.1 that b = (2Re a + F(2(Rea + CoA-' - E)) E So. We can write modulo S-1: b*#b = 2Rea - (3/2) = a + a* - (3/2)E, which implies 2CoA-1
a+ a*
for some cE S-1.
Then for w E S,
2Re (a(x, D),p,,p) = (a(x, D),p,,p) + (,p, a(x, D),p) = ((a + a')(x, D)w, w) = (b"#b(x, D)w, V) +
(23
'Y' 1P
+ (c(x, D)w, V)
> Ilb(x, D)wll0 + 2EIlwll0 - II c(x, D)wlII/211wI1-1/2 >_ Ellwll0 + (211w110 - C112114' 1/2)
for some constant CI/2 since c E S'. Thus the result will finally come from the estimate CI/2IIwIL1/2 !5 211wII0 + CNI1wII2-N
with CN =
(2C)2N
which can be proved as follows: when CI/2A-V) > /2, one has then
C1/2A-V) = so that CI/2A-1 <
+G,NA-2hr
from which one gets the estimate after multiplication by I,p12 and integration. I
58
Pseudofifferential Operators
To end this section, we simply point out that the result of Theorem 3.9 is still true with N = 1/2, when a is replaced with 0 in the assumption. The proof of this stronger version, known as the sharp Girding's inequality, does not require more theory than what we have here, but it is too long and technical to be given in this elementary course. Thus, we simply refer the interested reader to Hdrmander [8, Theorem 18.1.14]. Even sharper estimates are due to Melin and to Fefferman and Phong (see references in the Notes to Chapters 2 and 3).
3.3
Invariance under a change of variables
The first obstacle one meets when one wants to prove the invariance of this theory under a change of variables is the lack of invariance of the spaces S, S. and H8 where everything was done up to now. However, these spaces are clearly invariant under a linear change of variables. In this framework, let us introduce the following notation: if X(x) = Ax + b is an invertible linear change of variables in R',' and if V E 8, one defines the transform V. E S of ep under the change of variables X by the formula px(y) = 'p o X-'(y) = cp(A-'(y - b)). When performing the change of
variables y = Ax + b in the integral, one finds that for all V and /i E S. (W,,, ,O) = where IX'I = IdetX'I = IdetAI (also denoted by JAI) and z/ix-. (x) = V, o X(x) = ry(Ax + b). Therefore, if u E S', we define a
distribution ux E S' by the same formula (ux, cp) = (u, JX'Jcpx-. ). Example 3.5(i) allows us to guess what the symbol of the transformed operator will be. Indeed, one must have ax(y,1J) = e-'(y,'')ax(y, D)e'(y"n), and this leads to the following computations:
ax (y,rl) _
(e-'(Az+b,n)a(x,D)ei(Ax+b,q)) Ix=A '(y-b) (e-t(a,A7)a(x,
D)e'(x,`Av7)llx=A-'(y-b)
=
a(A-'(y - b), Arl)
And, indeed, one can state the following proposition. PROPOSITION 3.10
Let X(x) = Ax + b be an invertible linear change of variables in R. Then a E S'° if and only if ax E S"' where ax is defined as ax (y, r!) = a(A-' (y - b), Arl) = a(X-' (y), tX'rl) 'This only means that the matrix A is invertible. Likewise, we say that X : n 0x is an invertible C' change of variables to mean that x is one to one, indefinitely differentiable, and that its Jacobian matrix X' is invertible at every point of Q.
Invariance under a change of variables
59
Moreover, one has for u E S'
(a(x, D)u)x = ax(y, D)ux PROOF We have eA(77) < A(%,7) < A(71)/e for some e > 0 depending only on the matrix A. Since the derivatives of ax are equal, up to multiplicative constants, to the corresponding derivatives of a at the point (A-' (y - b). Ari). we thus get the equivalence a E S' ax E S"'. For cp E S, one can write - '(y,>>)cp(A-'(y - b)) dy
cPx(77) =
= e-i(b.77) f e-i(x,An)W(x)IAI
dx = e-"(b,n)43(Arl)I AI
through the change of variables y = Ax+ b. Similarly, the change C=%? gives
ax(y, D)cox(y) = (27r)-" f e'(y,n)a(A-'(y - b),
= (27r)-" f
ei(A-'(y-b),t)a(A-'(y
Arl)e-'(b.,,)
(All)IAI dr7
- b),d
= (a(x, D)cp)x(y) Taking the scalar product with a V) E S, we get the identity a* (x, D)(z'x-,) _ ((ax)*(y, D)1i)x-c, which gives ((a(x, D)u)x+'f') _ (a(x, D)u, IAI x-c) _ (u, I AI a* (x, D)('x-c ))
_ (u,IAI((ax)*(y,D)'+b)x-c) _ (ux,(ax)*(y,D)'+b)
_ (ax(y, D)ux,'c') The case of a nonlinear change of variables is much more intricate. Indeed, since such a transformation is usually only locally defined, we first have to define
the local action of a pseudodifferential operator if Q is an open set of R" one says that a E Sa(S2) if a is defined on fl x R" and satisfies cpa E S' for any cp E Co (fl). Then, for a E Sa(Sl), cp E Co (SZ), and U E S', the distribution cpa(x, D)u E S' satisfies supp (cpa(x, D)u) C supp cp, and therefore one can define an operator a(x, D) : S' - D'(cl) by the formulas (a(x, D)u, cp) =
D)u, 1) = (u, (<pa)If=O)
for cp E Co (Sl)
where the last equality comes from Example 3.5(i). When restricting this operator to e'(Sl) C S', one gets an operator a(x, D) : V(fl) D'(S2) for which a good theory of invariance under a change of variables can be expected. Indeed, if X : fl --+ S2x is an invertible C°° change of variables, one defines
cpx E Co (lx) by the formula cpx(y) = cp o X-'(y) for any cp E Co (fl),
Pseudodiferentia/ Operators
60
then, as above, uX E D'(1 ) by the formula (uX, cp) = (u, for any u E D'(1), where IX'I = I det X'I is now a C°° function. Again, in view of Example 3.5(i), a natural guess for ax would be a,, (X(x), rl) = e-i(X(=).n)a(x,
but this formula does not mean anything since e'(x(2),7) is defined only on SZ and does not have a compact support. Thus, to get a good definition of ax, we will have to modify this expression by introducing cut-off functions. A theory of oscillatory integrals, quite similar to the results given in Section 2.2 but more general and known as the stationary phase method (or formula), is the key of the following result, the proof of which is out of reach in an elementary introduction such as ours (we refer the reader to Hbrmander [8, Chap. 7 and Theorem 18.1.17]). THEOREM 3.11
Q. be an invertible C°° change of variables where Sl and Q. are two open sets of R". Then a E S. (Q) if and only if (a#cp)X E Sa(QX) for all V E Co (Q), where we set Let X : Sl
(a#W)X(X(x), n) =
e-i(X(2),7)a(x,
D)(o(x)e`(X(Z).n)).
Moreover, for any u E E'(1) and cp E Co (St) such that cp = 1 on supp u.
(a(x, D)u)X = (a#,)X(y, D)ux Finally, one can even write an asymptotic expansion for (a#cp)X which shows in where
particular that (a#cp)X(y,q) = (a#cp)(X-'(y),'X'rl) modulo S tX' is taken at the point X-' (y).
Even if the proof of this theorem is difficult in its full generality, there is at least one situation where it can be written through elementary computations:
when a(x, D) is a differential operator, i.e., when a(x, ) is a polynomial in . The student is invited to provide, in this case, the details of the proof (which only makes use of the chain rule). Such a result on the effect of a change of variables naturally leads to a theory
of pseudodifferential operators on a manifold M of dimension n. Since we cannot provide here the basic definitions of analysis on manifolds, we simply remind the student of some of its features: if p and ' are two C' functions 8v'i of defined near a point m on the manifold M, then the equality their gradients at m for a choice of local coordinates y implies the same equality O2V = a2' for any choice of local coordinates x. The chain rule even gives the m o r e precise result O (cp o X) (xo) = tX'(xo)(8yW)(yo) (here xo and yo = X(xo) are the coordinates of m in the two systems), which also shows that the vector space structure of gradients at m of C' functions is preserved after a change of variables. This intrinsic vector space of gradients at m of C' functions, T,, M,
Exercises
61
is called the cotangent space of M at m (it is naturally the dual space of the tangent space at m), and the set of all (x, ) with x E M and E Ti M is called the cotangent bundle over M and is denoted by T'M. In the construction of a theory of pseudodifferential operators on a manifold M, the symbols will be functions defined on the cotangent bundle T' M, but the asymptotic expansion for (a#W)x (see Theorem 3.11) suggests that a symbol
a will be intrinsically related to the mth order operator a(x, D) only modulo Sr'. In particular, in the case of polyhomogeneous operators, the principal symbol is a true function defined on T*M, but the following terms are not invariant under a change of variables. This phenomenon is very easy to check on simple examples of differential operators, and we recommend these computations as a useful exercise (take for example a(x, D) = D1 - D2 and y = X(x) defined by y, = x,, 112 = X2 + (xI/2)). To go a little bit further in this theory, we should add that the next problem is then to define the compound of two pseudodifferential operators with symbols a and b E Sa(S1). Indeed, since a(x, D) and b(x, D) are operators from V(Q)
into D'(S1), one cannot in general define a(x, D)(b(x, D)u) for u E P(Q). Thus, the operation # will be restricted to a subclass of Sa(1l), namely, the subclass of operators that map V(Q) into E'(1l). The use of adjoints will then lead to a smaller subclass of operators, the so-called properly supported operators, which can be extended as operators from V(Q) into D'(l), and the necessity of dealing only with properly supported operators is balanced by the fact that any operator in Ik'10C(f1) is equal to a properly supported operator modulo 'i Therefore, the counterpart of the use of adjoints and compounds will just be to achieve the calculations modulo smoothing operators. Actually, as far as we are interested only in local properties of solutions of partial differential equations, it is probably simpler to substitute a practical use of cut-off functions in place of these theoretical constructions, and that is the point of view we will take when presenting applications in Chapter 4.
Exercises 3.1
Remember that a(x, D) is said to have the local property if supp (a(x, D)u) C
supp u for all u E S'. The goal of this exercise is to prove that differential operators are the only pseudodifferential operators possessing this property. (a) Let a E S-" such that a(x, D) has the local property. Let t G E S and xp E III" ; for a fixed function 1p E Co such that V = 1
in B1, set fork E Z+ tPk(x) = (1-cp(k(x-xo)))iP(x). Using Proposition 1.14(ii) and Theorem 3.6, show that la(x, D)(r& - v'k)Io < C111y -'kIIo for a constant C independent of k E Z+. Then prove that a(x, D)r/ik(xo) = 0 for all k and finally that a(x, D)i(xo) = 0. Show that a(x, D)ri = 0 for all tp E S. (b) Let a E S' for some m < k E Z+ such that a(x, D) has the local property.
Pseudodifer+ential Operators
62
Show by induction that for any a, p E Z+, (c7z 3 a)(x, D) has the local property. Similarly, show that a' (x, D) also has the local property. For E S, write a Taylor formula up to order k+n, then use Example 3.5 and question (a) to prove that a(x, D)'lb(x)
r
a(x, 0)D°,r(x)
fal
3.2
Show that the symbol a is a polynomial in (or in other words, that a(x, D) is a differential operator). The elliptic estimate. Throughout the exercise, a E Sm will be assumed to be elliptic. One wants to prove that there exist constants C, and CN,8 such that u E H-N and a(x, D)u E H' imply it E H'+m with the estimate flulls+m < C.11a(x, D)ulla + CN.,II uII -N.
Questions (a) and (b) provide two independent proofs, while in question (c) it is
proved that a(x, D)u E H' no longer implies that it E H'+' if one does not assume u E H-0O (but of course one still has it E HL m). (a) In this question one proves only the estimate when it E S since the general case follows from this one by means of standard techniques of approximations (cf. Exercise 3.8(a) for example). Show that the symbol a'#a2'#a satisfies the assumptions in Theorem 3.9, and show that this implies the elliptic estimate for all it E S. (b)
By using the inverse a" of a, show that it E H-N and a(x, D)u E H'
imply u E H'+m with the elliptic estimate. In this question, it is any temperate distribution. Show that a(x, D)u E H' implies it E H'+m and that the elliptic estimate holds with CN,3 = 0 if a has an exact inverse b (i.e., b#a = 1). However, show that the Laplacian operator 0 = Fn _I has an elliptic symbol of order 2, that the function I is not in H-°°, and that A has no exact inverse. Another proof of continuity in Sobolev spaces. The major part of the proof of Theorem 3.6 was devoted to establish an inequality IIb(x,D)VGIIo 5 CbI) 'IIo. 1' E S, for any b E S°. The present exercise provides a different proof of this fundamental result. E S. (a) One sets XE(x) = f -,(1 (c)
3.3
+then, for
J Show that if E L2(lRIn) with II'I'IIo = 2-"/'(27r )n110110. ''(x, ) =
XE(x - y)'tl'(y) dy.
Lemma 1.17 to write f
(Hint: Use
as an integral involving Xo and Vii.) 41(x, )) E Using the same method, show that for a E {0, I in, M. L2(R2n) with a norm equal to 2-(n+i°1)/2(27r)nI[ IIo. (r1n ,(1 +OE,)) 4Y(x,e) = Show that Similarly, for V E S, show that the function
(2ir) " / x-:( - 17)0(77) d+ satisfies 7 E L2(1R2") with 11 0110 = 7rnJ2IIWIIo and
C
(1 - a=i) I a /
(x, e) =
W(x).
63
Exercises
(b)
0,6b is bounded for all a, 0 E {0,1 Let b E C2' (R211 ) be such that (any symbol of order 0 satisfies such an assumption, but also symbols taken in much wider classes, as the So,o studied in Exercises 2.9 and 3.6). One sets for gyp, V) E S
(b(x,D)tb,,p) = (2ir)'" J Use results from (a) and integrations by parts to show that (b(x, D)ip, cp) is also equal to
(2tr)-" ` (-l)I'
` /f J Ca
rt,/3,yE{0,11^
Conclude that b(x, D)1 E L2(R,) with an estimate Ilb(x, D)tlll0 <- CbIJV'IIo
where the constant Cb will be computed explicitly. 3.4
An extension of Schur's lemma. Assume as in Lemma 3.7 that K(x, y) is a measurable function defined on R" x R" and satisfying
I K(x, y)l dx < C and
J
E Lp(R") (where I < p < oc)
for some uniform bound C, and define for
1P(x) = J K(x, y)t4'(y) dy
whenever K(x,y)tp(y) E L'(dy), and
oo otherwise. Show that
I(x)I C (JIKixv)iIYw'dY)
l/p
and conclude that cp E LP with NormLp (gyp) < C NormLP (w). (Remark in addition
that for p = I you need only the assumption f I K(x, Y) I dx < C, while only f I K(x, Y) I dy < C is needed when p = oo.) REMARK
This exercise thus shows that the operators a(x, D) are bounded on
the Lebesgue spaces LP when a E S-"-'. One can even prove that this is still
3.5
true when a E S° and p E (1,001(cf. Coifman and Meyer (6, Theorem 9, p. 38]), but the proof is much longer. L2 continuity for certain symbols decreasing at infinity in x. Assume that b E
C"+' (R2i) satisfies f li b(x, )I dx is bounded for all Ial < n + I (remark that no smoothness is assumed with respect to t), and let ' E S. Show that b(x, {) c({) E L' (dC) for almost all x E R" , and that f ei(z.f)b(x, b(x, D)tIc(x) =
)t 4
(21r)-n
defines a function b(x, D)V) E L' (R"). Show that b(x
(tl) =
f
K(r1,
dC
will be written explicitly.
where the kernel Show that .\n+1
-17)K(q,.) is bounded and that ((
J IK(t1,C)Id71
Pseudod{,ferential Operators
64
for some uniform bound C, and conclude that b(x, D)* E L2 With IIb(x, D)iO IIo < C,II'IIO. 3.6
Continuation of Exercise 2.9. (a) The S,7,0 calculus. For any symbol a E S,7,, (classes of symbols defined in
(b)
Exercise 2.9), one defines an operator a(x, D) E if' o by the formula given in Theorem 3.1. Extend the results of Chapter 3 up to Theorem 3.6 and Corollary 3.8 to these classes of operators. Hypoellipticityy. Let a and c E Sp o be such that c#a - I E SP°. For any 'p E C.1, choose a ' E Co such that ?P = I on supp gyp. Show that there exist b E SPo and r E SS,°° such that 'p = (b#r/i#a) + r, then prove that a has the property of hypoellipticity:
,
a(x,D)uES'nC°°
uES'nC°°.
.
More precisely, prove sing supp (a(x, D)u) = sing supp u for any u E S'. (c)
,
Local solvability. When a has a right inverse (i.e., a#b - I E S,-.' for some b E SP o). then the equation a(x, D)u = f is locally solvable. This last result will be proved in Exercise 4.1.
3.7
Converses of Theorems 3.6 and 3.9.
(a)
"Microlocalization" at (y, q). Let a E S' for some rn E R and 'p(x) _ (2n)-"/4 then simply writing ) for .S(ri), set p,.,,(x) = V((x e-I=I2i4,
Compute
then show that
and
a(x. D)'p,,.»(x) =
e'(:.n)(ba.,,(z,
D)P(z))i:_(Z-v)a'n
where br,,, E S"' is given by b,,,,, (z, () = a(y + A-1J2z, )? + Coming back to the notation \(77) instead of A, show that for any s E 1[P,
a°(17 + /2(i)!)
(treat the situations 12171 ? A'/2(1))I(I
and 12771 < a'/2(;7)I(I separately), then prove that b,.,,(z,() = a(y,ri) + cv,,,(z, () where c,,.,, satisfies an estimate
\(Z))
ICN,7(Z, ()I
uniformly for y E R', and finally conclude that (a(x, D),pv,,,,'py.,,) = (a(y, n) + r(y,
17))11
vb.n I10
where the function r satisfies an estimate Ir(y,ri)I < CA--(1i2)(77) uniformly for y E R". (b) Applications. Let a E S2' and assume that for some f > 0 and b > 0, 2Re (a(x, D)'p.'p) > CIIV112 - CIpIIm_6 for V E S
(cf. Theorem 3.9). Show that b = (A-'"#a#A-"') + CA-26 satisfies 2Re (b(x, D)'p,'p) > EII'v1l2
for 'v E S,
and use results of part (a) to show that there exists a constant Co such that CoA'm-' > ,\2m /3. Re a + Let a E S'" and assume that for some s and e E R, IIa(x, D)'p1l° < CIkvII°+e
(cf. Theorem 3.6). Show that b = IIb(x, D)'pIIo <- CII'pjIo
for p E S satisfies for cp E S,
65
Exercises
3.8
+Am-t-(1/2)). Then and use results of part (a) to show that IA-CaI < C'(1 use results of Exercise 2.8(b) to conclude that A -'a is bounded and that a E St+ = nm>tSm Friedrichs's lemma and subellipticity. (a) Friedrichs's lemma. If V is a unit test function, p, will denote the symbol cp(el;)(E S-°°), and the corresponding operator c',(D) is called a Friedrichs mollifier. then show that u E Show that for it E H' if and only if II,p,(D)uII, is bounded uniformly fore E (0, 1). Using the proof of Example 3.5, show that for a E S' and ) E S,
(a*#cp,
- 0,#a')(x, D)'+b(x) = (2ir)-n
r e'(=-v.() (a
(x, ) - a(y, ))
dy d ,
and show that this quantity can be rewritten (b#O )(x, D)t1(x) +
(b)
fK(xY)l&(Y)dy
where b E S° and K, (x, y) _ (27r)-n f a+(x-v.() (a(x, 0 -a(y, fOK) A. Then use a Taylor formula and integration by parts to show that this kernel K, satisfies the assumptions of Schur's lemma (Lemma 3.7) with a constant C independent of e, and finally prove that if it E L2, cp,#a)(x, D)uII o is bounded uniformly fore E (0,1). Use Theorem 1.13 to show that supp u C BR implies that ;!,(D)u E C" with Supp (,p,(D)u) C BR+,. Subellipticity and hypoellipticity. A differential operator
a.(x)Do
a(x, D) _ aI<m
with complex-valued coefficients ao E H°° is said to be subelliptic at x° if there exists a neighborhood 0 of xo, a b > 0, and a constant CO such that
IIt1IIm-1+6 C Co(IIa(x, D)'Ilo + ft1IIm-1) for t' E Co (Q). Here, a(x, D) is assumed to be subelliptic at xo. Show that if w has a compact closure in fl, there exist constants (C,),ER such that for any s E R for zV E C0 00(w). IIYIla+m-1+6 !5 C,(IIa(x, D) 'II, + Use the functions 7P, = p,(D)(Xu), where 0,(D) is a Friedrichs mollifier as in part (a) and X is a cut-off function, to show that if it E H'+'4-' and
a(x, D)u E H' in some ball centered at xo, then it E
H'+--1+6 in
any
smaller ball.2
Show that if it E S' satisfies a(x, D)u E C° near xo, then u E C°° near xo (property of hypoellipticity). (c)
A subelliptic operator. For k E Z4., let a2k(x,C) = 1 + ix2 kl;2. For x1 (xik+' - y2k+1)/(2k + 1), then for t2 E R, and y E R define B(xi, y) = eB(x1,v)(2 if 52(x1 - y) < 0 K2(x1,y) _ 0'(z if {2(x l - y) ? 0,
2We say that u E H' in a ball B if there exists a v E H' such that it = v in B.
66
Pseudod(ferential Operators
and finally for 0 E S = S(W), set
(y,l;z) = J e-"={=,L(y,z)dz,
/ ?0(xi,t2) = J
and
K,O(xi,x2) = (27)-i / eix2fz,I0(xl,6)42.
J
Show that for any ?1' E S, Kii is aCOO function and -0 = Ka2k(x, D) V,. Show that there exists a constant C such that
f IKe2(xj,y)I dx1 <
and
where 6 = 11(k + 1). Similarly, show that
fxlK(xiY)Ida
i
<_ CI&I
and
/ J
dy 5 Cj6l-
Show that for any r/' E S,
J
(1 +
then show that
d < C J (lazk(x, D)v(x)I z + IVG(x)I2) dx, L'(x1,£z) =
f iI+G( )Izd{
f
and (lazk(x,D)V,(x)Iz+IV,(x)J2)dx,
and finally conclude that there is a subelliptic estimate for azk(x, D). Use the function u(x) = ((x,k/2k) + ixz)31z to show that the operator azk_I(x, D). where CI +ix2k-16, is not hypoelliptic.
Notes on Chapters 2 and 3
67
Notes on Chapters 2 and 3 The notion of the pseudodifferential operator has old roots: as early as 1927,
Weyl [69] suggested associating to any symbol a(x, ) an operator a(x, D) defined by a variant of the formula we used in the text (see Theorem 3.1). One could also quote Hadamard [38] for his finite parts of (singular) integrals and his elementary solutions of elliptic or hyperbolic second-order equations. However, the direct origin of the theory lies in the study, taken up by Mikhlin [58], Calderdn and Zygmund [27], Seeley [62], and others, of singular integrals occurring in elliptic problems. The consideration of an algebra containing both singular integral operators and partial differential operators then led Kohn and Nirenberg [48] (see also Hormander [41 ]) to the definition of pseudodifferential operators and the proof of their basic properties. These were essentially the operators with polyhomogeneous symbols as defined in Chapters 2 and 3. Eventually, the theory of pseudodifferential operators was extended in a lot of various ways. We now describe some of these extensions (with references), but we apologize for the incompleteness of the picture; the applications will be discussed in the Notes on Chapter 4. First, it was natural to adapt pseudodifferential operators to the analytic framework. This was done by Boutet de Monvel and Kree [21 ] and Boutet de Monvel [20], while an algebraic approach was developed by Sato et al. [60]. The theory presented in Treves [13, Chap. 5], equivalent to that of [211, makes use of the "analytic cut-off functions" of Andersson and Hormander (as in [46]). Another approach, avoiding the use of such cut-off functions but based on the methods of stationary (complex) phase, was initiated by Bros and Iagolnitzer [22] and developed by Sjostrand [63]. A link between these divergent points of view was established by Bony [19). The student must be informed that none of the references quoted here is elementary or easy to read. Another generalization led to the so-called Fourier integral operators. If the definition of O(C) is carried into the formula of Theorem 3.1, we can formally write
a(x, D)v(x) =
(21r)-"
f
dy A.
Fourier integral operators are essentially given by the same formula where th exponent (x - y, ) is replaced with a more general phase function 4 (x, y, ). The use of such a formula to represent solutions of hyperbolic Cauchy problems appears explicitly in Lax [49], but it has a long tradition that goes back at least to Poisson [59] . These operators were also used in Hormander [41] to prove the invariance of pseudodifferential operators under a change of variables, then by Egorov [32] to prove the same invariance under the more general canonical transformations. A systematic presentation of the theory, including the introduction of very general oscillatory integrals, is given in Hormander [45] and Duistermaat and Hdrmander [31]. The interested student will find a nice expo-
68
Notes on Chapters 2 and 3
sition in the lecture notes of Duistermaat [30] (see also Treves [68, Chaps. 6,8], Taylor [11, Chap. 8], and Kumano-Go [9, Chap. 10]). Further developments of the theory, allowing phase functions with nonnegative imaginary part, can be found in Melin and Sjostrand [56,57]. In [43], H6rmander enlarged the algebra of pseudodifferential operators to include the fundamental solutions of hypoelliptic operators of constant strength. This was obtained by allowing more general symbols, called symbols of type
p, 6, and the proof of continuity of these new operators was completed by Calderdn and Vaillancourt [25,26]. An elementary discussion of the LP and Lipschitz theory of pseudodifferential operators is given in Folland [34], and a systematic investigation of minimal conditions implying continuity is taken up in Coifman and Meyer [6], where nonsmooth symbols are considered (see also Stein [66] and Bony [3]). In another direction, the study of locally solvable and subelliptic operators led Beals and Fefferman [2], Beals [16], and H6rmander [47] to very general classes of (smooth) symbols which can be adapted to the operator being studied. One by-product of these generalizations is an improvement by Fefferman and Phong [33] (see also H6rmander [8, Theorem 18.6.8]) of the Girding inequalities (i.e., inequalities of the type considered in Theorem 3.9) previously obtained by H6rmander [42], Lax and Nirenberg [50], and Melin [55].
4 Applications
Introduction There are so many applications of pseudodifferential operators that it seems impossible to describe them completely. Books have been written on this subject (cf. [8,9,11,13]) but none claims to give a general description of all possible applications. Moreover, this introductory course was not intended to develop applications but only to describe the very basic elements of the theory itself. However, it would have been very artificial to drop the applications completely, since the main motivations of a theory lie in its ability to solve problems.
Thus we now present a few applications without pretending that they well represent the power of the theory. Our goal here is just to give examples of how to use pseudodifferential operators to solve problems of partial differential equations, and we hope that the very few results described will convince the reader of the great convenience of the whole theory. In this chapter we will change the style of exposition slightly to avoid overly long developments: we will give fewer details, more freely use results from functional analysis, and sometimes give only references instead of proofs. The first section is devoted to the study of local solvability of linear differential equations. The elementary result we present in Section 4.1 was essentially obtained before (sic!) Nirenberg and Treves [10] began to use pseudodifferential
operators to take up this problem. However, it is thanks to this tool that we are able to give here a short proof of it. In Section 4.2 we describe the bases of the study of microlocal singularities of solutions of partial differential equations.
This topic could be considered as a part of the theory itself rather than as an application, and it is certainly one of the main motivations of this whole construction. Finally, in the last section we illustrate these latter results in the study of the Cauchy problem for the wave equation.
69
Applications
70
4.1
Local solvability of linear differential operators
Consider a linear differential operator a(x, D) = a,(x)D° with complex-valued C°° coefficients aa. Remember that its principal symbol p(x, ) = a complex-valued COD function on T`R". We want to study the following property: the operator a(x, D) is said to be locally solvable
at xo if there exists a neighborhood Sl of xo such that a(x, D)u = f has a solution u E D'(cl) for any f E Co (0). It can be proved that this property is equivalent to some a priori estimate' for the operator a' (x, D). The next problem then is to translate this a priori estimate into a checkable geometric condition on the symbol a. To state such a condition, we introduce the Poisson bracket {p, q} of two complex-valued C1 functions on T`R" defined as
{p, q}(x, ) = (O p(x, ), 05q(x, )) - (axp(x, ), (9fq(x, c)).
This quantity appears naturally as the principal symbol (up to a factor i) o the commutator [a(x, D), b(x, D)] = (a#b - b#a)(x, D) when p and q are the principal symbols of a(x, D) and b(x, D) (cf. the asymptotic formula in Theorem 2.7). In [7, Chap.6], Harmander proved that the principal symbol p of a locally solvable operator a(x, D) must satisfy {p, p} = 0 on p = 0, but the proof is too long and technical to be rewritten here. In this section, we will just prove a converse of this result, the statement of which requires the following definitions. The operator a(x, D) is said to be of principal type at xo if the c-gradient of its principal symbol at xo vanishes only for = 0. It is said to be principally
normal at xo if there exists a function q E C°°(T'R" \ 0) homogeneous of degree m - 1 in C such that the principal symbol p satisfies
{p, p} (x, ) = 2iRe (q(x, i;)p(x, )) for i; E R" \ 0 and x close to xo. REMARK
In order to prove that a(x, D) is of principal type at xo, it is sufficient
to check that Ot p(xo, C) 96 0 when p(xo, l;) = 0 in view of Euler's identity p(x, i;) = (I /m) (8fp(x, i; ), C). In order to prove that a(x, D) is principally normal at xo, it is also sufficient to check that property near the zeroes of p. p} = 2iRe (qp) near (xo, to) if p(xo, to) -A 0 Indeed, one can always write because it suffices to take q = {p, p}/2ip as long as p 0 0. Thus, if one can also write {p, p} = 2iRe (qp) near any zero of p, the compact set K = {(x, £) E R" x R"; x = xo and 1i; I = 1 } can be covered by open sets S2, where one has {p, p} = 2iRe (qjp). Then, using the partition of unity constructed 'The Latin words a priori mean "before"; an a priori estimate for an operator A is an inequality between a norm of A,p and a norm of V proved for a whole class of functions ,p (typically for all w E CO '(0) for some 0) before assuming anything on ,p; it is only eventually that it will be used for special
Local solvability of linear differential operators
71
in Lemma 1.5, the function qo = > ojq, satisfies {p, p} = 2iRe (q0p) in a neighborhood of K, which finally gives {p, p} = 2iRe (qp) for f E ' \ 0 and x close to xo by homogeneity if we set q(x, ) = I I m_ I qo(x, /ICI). Principally normal operators obviously satisfy Hormander's condition {p, p} = 0 on p = 0. The converse will be discussed at the end of this section (see Corollary 4.4 and its comments). I The main result of this section is the following. THEOREM 4.1
Let a(x, D) be an mth order principally normal operator of principal type at xo. Then there exists a neighborhood S2 of xo such that the equation a(x, D)u = f
(in S2) has a solution u E L2 (Q) for any f E H'-'.
Example Operators with real or constant coefficients in the principal part satisfy {p, p} 0, and therefore they are principally normal (take q = 0). Elliptic operators have nonvanishing principal symbols (except at = 0), so that - thanks to the
previous remark - they satisfy the assumptions of Theorem 4.1. The student will find other operators satisfying these assumptions by solving the following questions. If a is of principal type, is a' also of principal type? If a is principally normal, is a* also principally normal? Is a#b of principal type when a and b are of principal type? What about the converse? Assuming that a#b is of principal type, that a and b are principally normal, and that the order of c is smaller than the order of a#b, does a#b + c satisfy the assumptions of Theorem 4.1? 0 In the proof of Theorem 4.1, our first step will be to show that the principal type and the normality correspond to some a priori estimates, but before stating and proving this result, we give a lemma that provides estimates for functions
with small support. In the following, we denote by S26 = {x E R"; Ixi < b} the open ball of radius b > 0. LEMMA 4.2
For allb>0andmEZ
,
for all p E Co (SZ6).
II'PIIm <_
Moreover, if Q and R are differential operators of orders m and 2m respectively, there exists a constant C such that for all E Co (S26) and
I(ixj p,RV)I S C6IIPIIm
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72
PROOF The first inequality is proved by induction. Indeed, it is sufficient to prove it for m = 0 in view of Proposition 1.14. and since one can write Ilsall0 = (tG, gyp) = (D1 (ixl (p), (p) - (ixI (D1 gyp), gyp)
= (ix1 p, Di p) + (Dip, ixl'p) S 211ixivIIoIlDiwIIo <- 2blIwIloflw
.
For the second inequality, let us write Q(ixjy2) = [Q , ixj]yp+ix3 (QV), which gives
5 II[Q,ixJ]'IIo + llix2(Q')Ilo <
CbII'PII,n
since [Q, ixj] has order m - 1, and we get the result from the first inequality. Finally, for the third inequality, we write R as a sum of differential monomials
of order 2m, i.e., R = Ek QkQk for some mth order operators Qk and Qk, and the result comes from the second inequality since
I(ix.,p, Rw)l = E(Qk(tx.JV),Qk' ) k
1
k
PROPOSITION 4.3
Let a(x, D) be a linear differential operator of order m. Then, (i) If a(x, D) is of principal type at 0, there exist a bo > 0 and a Co such that for all 6 < bo and E Co (S26 ),
n-I < Cob(Ila(x,
Ila*(x,
D) is principally normal at 0, there exist a 6 > 0 and a C such that for all p E Co (f2b), Ila(x,D)Vblo <- C(Ila'(x,D)
If a(x, D) is both principally normal and of principal type at 0, there exists a 6 > 0 such that for all p E C o (11 ), D)VIIo
PROOF
(i) Let A=a(x,D), Qj=[A,ixj]=(8f,a)(x,D), B=Ej_I Q,Qj=
late pI2 modulo S2ni-3, and since A b(x, D). Thanks to Theorem 2.7, b = EJ=1 is of principal type we have by homogeneity I at,p(x, )I2 > 2E1e12m-2 for some e > 0 and all x in some S22N. Therefore, the symbol b + EA2m-2#(1 - V)) satisfies the assumption of Theorem 3.9 (GArding's inequality) if 0 E Co (126o)
and 0 < 1. Moreover, if 0 = 1 in !
and 6 < bo, (1 - V,)Ip = 0 for all
Local solvability of linear differential operators
73
W E Co (f26) then (b(x, D) + EA2m-2(D)(1 - 0))y7 = Bye. It follows that we have n
2Re (By7, p) ? EIIsvIIm-i - C'IIVIIm-2
for some constant C. On the other hand, for each operator Qj one can write
= (A(ix.iV) - ix3(Ap),Q3p) (ix3sc, A`Q, ) - (ixj (Afw), Qif')
(ixi'p, [A',Qi] 'p) + (Q(ix.,'p), For V E Co (06), this can be estimated by using Lemma 4.2: IIQ2wII <
-i + Ci,26Ik IIm IIA',pIIo + CC6(IIApII0 1 + IIA'wllo + IpIIm_1),
and since the same Lemma 4.2 also implies IIVII -2 <
C
E E IIQ,"'IIo + f j=1
Cob(IIApII0 +
this gives II,pII;,
)
as required.
(ii) Let us modify the function q near = 0 so that q is now C'° everywhere while the relation {p, p} = 2iRe (qp) holds only for ICI > I and x in some f226. Then for '0 E Co (ft26) we set
b=zbaESt,
c= q+i{a,ii} E S'-1
and r = b'#b - b#b* - b#c' - c#b' E S2m.
Actually, the symbol r is in S2+,-2 because by using the asymptotic formulas of Theorem 2.7, we can write modulo S2,n-2: b' = b (where we use the notation a(x,E) = F_i 8,x,8{,a; similarly, we will use the notation ax = 8.'a and at Ota), then r = (b - ib(1,4))b - i(b{,bx) - b(b -
+ i(bt,bx) - be - cb
_ -i{b, b} - 2Re (eb) = -i({b, b} - 2iRe (eb))
_ -iil12({a, a) - 2iRe (qa)) = -ii2({p, p} - 2iRe (qp)) and this is identically zero for ICI > 1.
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Applications
Therefore, if we take such that 0 = 1 in S26 and we write A, B, Q, and R for a(x, D), b(x, D), c(x, D), and r(x, D), then Bp = Acp and B'cp = A'cp for all cp E Co (Q6) since A has the local property, and we can write
IIAwllo== (B'Bp,') = (Rye, p) + (BB'V,1v) + (BQ'So,,v) + (QB'1v, o) (Rp, gyp) + IIA',pIlo + 2Re
A'4p)
<_ IIRwlI1-mllcvllm-1 + 2IIA'1pl1o + !IQ'1vllo
< 21IA'Vllo +
as required, since R E W2",-2 and Q' E IV--1. (iii) Finally, when both assumptions hold, we get from (i) and (ii) that for
small 6 > 0 and for p E C0,A), IIVII n-1 < C1b(lla'(x, D)wllo +
Ilwll;,,-1)
for a new constant C1, and if we now choose 6 < 1 /2CI, one has IlVllm.-1 = 2llcvllm-1
-
11o111
M-1
< 2C16(Ila'(x, D)cpllo + lllvllm-1)
- IIcIIm-1 < Ila'(x, D)wllo
PROOF OF THEOREM 4.1 The property of local solvability now follows from the a priori estimate thanks to a classic argument from functional analysis. Indeed, using a translation, we can assume that x0 = 0, and we take b > 0 as in Proposition 4.3(iii). Then, the operator a*(x, D) is injective on Co (S16), and we can consider its inverse (A')-1 which is well defined on the space
E = {zL' E Co (Slb); 3cp E Co (16) with iP = a'(x, D)y'}.
For f E H1-" we define on E the semi-linear form U(ii) = (f, (A')-1 ') which satisfies
IU(0)I =1(f, p)l <_
11f111-m1l'vllm-I <_ 11f111-rIla'(x, D)cp110 =11f111-mll'Ilo
where we used the estimate (iii) of Proposition 4.3 for V = (A')-1 0. Thus, U is continuous for the L2 norm. We can then extend this form as a continuous form on L2(1)6) (by the Hahn-Banach theorem, or in a more elementary way by the following Hilbertian arguments: by continuity to the closure F of E, then by setting U(O) = U(rrF*) where lrF denotes the Hilbertian projection on the closed subspace F), and by using Riesz's representation theorem we get the existence of a u E L2(06) such that (u, V') = U('r') for 10 E E, i.e., (u, a* (x,
(f, w)
for all 1v E Co (Slb ),
which means a(x, D)u = f in S26 by definition.
Local solvability of linear differential operators
75
To close this section, we finally compare Hdrmander's result quoted at the beginning of this section with the result of Theorem 4.1.
The first situation we consider is when the sets Rep = 0 and Imp = 0 intersect transversally. Technically, this leads to the following transversality assumption: p(xo, ) = 0 and # 0 = Re OO p(xo, l;) and lm OE p(xo, e) are linearly independent. Then we have: COROLLARY 4.4
Let a(x, D) be a linear differential operator with principal symbol p satisfying
the following statement: the real and imaginary parts of the -gradient of p are linearly independent at (xo, ) for all solutions 0 0 of p(xo, ) = 0. Then a(x, D) of principal type at xo, and the following three conditions are equivalent: (i)
(ii)
(iii)
a(x, D) is principally normal at xo. a(x, D) is locally solvable at xo. a(x, D) satisfies Hdrmander's condition { p, p} = 0 on p = 0 in a neighborhood of xo.
PROOF Under the transversality assumption, a(x, D) is obviously of principal type at x0 in view of the remark following the definitions given above. The implications (i) = (ii) (iii) follow respectively from Theorem 4.1 and (1) from Hdrmander [7, Theorem 6.1.1], so that we only need to prove (iii) under the transversality assumption. Again, thanks to the remarks follow-
ing the definitions given above, it is sufficient to prove that one can write {p, p} = 2iRe (qp) near the zeroes of p; at such points, the transversality assumption shows that Rep and Imp can be taken as local coordinates in R2" and a Taylor formula thus gives 1
1
2i{p,p} =
2i{p",p}IP=o+giRep+g2Imp
for some coefficients ql and q2 E C°°(R2n). Finally, Hdrmander's condition (iii) allows to rewrite this relation {p, p} = 2iRe (qp) by setting q = q, + iq2.
The next step would be to weaken the transversality assumption in that corollary, and a natural substitute is to require that the c-gradient of p never vanishes at zeroes of p, in other words, to assume that a(x, D) is an operator of principal type. In this situation, Hdrmander's condition is no longer equivalent to the normality of a(x, D). As a matter of fact, it can even be proved that Hdrmander's condition is not sufficient and the normality is not necessary for the local solvability. However, a more precise characterization of local solvability is known for these operators of principal type (the so-called condition (P)), but the proof is much more difficult and requires the introduction of much wider classes of pseudodifferential operators. Indeed, this result, mainly due to Nirenberg and
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76
Treves [10] and Beals and Fefferman [2], was one of the main motivations in later developments of the theory of pseudodifferential operators. When any transversality assumption is removed, the question becomes even more difficult because the lower order terms then play an important role, and there is no complete answer to this question.
4.2
Wave front sets of solutions of partial differential equations
In Corollary 3.8 we established the pseudolocal property: pseudodifferential operators do not increase singular supports. In particular, operators of order -oc erase singular supports while elliptic operators preserve them. In this section, we are going to present the notion of wave front set of a distribution, a refinement of the notion of singular support that is well adapted to the study of singularities of solutions of partial differential equations. As explained at the end of the next section, this point of view will lead to much more precise results than when one considers only singular supports. The convenience of this notion lies in the following basic three properties: (i) One can compute exactly the singular support from the wave front set (Theorem 4.6). (ii) The action of a pseudodifferential operator does not increase wave front sets (microlocal property) and actually preserve them where there is some ellipticity (Theorem 4.7). (iii) There is a propagation property for wave front sets of solutions of partial differential equations (Theorem 4.8). Before defining the wave front set of a distribution u E S', recall a characterization of its singular support: xo 19 sing supp u if and only if there exists a neighborhood S2 of xo such that Vu E C°° for all cp E Co (S1). The definition of the wave front set will be very similar to this characterization, but it will first require us to "microlocalize" the notion of a pseudodifferentia] operator, i.e., to consider localizations in the cotangent space.
A subset r of T* R' \ 0 is said to be conic if
E F and p > 0
imply (x, p) E F. If r is a conic open set in T' R" \ 0, a E S'" is a compact set K C F such that supp a C {(x, p ); (x, 1;) E K and p > 0). (Using the asymptotic expansion for the operation # and Lemma 2.2, one gets that if a E and b E St, then a#b and b # a can be written as sums of a term in ScornP (F) and a term in S-OD.) a E S°° is said to satisfy a E Sa(F) if ab E S'" for all b E S°°,,,p(r) (this also implies that ab, a#b, and b#a are in S°'+t for all b E S ...P(17)). The symbol a E S"' is said to be elliptic at (xo, Co) E T*R" \0, or (xo, o) is said to be noncharacteristic for a, if there exist a b E S-'" and a conic neighborhood I' of (xo, o) such that ab - I E S;.' (t). The student will check that the proof of Theorem 2.10 can then be adapted to construct a b E S-' such that a#b- 1 and b#a -1 are in Sj;°°(F), maybe for a smaller F. Finally, the set of characteristic
said to satisfy a E S
Wave front sets of solutions of partial differential equations
77
points for a will be denoted by Char a, and from its definition it is thus a conic closed subset of T'R1 \ 0.
Example If a(x, D)
p(x,.) =
a,, (x)D' is a differential operator with principal symbol
a,,(x)t , the characteristic set of a is simply Char a = { (x, l;) E T* R' \ 0; p(x, t;) = 0}. Indeed, if p(xo, to) # 0, one can define EJa1_,,,
b(x, t;) = 1 /p(x, f) in a conic neighborhood r of (xo, CO) since p is homogenous,
and define b E S-"` anyhow out of r, and it is then clear that ab - I E Si-.' (I'). Conversely, if p(xo, to) = 0 and b E S-'°, a(xo, µl o)b(xo, 14o) = O(µ-1) and this shows that ab - 1 ¢ Sj;' (F) for any conic neighborhood r of (xo, co). 0 The wave front set of a distribution is then defined as follows.
DEFINITION 4S Let u E S'. One says that the point (xo, to) E T'R" \ 0 is not in the wave front set of u, or (xo, to) 0 WFu, if there exists a conic neighborhood r of (xo, to) such that a(x, D)u E C°° for all a E S P(r). From its definition, WFu is thus a conic closed subset of T*R \ 0. The wave front set is related to the singular support through the following result.
THEOREM 4.6 PROJECTION THEOREM
Let u E S'; then one has sing supp u = {x E lR
;
there exists a C # 0 with (x, f) E WFu}.
PROOF Let xo ¢ sing supp u, V E Co such that Vu E C°` and cp = 1 in a neighborhood f of xo, and r = St x (Rn \ 0) which is a conic neighborhood of (xo,l;) for every t 0. Then if a E S(r) one writes
a(x, D)u = a(x, D%pu) + a(x, D)((l - V)u). Since Vu E Co, one has a(x, D)(Vu) E S. Moreover, a(x, D)((1 - cp)u) _ b(x, D)u where b = a#(1 - c') E S-OC thanks to the asymptotic expansion of the operation # since the supports of a and (1 - cp) do not meet. It follows that
a(x, D)((l - V)u) E P (cf. Corollary 3.8) and a(x, D)u E C°°. Conversely assume that (xo,1;) it WFu for all t E R'a \ 0. For every such £, there exists a conic neighborhood r(t) of (xo,t;) as in Definition 4.5. The compact set { (xo,1; ); It I = 1 } is covered by these neighborhoods and one can find a finite number of them F1,..., rk and some functions cps E Co (F3) such that Vi (x,.) = 1 in KE = {(x, t;); Ix - xol + IIt;I - 1I < e} for some e > 0 (cf. Lemma 1.5). Choosing also a function -0 E Co(k') satisfying 'o = 1 near t = 0, one sets aj(x,l) = (1 -''(t;))cp?(x,t;/I1;I) E S.' P(FD). Therefore
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78
F,, aj 1 for Ix - xoI < e and E R1. We then take S2={x;Ix-xoI <e}; for any WECo (Q),cp=v7P+J:,ypajand one has vp
aj (x, D)u.
One has i4i(D)u E P since z' E S-'° (cf. Corollary 3.8), and aj (x, D)u E C°° by assumption, so that cpu E COD.
The next result links singularities of a(x, D)u and singularities of u. THEOREM 4.7 MICROLOCAL PROPERTY
LetuES'andaES°°; then WF(a(x, D)u) C WFu C WF(a(x, D)u) U Char a. PROOF Assume that (xo, o) WF(a(x, D)u) U Char a. Then, there exist a conic neighborhood r of (xo, eo) and a b E S°O such that
c#a(x, D)u E C°°
for all c E S mp(r) and b#a - 1 E S
(F).
Then for any d E S°°,,,P(17). one has d#(1 - b#a) E S-O° and one can write d#b = c + r with c E S P(r) and r E S-0°, from which one gets
d(x, D)u = d#(1 - b#a)(x, D)u + c#a(x, D)u + r#a(x, D)u. The first and third terms are in P since the operators are in %V°°, while the central term is also in C°° by assumption, so that d(x, D)u E C°°. Assume now that (xo, fo) it WFu, which means b(x, D)u E C°° for some conic neighborhood r of (xo, to) and all b E S a,,,p(I ). Then, for all c E c#a = b+r with b E S and r E S-O°, from which c#a(x, D)u =
b(x, D)u + r(x, D)u where the two terms are smooth since r E S-°° and b(x, D)u E C°° by assumption.
I
Let a(x, D) = aQ(x)D° be a linear differential operator with realvalued coefficients aQ E C°° in the principal part (i.e., for jai = m), and u E S' a solution of a(x, D)u = f where f E C°°. Thanks to Theorem 4.7 one has WFu C Char a = { (x, ); p(x, C) = 0} where p(x, C) = a,, (x)Ca is the principal symbol of a. Now, we are going to prove a more precise result, namely that WFu is actually a union of curves, called "bicharacteristic curves," contained in Char a: this property is known as the "propagation of singularities." Let p E C°° (T' R) be a real-valued function; its Hamiltonian vector field Hp = (8Ep, 8x) - (,9.p, .Qt) is clearly tangent to the level surfaces of p since one has Hpp = (0g, 8xp) - (8xp, O p) = 0. Its integral curves, that is, the
Wave front sets of solutions of partial differential equations
79
solutions (x(t),z;(t)) of the equations dx/dt = Ot;p(x,l;)
dl;/dt =
(which exist locally thanks to the Cauchy-Lipschitz theorem) are therefore contained in the level surfaces of p. In particular, those curves that start at a point (x(0), e(0)) where p vanishes satisfy p(x(t), fi(t)) = 0. and they are then called bicharacteristic curves of p. If p and q E C°° (T'1R') are real valued and if
p(xo, o) = 0 while q(xo, o)
0, then one has Hpq = qHp on p = 0 near
(xo, i;o) so that bicharacteristic curves of p or of pq are geometrically the same near (xo, CO), although parameterized in a different way. One can then state the propagation result. THEOREM 4.8 PROPAGATION THEOREM
Let a(x, D) =
a,(x)D° be a linear differential operator with a real-
valued principal symbol p(x,t) = c m If u E S' satisfies an equation a(x, D)u = f with f E C°O, then its wave front set is a union of I
bicharacteristic curves of p. Before giving the (rather long) proof of this theorem, we will prove a technical lemma that forms the basis of the study of the Cauchy problem for hyperbolic operators with variable coefficients. In return, this latter problem will then be the typical situation where the result of Theorem 4.8 can be used, as we will see in Section 4.3. If I is a compact interval of liP, we will use the notation C°(I; Hk) for the space of functions w(t), which are continuous functions of t valued in the Sobolev space Hk. Similarly, we will use C' (1; H k ) and C°(I; S) in the same way. LEMMA 4.9
Let b E S' be a symbol satisfying b - b" E S°, s > 0, and I = [-s, s]; then for any k E Z one has sup II e``tw(t)II k <- II e"8w(s)II k + 2
J
1Ie"'(Ot - ib(x, D))w(t)II k dt,
and
IIe-"W(t)IIk SUP
< Ile1'w(-s)II k + 2 if
IIe-,t(Ot
- ib*(x, D))w(t)IIk dt
for a constant p depending on k and for all w E CO (1; Hk+') n C' (1; Hk). Moreover, for any g E C°(1;S) and X E S, the Cauchy problem (Ot - ib(x, D))w(t) = g(t)
w(s)=X has only one solution w E UkC°(I; Hk), and this solution actually satisfies w E nkC°(I; Hk).
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Applications
PROOF We will give the proof in three steps. (i) Energy estimates. Since b - b` E So, one has for any w E C°(1; H' )
12Re (e"tib(x, D)w(t), e"tw(t))I = I (i(b - b')(x, D)e"t w(t), a"t w(t))I
< CIIe"tw(t)Ilo Thus one can write for w E Co(I; H') fl C' (I; H°) Me (e"tatw(t), e"tw(t))
dt Ile"tw(t)IIo =
> (2tt - C)Ile"tw(t)II2 + Me (e"" (at - ib(x, D))w(t), e"tw(t)) > -2lle"t(Ot - ib(x, D))w(t)IIoIIe"tw(t)IIo
if one has chosen p > C/2. Using the notation W(t) = IIet'tw(t)1io, M = suptEI W(t), and Int = f, IIe"t(Ot - ib(x, D))w(t) 11o dt, we can integrate this estimate over the interval It, s) then take the supremum for t E I, and this gives
M2 < W(s)2 + 2MInt, so that (M - Int)2 < W(s)2 + Int2 then M - Int < W (s) + Int, which is our first estimate with k = 0. For k 0 one has
((fit - i(ak#b#X-k)(x, D))Ak(D)w(t) = \k(D)(8t - ib(x, D))w(t), and since bk = Ak#b#X-k E S' still satisfies bk - bk E So, one can use the previous result (now p depends on k), which gives sup IIe"tAk(D)w(t)IIo <- IIe"-'Xk(D)w(s)Ilo tEl
+2
1IIe"t'\k(D)(8t
- ib(x.D))w(t)Ilodt,
and this is exactly our first energy estimate. Finally, the second estimate in terms of w(-s) and b' can be obtained in the
same way by changing tin -t since b' also satisfies b* - (b*)* E So. (ii) Uniqueness. If the Cauchy problem had two solutions in UkC0(1; Hk) for the same data g and X, then their difference w(t) would satisfy w E C°(1; Hk+' ) for some k E Z, Otw(t) = ib(x, D)w(t), and w(s) = 0. The equation for c9tw(t)
shows that w E C' (I; Hk), then the energy estimate implies that w(t) = 0 for t E 1, and this is the uniqueness property. With the second energy estimate, one would prove in the same way that the Cauchy problem
f (8t - ib'(x, D))ip(t) ='p(t) 0(-s) = 0 has at most one solution ' E UkCo(I; Hk)
Wave front sets of solutions of partial 4fifferential equations
81
(iii) Existence and smoothness of the solution. Actually, we will show that
for any g E C°(I; S), X E S, and k E Z, the Cauchy problem with data g and X has a solution wk E C°(1; Hk-' ). However, thanks to the uniqueness property we got in step (ii), all these solutions wk are equal to a unique solution
to E lkC°(I; Hk). For any cp taken in the space
E = {(8t-ib'(x,D)) (t);IP E Co (RxWz) with supp TP C {(t,x);t > -s}}, the Cauchy problem
(t - ib' (x, D))z'(t) = cp(t) ,0(-s) = 0 has a solution E C°(I; S), namely the restriction to I of the Vi given in the definition of E. This solution is unique according to step (ii) and it satisfies the energy estimate SUP El II (t)II-k < Ck f j
dt for any k E Z. Then,
for g E C°(I; S) and X E S, one defines a semilinear form W on E by
W(A _ (X, -005)) -
J
(9(t), 0(t)) dt
where zji is the unique solution of the previous Cauchy problem. One has
IWMI <-
(iiIk + J
)
119(t) Ilk dtSUP II
tE/
tII-k
J
IIv(tII-k dt,
so that W is continuous for the norm of the space L' (I; H-k), and thanks to the Hahn-Banach theorem, there exists an element w E L°° (I; Hk) (L' (I; H-k))' such that
(X, '(s)) - j(9(t),ib(t)) dt = for all
r
Jr
(w(t)(t - ib(x,
dt
ECC (RxR')with supp+P C{(t,x);t>-s}.
When we restrict ourselves to functions 7P E Co with supp zb C H = {(t, x); Iti < s}, the term (X, -i(s)) vanishes and the previous equation then means that (t-ib(x, D))w(t) = g(t) in Q. It follows that 8tw E L°°(I; Hk-') whence w E C' (I; Hk-' ), and similarly w E C' (I; Hk-2). Moreover, for any cF E Co (l[l;") one can construct a 10 E Co (R x R") with supp C {(t,x); t > -s} and V'(s) = cp, so that the integration of f, (w(t), O (t)) dt by parts gives (w(s), co) = (X, cp), and this proves that the function w E C°(I; Hk-') is a solution to the Cauchy problem with data g and X. I REMARK Of course, we could have solved the Cauchy problem with data on
t = -s as well since all the assumptions are preserved when reversing the time. To use this result in the study of the Cauchy problem for a hyperbolic
82
Applications
operator with variable coefficients, we would have to consider more generally symbols b depending also on t, but this would not affect the proof very much (cf. Hdrmander [8, Chap. 23], where we took the proof of Lemma 4.9). 1
Since WFu is a closed set and a bicharacteristic curve ry is a connected set, the only fact to prove is that WFu fl y is an open subset of y. Thus, given a E WFu C Char a, we want to prove that the bicharacteristic curve (x(t),.(t)) starting at this point is locally contained in WFu. We will proceed by contradiction, assuming that this is not true. does not modify WFu Since the action of an operator elliptic at near this point (cf. Theorem 4.7), we will deal with the distribution v = ),'- t (D) (pu) rather than with u itself where the function z!, E Co satisfies 0 = l near x0. The advantage is twofold: on one hand, we have iu E Et from which we get r(x, D)v E S for all r E S-°` (cf. Corollary 3.8). On the other hand, if cp E Co is chosen such that p = 1 near xo and V) = 1 near PROOF OF THEOREM 4.8
supp gyp, the distribution v will satisfy an equation b(x, D) v = pf E Co where
b = coaA` is a first-order polyhomogeneous symbol (the three factors are polyhomogeneous) whose principal symbol q(x, t:) = cp(x)p(x, t )JC'Jt -' is real valued and has the same bicharacteristic curves as p. WFv = WFu, and this will Our contradiction will then be that (xo, Co) follow from the construction of a symbol co E S° elliptic at (xo, eo) and such that co(x, D)v E C. To prove this latter property, we will construct a symbol
c(t, x, ) satisfying c(0, x, ) = co(x, C) and such that w(t) = c(t, x, D)v is a solution of a Cauchy problem as in Lemma 4.9. Since this w E C°(I; H-N) for some N E Z, it will then follow from Lemma 4.9 that w E lkC°(I; Hk), so that co(x, D)v = w(0) E Cc". The symbol c(t, x, C), a C' function of t valued in So, is going to be constructed so that
r(t) = 8tc(t) - i(b#c(t) - c(t)#b) is a continuous function of t valued in S'°°. To achieve this, we just have to choose a co homogeneous in of degree zero, and to look for a polyhomogeneous symbol c(t, x, ) - F_, c, (t, x, ) where the c) are homogeneous in a of degree -j. Following this point of view and ordering the terms in the asymptotic expansions of b#c(t) and c(t)#b according to their degrees of homogeneity, we then have to solve the sequence of problems
{
8tco-Hgco=0 co (0,x,0 = co(x,0,
and for
j >0
9c, -Hqc. =rj c.7 (O,x,0 = 0
where the vector field Of - Hq is real. These problems can be solved in a same neighborhood of (xo, o) thanks to the Cauchy-Lipschitz theorem, and their solutions cj(t) are there homogeneous in l;' of degree -j and have their support contained in the support of co(t). If we have chosen co E Soomp(F) for
The Cauchy problem for the wave equation
83
some small conic neighborhood r of (xo, co), it thus follows that for small t, the c3(t) can be extended by homogeneity as symbols c,(t) E S (F), and the symbol c(t, x, ) is then constructed as in Lemma 2.2. Since we assume that the bicharacteristic curve is not locally contained in WFu, there is a point (x(s),C(s)) ¢ WFu for an s in the domain of definition of c, and one has
supp c(s) C supp co(s) C e'H9 supp co where e'H, supp co denotes the image of supp co under the flow of the Hamiltonian vector field HQ. Therefore, if the (conic) support of co has been chosen
sufficiently small around (xo, to), the support of c(s) will be small around (x(s), C(s)) and we will have w(s) = c(s, x, D)v = X E C'° (and actually X E Co since the support of c(s) is compact in x). Thanks to Lemma 1.16, the distribution ipu is in some Sobolev space so that
v = a'-1(D)(i)u) E H-N for some N E Z+ and then w(t) = c(t, x, D)v E C°(I; H-N) since c E C°(I; S°). Moreover, this distribution w(t) satisfies Otw(t) - ib(x, D)w(t) = g(t) where
g(t) = r(t, x, D)v - ic(t, x, D)b(x, D)v
by construction of c(t), and these two terms are in C°(I; S) because r E C°(I;S-OQ) and v = ,\'"-I(D)(,u) with r1'u E £' (cf. Corollary 3.8), and b(x, D)v = V f E C000. It follows that w is a CO(I; H-N) solution of a Cauchy
problem as in Lemma 4.9 (indeed, b E S' satisfies b - b' E S° since it is polyhomogeneous with a real principal symbol), therefore w E lkC°(I; Hk), then ca(x, D)v = w(O) E HOC C C°° as claimed. I
43 The Cauchy problem for the wave equation This section actually has little to do with pseudodifferential operators. Following
an idea of Treves [121, the Cauchy problem for the wave equation is treated through very simple computations, which allow us to prove the main results. We have added this section because the last theorem provides a pleasant illustration of the general results given in the previous section.
Consider in the space R"+' = {(t, x); t E R, X E R" } the wave operator,
i.e., the operator u '-4 Ou - Du where 0 = E' 102 with 0, = 0/Ox, is the Laplacian operator in R". The present treatment will be completely dissymmetric in x and t: we will consider solutions u such that for each fixed t, u(t) E S' (space of temperate distributions in x only). The spaces Ck(R, H') and Ck(R; S) are the standard spaces of Ck functions of t valued in the Hilbert space H' (Sobolev space in x only) or in the Fr6chet
space S (Schwartz space in x only). We will write u E C°(R;S') if for each fixed
E S the expression (u(t), gyp) is a continuous function of t, and actually,
Applications
84
according to general results from functional analysis, these separate continuities
imply the continuity of u : R x SE) (t, cp) '-. (u (t), V). As a consequence of this remark, if u E C°(R, S'), the formula (U(t), cp) =
J0
t(u(s), cp) ds
for p E S
defines a U E Co (R; S') also. Then, for k > 0 we will write u E Ck (R; S' ) if for each fixed cp E S, (u(t), cp) is a differentiable function of t such that the formula
Ptu(t), 0 = a (U (t), V) defines a distribution Btu E Ck_ I (R; S'). It is clear that the previous U(t) _ fo u(s) ds satisfies U E Ck+' (R; S') if u E Ck (R; S'). Most of the proofs of this section will be based on the following, very elementary lemma on trigonometric functions. LEMMA 4.10
The functions
C(t,()
and
S(t,() =
s ICI
are smooth on R"+' and for any k E Z+ and fixed t, one has 8i C(t) E P and Ot S(t) E P as functions of ( E Rn. Moreover, they satisfy 8tC(t) = (I - A2)S(t) and 8tS(t) = C(t), they can both be extended to the whole of Cn as entire functions with the estimates I C(t, OI 5 eltl hmCI
and
I S(t, ()I <- Itleltl IIm(I,
and finally they also satisfy (for ( E Rn )
IC(t)I < I
and
XIS(t)l < (1
+t2)1/2.
Ek>o(-t2IeI2)k/(2k)! so that C Indeed, we can write C(t, f) = is obviously smooth and can be extended as an entire function by setting PROOF
C(t,() _ Ek>O(-t2(2)/C/(2k)! Where (2 = (j + +(n. Moreover, C(t,() _ (e'tz + e-'tz)72 if z E
has been chosen such that z2 = (2, and since n
n
2(Im z)2 = Iz2I - Re (z2) 5 E(I(; I - Re ((()) = 2 E(Im(()2 = 2IIm(12, j=1
j=1
one gets the estimate IC(t,()I < ellmt=I < eltlllm(l. The Paley-Wiener theorem (Theorem 1.13) then shows that C(t, () is the Fourier transform of some distribution c(t) with support contained in the ball of radius Iti, and since D C(t, () is the Fourier transform of (-x)ac(t) which also has its support in the same ball, D C(t, f) also has a polynomial growth at infinity, i.e.,
85
The Cauchy problem for the wave equation
C(t) E P. One can similarly prove the same facts for S, or simply remark that S(t) = fo C(s) ds. (1 - \2(C))S(t'C) and The relations C(t,1;) are immediate and they also imply that Ot C(t) E P and 8i S(t) E P simply by induction. Finally, the estimate AIS(tt )I < (1 +t2)'/2 comes from a2(C)IS(t,C)I2 =
sin tici + sine tIeI < t2 (sup sER
2
nl s
J
+ 1 = 1 + t2.
1
The first result we give here is an existence and uniqueness theorem for the Cauchy problem. It also shows that the smoother the data, the smoother the solution. THEOREM 4.11
Let uo and ul E S' and f E Ck(R;S'). Then the Cauchy Problem (8c
- A)u(t) = f (t),
u(0) = uo,
8iu(0) = ui
has a unique solution u E C2(R; S'). Moreover, this solution satisfies
u(t) = C(t)vo + S(t)ir + J S(t - s) f (s) ds, c
0
u E Ck+2 (R; S') and the following: (i)
(ii)
If uo and ui E S and f E Ck(R,S), then u E Ck+2(RS). If uo E Hs+', ul E Hs, and f E C°(R; Hs), then u E C°(R, H3+') fl C' (R; Hs) f1 C2(R; Ha-1).
PROOF Through Fourier transformation in x only, the operator -0 = - Ejol 82 is transformed into the operator of multiplication by the function Fn q2 = )12(1;) - 1. Thus, this Cauchy problem is equivalent to the j_l following:
(8r + (J12
- 1))u(t) = f (t),
u(0) = uo,
8tu(0) = ul,
where the only variable is t since there is no derivative with respect to the j's (they are merely parameters). In view of Lemma 4.10, it is now obvious that the given formula for u(t) defines a temperate distribution that solves this problem. For the uniqueness, we have to prove that u(t) must be given by this formula. Indeed, if u E C2 (K S,) is a solution of the problem, we can use the distribution U(s, t) = C(t-s)u(s)+
S(t - s)8tu(s) to write ft(t) = U(t, t) = U(0, t) +
= C(t)Lo + S(t)ii +
0sU(s, t) ds
J0
J0
t
S(t - s) f (s) ds
Applications
86
since 83U(s, t) = S(t - 8)(O u(s) + (A2 - 1)u(s)) = S(t - s) f (s). To prove u E Ck+2 (fly; S') we simply compute
Otu(t) = (I - A2)S(t)uo + C(t)ui + f tQt - s) f (s) ds, 0
Ot u(t) = (1
- A2)u(t) + j (t)
and, because f E 0(1R; S'), this gives the result. Finally, the claim (i) is obvious in view of the previous formulas, and for (ii) we remark, using estimates from Lemma 4.10, that the function Ae+1
A8+, IC(t)'uol + as+, IS(t)u, I + \8+1 I f t
I u(t)i
S(t - r)f (r) dr
o
t
AS(t - r).A8 f (r) dr
< a8+, Iuol + (1 + t2)1i2a81u, I + J0
is square integrable in (for the integral, use the Cauchy-Schwarz inequality), so that u E C°(K- H8+1). The proof of u E CI (X- H8)f1C2(l H8-,) is similar when using the previous formulas for Btu and 8i u. I
When f = 0, the same computations lead to the property of conservation of energy. COROLLARY 4.12
If uo E H8+1, u1 E H8, and f = 0, the energy E.(t) = IIOiu(t)Ils +
It
IID,u(t)II
=1
of the solution u E C°(X- H8+1) n C' (R; H8) obtained in Theorem 4.11(ii) is independent oft and therefore is equal to E8 = 11u, I I2 + F- , I I D, uo 112 PROOF
Using the expression of u given in Theorem 4.11, n
ID-,U(t)12 (1- A2)S(t)uo + C(t)u112
l0tu(t)I2 + 9=1
+ (A2 - 1)IC(t)uo + S(t)u112 (A2
= (C(t)2 + = Iu1 I2 + (A2 -
- 1)S(t)2)(1u112 + (A2 - 1)Iuo12) 1,&o 12
since C(t)2 + (A2 - 1)S(t)2 = cost tI l + sin2
1. The conservation of energy follows by multiplying by \2a and integrating in . I
The Cauchy problem for the wave equation
87
The last result will describe the phenomenon of finite propagation speed. It shows that a modification of the data cannot immediately affect the values of the solution far from the domain where the data are modified. For the sake of simplicity, we continue to assume f = 0. THEOREM 4.13
Let uo and ul E S', f = 0 and u E C°°(IRY;S') the solution of the Cauchy problem as in Theorem 4.11. Then (i)
If supp uo U supp ul does not intersect the closed ball l y E 1Rn ; l y - x I <
ItI}, then (t,x) ¢ supp u. (ii) If sing supp uo U sing supp ul does not intersect the sphere {y E RI; (y - xI = ItI}, then (t, x) ¢ sing supp u. PROOF We will assume x = 0 and t > 0 since the wave operator is invariant under translations in x and reversion of the time. (i) Let us denote by Bi the closed ball of radius t and by Qt the open ball of the same radius; if supp up U supp ul does not intersect the closed ball BI, we actually have uo = ul = 0 in some 11T for a T > t. Then we will prove that for 0 < s < T, u(s) vanishes in QT-,, and this will obviously imply that (t, 0) 0 supp U. Thus let 0 < s < T be fixed and let E Co (S2T_s). From Theorem 4.11 we know that u(s) = C(s)uo + S(s)ul, and we can write
(u(s),,p) =
(2ir)-n(uo,C(s)o)+(2ir)-n(ul,S(s)'4
Actually one has supp C BT_e_, for some positive c, and thanks to the Paley-Wiener theorem (Theorem 1.13), 0 can be extended as an entire function I(I2)-Ne(T- c)1Im(I for all N E Z+ satisfying estimates lo(C)l < CN(1 + and some sequence of constants CN. Using the results of Lemma 4.10, we see that the functions 4o = C(s)y and 4Dl = S(s)y' can also be extended as entire functions satisfying estimates
CN(1 +
I(I2)-Ne(T-E)IhnCt
for all N E Z and a new sequence C. Again using the Paley-Wiener theorem, it follows that there exist two functions Wo and cpl E Co with supp cpy C BT_,, cpo = C(s)o and c = S(s)cp, so that we can write (u(s), p) = (27r)-'(u0, 0) + (27r)-n(ul, 01) = (uo, po) + (ul, WI),
and this is zero since uo and ul vanish in S1T. We thus get u(s) = 0 in SIT-., as claimed.
(ii) We will first prove that u is smooth in a neighborhood of {(s, y) E R x 1Rn; s = 0 and jyj = t}. Indeed, if lyI = t, one has y ¢ sing supp uo U sing supp ul by assumption, and one can find two functions vo and vi E S such
that no = vo and ul = vl in some neighborhood of y. The Cauchy problem
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Applications
with data vo and v, has a solution v E C'°(R;S) (cf. Theorem 4.11(1)), and thanks to the part (i) already proved, (0, y) ¢ supp (u - v) since u - v is the solution of the Cauchy problem with data uo - vo and ul - v, the supports of which do not intersect {y}. Therefore u is smooth near (0, y). To prove that (t, 0) ¢ sing supp u, we now use the results of the previous section. If u were singular at (t, 0), there would be a direction (r, f) E R" +I \ 0
such that (t, 0; r, ) E W Fu and since (o - O)u = 0, this point (t, 0; r, ) should lie in the characteristic set of the wave operator by Theorems 4.6 and 4.7. This characteristic set is described by the equation 1t 12 - rZ = 0 so that we get
r = ±jjj # 0. According to the propagation theorem (Theorem 4.8), the whole bicharacteristic curve starting at this point should then lie in WFu. This bicharacteristic curve (s(r), y(r); a (r), q(r)) is the solution of ds
dr
= -2a(r)
dy
dr
= 2q(r),
da
dq
dr = dr = 0,
(3(0), y(o); o(0), 77(0)) = (t, 0; r, t)
i.e., (s(r), y(r); a(r), q(r)) = (t-27-r, 4r; r, ). For r = t/2r we get s(r) = 0 while y(r) = tC/r satisfies l y(r)l = tle/rl = t, so that this point, which is in sing supp u because of Theorems 4.7 and 4.8, lies on {(s, y) E JR x ]It"; s = 0 and jyl = t} where we proved that u is smooth. This contradiction shows that
(t,0) ¢ sing supp u.
I
A consequence of Theorem 4.13(ii) is that one has the following estimate for the singular support of the solution u:
sing supp it C
U
W, X). Ix - yi = (tl },
yEE.UE. where E, = sing supp uj. In other words, the singular support of u is contained in the union of all light cones with vertices located at the singularities of the data. However, the reader will easily understand that the results of Section 4.2 give much more than this estimate. Indeed, while this inclusion is the most precise one that can be proved when
one knows only the singular supports of the data, it is not true that it is an equality. Actually, the singular support of u is much smaller than this union of cones, at least in most of the cases. On the contrary, if we take the point of view of wave front sets, we can give an exact description of the location of singularities of it (and therefore of its singular support). Indeed, denoting the (polyhomogeneous) pseudodifferential operator of symbol ICI by IDJ, it can be proved that (WFu) fl {t = 0} is equal to
{(0,x;(x,C) E WF(ul +ilDIuo)} U {(0,x;
(x,C) E WF(ui - ijDjuo)}
89
Exercises
and then the exact description of WFu follows by using the propagation theorem (Theorem 4.8); cf. Exercise 4.7. We hope that this simple example will convince the reader of the great usefulness of the microlocal point of view in the study of singularities of solutions of partial differential equations. To close this course, we simply point out that very similar results have been proved for the nonlinear hyperbolic Cauchy problem, and the essential tool in the proof was a well-adapted variant of the theory of pseudodifferential operators. For such results, we refer the reader to Bony [31 and Chemin [5].
Exercises 4.1 Local solvability of (approximately) invertible operators. Assume that a and b E
S°° satisfy a#b-1 E S-°° (or more generally a and b E So with aft-1 E So.°`. as in Exercises 2.9 and 3.6).
Let H6 = {x E R"; txI < b} as in Lemma 4.2; show that for any E > 0 there is a 6 > 0 such that 11,P11_1 < ik1'lIo
for all WE Co (t1 ).
(Hint: Show by contradiction that otherwise you could find a square integrable function with support equal to {0}.) Show that you can find a 6 > 0 and a function E Co (f26) with o = I near 0 and such that the operator c(x, D) = p(x) (I - a#b(x, D)) satisfies
IIc(x, D)iPIj-, 5 1 M-
for all 1P E Ca (S26).
Let s E IR and f E H"; explain why the formulas W'o = c(x, D) f,
ip,+i = c(x, D)>yr,
and ?G., _
TV
3>o
define functions ?, = Co (f26) (for j = oo, first prove that ?Pa E H-1, then observe that tj°° = o + c(x, D)?Pa). Finally, show that the formula
u = b(x,D)(f +tV) defines an H"-' distribution solution of a(x, D)u = f in a neighborhood of 0 (here, m denotes the order of b).
4.2 Let b : 1(P -. R be a C°° function, and let a(x, ) = t;i + ib(xi )l;2. We want to study the local solvability of a(x, D) at the origin of R. (a) Determine conditions on b equivalent to the following properties:
- a(x, D) is of principal type. - a(x, D) is principally normal. - a(x, D) satisfies Hdrmander's condition p = 0 = {p, p} = 0, where p is the principal symbol. Then describe for what functions b the operator a(x, D) is locally solvable (resp. is not locally solvable) at the origin thanks to the results given in
Applications
90
Section 4.1, but exclude the case where b would have a zero of infinite order at x, = 0. (b) Assume that b does not change sign for lx, I < E < 1, and set 9 = (-E, E) X (-E, E).
Take B(x3) = fo' tlb(t)I dt and prove that the operator c(x, D) defined by c(x, D)t,(x) = eB(x D)(e-B(nl)?i)(x) satisfies the a priori estimate II
for all
'IIo
E CO '(Q).
(Hint: Integrate by parts the scalar product Im (c(x, D)tli, (x, ± D2)tP).) Show that there exists a constant C such that IkcIIo <_ Clla'(x, D),pll,
for all W E CO '(9).
(Hint: Use 0 = eB(n1)gyp.) Conclude that the equation a(x, D)u = fin f'? has a solution u E H'3 for all f E H°. The converse of this result will be proved in the next exercise. 4.3 Necessary conditions for local solvability. As was said in Section 4.1, the property of local solvability for an operator a(x, D) is equivalent to an a priori estimate for the operator a' (x, D). In this exercise, it is proved that the solvability property implies such an estimate, and then this result is used to complete the study of the operator D, + ib(x, )D2 at the origin of R2. (Hormander's theorem [7, Theorem 6.1.11, i.e., (ii) r (iii) in Corollary 4.4, can also be proved by using this estimate.)
(a) Let a(x, D) = LloI<m a°(x)D° be a differential operator with complex-
valued coefficients a° E H°"(R"). Let K be a compact set of R", denote by Co (K) the space of C"0 functions with support contained in K, and consider on CI (K) the topology defined by the Sobolev norms II 11. for s E 7+. Show that COI(K) is a Fr+ chet space.
With f E C, 00(K) as above, assume that a(x, D)u = f has a solution u E D'(Sl) where Il is a neighborhood of K. Show that there exists an r E Z+ (depending on f) such that I(w, f )l <_ rll a*(x,
for all p E CO '(K).
Show that 52,. = if E Co (K); I (gyp, f )I > rll a' (x, D)Vll r for some
p E CI(K) I is open in CO(K). Assume that a(x, D)u = f has a solution u E 1Y (Q) for all f E Co (K) where K C S1 are two neighborhoods of the origin (K is compact). Use Baire's theorem to prove that there exist fo r= CI (K), c > 0, and r and s E Z+ such that
Ilf - foil, < 2E =* f V flr. Finally, prove that if a(x, D) is locally solvable at the origin, there exist a compact neighborhood K of the origin, a constant C. and two integers r, s E 7L+ such that
for all cp,fECo (K). 1(p,f)l
91
Exercises
(b) An example. Assume that a(x, D) = D, + ib(x, )D2 is locally solvable at the origin of R2 (here, b is a smooth, real-valued function as in Exercise 4.2),
and let K, C, r, and s be as in part (a). Assume in addition that b changes sign in every neighborhood of 0.
Show that K contains a rectangle R = [x_,x+J x [-E,EJ with a xo E (x_, x+) such that B(xl) = f b(t) dt does not change sign and satisfies 1131 < 1/2 on [x_,x+], and does not vanish at x_ or x+. Then set
O(x) = B(xi) - ix2 - (B(x,) - ix2)2
if B > 0 on [x_, x+],
O(x) = ix2 - B(xi) - (ix2 - B(xi ))2
if B < 0 on [x-,x+],
and show that in both cases a* (x, D)ct = 0 and Re 0(x) > x2+(1/2) B(xj) in R. Choose (carefully) two functions Wo E Co (R) and f E Co (with <po = 1
near (xo,0) and f(0,0) = 1). Then use Vµ = e'µmcpo and f,(x) = f (U(xi - xo), µ2x2) to show that the a priori estimate of part (a) cannot hold for all p and f E Co (K). Conclusion? 4.4 In part (a) of this exercise, we prove that if u E S' and (xo, o) E T' R" \ 0, then WFu if and only if there exists a V E Co satisfying cp = 1 near xo and a conic neighborhood r of o such that the functions ,\kWpu are bounded in r for all k E Z+. This characterization is then used in part (b) to determine some wave front sets. (a) Characterization of the wave front set. Let u E S' and (xo, to) E T'1R" \0.
Use Remark 2.8 to show that if the symbol a E S°° does not depend
on x,then aEPand a(D)v=avforall vES'. Assume that for some wp E Co such that cp = I near xo, the functions Akcpu are bounded in some conic neighborhood r of to for all k E Z. Prove that there exists an a E S°° independent of x such that a#cp is elliptic
at (xo,lo) and E S, and conclude that (xo,lo) ¢ WFu. Conversely assume that (xo,1;o) V WFu. Show that one can construct a W E Co with W = I near xo and a symbol a E S°° independent of x with a(C) = 1 for large in some conic neighborhood r of c o such that a#cp(x, D)u E S, and conclude that the functions Apu are bounded in r for all k E Z+. (b) Applications. Using the characterization proved in (a), determine WF6
where b E S' is defined by (b, r[)) = (0) for ' E S. In the following questions, the dimension is n = 1. For tji E S = S(R), one sets
r
(pv ! , ) = lim ,-o+ J xl? X
r
`x + i0 '
Irm i_.0+
(x
E-.0+
lim
+G(x)
dx,
X
f !(x)
dx
f
dx.
x + if X_%(
Show that these formulas define temperate distributions satisfying 1
pv
1
1
x+i0+irb=
x- i0
-i7rb
Applications
92
and 1
=x-i0 = x+ i0 _
1
_
1
1.
xpv x x x Show that if V is a unit test function, then P(x) - (f 0(t) dt)V(x) is the derivative of a function tli, E S for all ' E S. Then, prove that if u E S' satisfies 02u = 0, then u is a constant. Show that DEpv(l/x) = -2irb, and observing that pv(I /x) = -pv(l/x), conclude that pv(l /x) is the L'° function with value iri on < 0 and value -ai on > 0. Compute also the Fourier transforms of 1 /(x + i0) and
1/(x - i0). Prove that if cp E Co satisfies p = 1 near x = 0, then ((cp(x) - 1)/x) E H°O, and use the characterization proved in (a) to determine the wave front
sets of the distributions pv(l /x), 1/(x + i0), and 1/(x - i0). 4.5 Continuation of Exercise 4.4: Products and restrictions of distributions. In this exercise, we want to discuss the possibility of defining the product of two distributions in R", or the restriction of such a distribution to a submanifold RP
of R". The properties of these notions we expect are that if b E P. then (" ,u)v = O(uv) = -i(vu) = (a/iv)u and ('tl'u)IRP = ('+LIRP )(uIRP ).
(a) Consider in R(n = 1) the distributions u = pv(1/x) and v = b, defined in Exercise 4.4(b), and ?P(x) = x, and show that you cannot define both (tpu)v and (z'v)u, nor both uI:=o and (t'u)j=..o with the expected properties.
To overcome this difficulty, one can assume some smoothness on u: indeed, uv is a L' function if u and v are L2 functions. Similarly, let (x, y) denote the elements of R" where in E RP and y E R"-', assume that u E H8(R") for some s > (n - p)/2 and show that the formula for Wp E S(RP)
(uIRP, o) = (21r) " J u(C,,7)W-(f) d6 dp
defines a distribution UIRP
E H''(("-P)/2)(RP) satisfying (r(iu),RP =
(tPIp,)(u1R,,) for all V, E HOO. In parts (b) and (c), we will investigate another way of defining products and restrictions where some geometric properties of the wave front sets are required instead of smoothness.
(b) Products of distributions. Let u and v E S' be such that WFun WFi = 0.
Relate WFv and WFv. Prove that for any in E R" there exists a function cp= E Co with V. = I in a neighborhood !l of in and a v ucpzii
bounded for all k E Z. Show that for any -0 E Co (fly), cp u(C)c that the formula (uv,'+1') =
(21r)_2"
E L'(R2") and r)) dt drl
defines a distribution uv E D'(12 ). Explain how to define a distribution uv E D'(R"), show that this definition has the expected properties, and determine the wave front set of the product uv. What products can you write with the distributions pv(1/x), I/(x+i0), and I/ (x - i0) (cf. Exercise 4.4(b))?
Exercises
93
(c) Restrictions of distributions. Let (x, y) denote the elements of R", where
xE RP and yER"-P, and set N'RP={(x,y;g,n)ET'R";y={=0}. that WFunN'RP=0. Prove that for any x E RP there exists a function gyp= E Co (R") with V.r = 1 in a neighborhood of (x, 0) and A';p u bounded in some conic neighborhood of q); = 0} for all k E Z+. In the following, Q.. denotes a neighborhood of x in RP where V_,Iv=o = 1.
Show that for any ip E Co (12=), pur E L'(R"), and show that the formula dt; drr
(uIRP, i,) = (21r)-° J
defines a distribution uIRP E D'(1l ).
Explain how to define a distribution uIRP E D'(RP), show that this definition has the expected properties, and determine the wave front set of the restriction.
4.6 Extend the theorems of Section 4.3 to the Klein-Gordon equation (8 - 0+µ) u(t) = f (t), where p is any fixed complex number. More precisely: Write the solution u(t) given by the method of variation of parameters and show that the coefficients of uo and 1i1 can be extended as entire functions U,,(t,() satisfying estimates
I or Itl). IUN(t,()l Prove the existence and uniqueness theorem, and a regularity theorem with Sobolev spaces. Assume f = 0 and it E R and prove the conservation of energy for a modified definition of the energy. Come back to any µ E C but keep the assumption f = 0, and prove theorems of propagation of the support and singular support. 4.7 Propagation of the wave front set in the wave equation. Let uo and u, E S', and let u be the solution of the Cauchy problem (' - o)u(t) = 0,
u(0) = uo,
Otu(0) = u,
(cf. Theorem 4.11). Denote by IDI the polyhomogeneous pseudodifferential operator of symbol and set
v = (8, +iIDl)u
and
w = (8, - iIDI)u.
Show that ((9t -zIDI)v, (Be+il DI )w, and (B,+iI DI )v+(Ot -iI Dl )w-2(8+0)u are smooth.
Show that WFu = WFv U WFw. Show that
l,VFvl,=o = {(0,x;
(x,t;) E WF(u, +iIDluo)}
and
WFwie=D = {(0,z;
(z,.) E WF(u, - iIDluo)}
(cf. Exercise 4.5(c)), then give an exact description of WFu involving only WF(ui + iIDluo) and WF(u, - ilDluo).
94
Notes on Chapter 4
Notes on Chapter 4 Singular integral operators were first introduced in the study of elliptic problems (see e.g. Calder6n and Zygmund [27]), and Calder6n [231 used them later to prove his celebrated uniqueness theorem. However, these operators did not play an essential role in these questions, and it is mainly the proof of the index formula by Atiyah and Singer [15] that convinced analysts of the importance of this tool. These first results were soon followed by many other theorems proved with the use of pseudodifferential operators theory. Among them, let us quote the boundary problems treated by Calder6n [24], some hypoellipticity and subellipticity results obtained by H6rmander [43,44], the results on microlocal singularities of solutions of partial differential equations as found in Hi rmander [45] and Duistermaat and Hormander 1311, for example. We now discuss more precisely the topics treated in Chapter 4. During the nineteenth and half of the twentieth centuries, the problem of existence of solutions of partial differential equations was reduced to find how many additional conditions had to be given in order to insure uniqueness of the solution. This led essentially to the development of the study of the hyperbolic Cauchy problem and of the elliptic Dirichlet problem. Things changed drastically in 1957 when Lewy [53] discovered a simple first-order equation (even with analytic coefficients) admitting no solution for most C°° right-hand sides. The first general results on local solvability are due to Hdrmander [40], whom we followed in Section 4.1, and a characterization in the case of differential operators of principal type was obtained in Nirenberg and Treves [10) and Beals and Fefferman [2]. (The corresponding result for pseudodifferential operators of principal type is still an open problem.) For operators that are not of principal type, some results have been obtained in the framework of nilpotent Lie groups (see Levy-Bruhl [52] and the references therein). It is difficult to decide who first proved that elliptic equations with smooth coefficients and right sides admit only smooth solutions, since this was proved in greater and greater generality by many authors. One main step was the famous
Weyl's lemma [70], while the first explicit statement of the result for general linear elliptic equations is probably that of Friedrichs [36]. On the other hand, the result on propagation of "discontinuities" in the hyperbolic Cauchy problem seems to be due to Hadamard [38]. The microlocal versions of these results were first given by Sato in the framework of hyperfunctions (see [60]), then in Hbnnander [45, Section 2.51 and Duistermaat and Hbrmander [31, Section 6.1] in the form we give in Section 4.2. The use of Fourier integral operators in the latter reference is replaced in our text with Lemma 4.9, which is very close to a theory of these operators. For the corresponding results in the analytic framework, we refer to Sato et al. [60], Hormander [46], or Sjostrand [63]. In the case of nonlinear equations, similar results were also obtained by Bony [3]. It is from the original memoire of Poisson [59], who treated the Cauchy
Notes on Chapter 4
95
problem for the wave equation, that the hyperbolic theory was developed. The results of Section 4.3 were essentially already given in Hadamard [38] for general second-order hyperbolic equations, but the microlocal point of view we use here simplifies the proof. For similar results in the nonlinear hyperbolic Cauchy problem, we refer to Majda [54] (existence of a local solution) and to Bony [3] and Chemin [5] (propagation of singularities).
Bibliography
References quoted in the text.
[1) Alinhac, S., and Gerard, P., Operateurs pseudo-differentiels et theoreme de Nash-Moser, Orsay Plus, Universite de Paris-Sud, 1989. [2] Beals, R., and Fefferman, C., On local solvability of linear partial differential equations. Ann. Math. 97, 482-498 (1973). [3] Bony, J-M., Calcul symbolique et propagation des singularites pour les equations aux derivees partielles non-lineaires. Ann. Sc. Ec. Norm. Sup. Paris 4 14, 209-246 (1981). [4] Bony, J-M., and Lerner, N., Quantification asymptotique et microlocalisations d'ordre superieur 1, to appear. [5] Chemin, J-Y., Regularite de la solution d'un probleme de Cauchy fortement non-lineaire a donnees singulieres en un point. Ann. Inst. Fourier Grenoble 39(1), 101-121 (1989) and references therein. [6] Coifman, R., and Meyer, Y. Au-dela des operateurs pseudo-differentiels. Asterisque 57, Paris 1978. [7] Hormander, L., Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. [8] Hormander, L., The Analysis of Linear Partial Differential Operators, Vol. I (Chap. 7) and Vol. III (Chaps. 18 and 23), Springer-Verlag, Berlin, 1983/85.
[9] Kumano-Go, H., Pseudo-differential Operators, MIT Press, Cambridge, MA, 1981. [10] Nirenberg, L., and Treves, F., On local solvability of linear partial differential equations: I. Necessary conditions; II. Sufficient conditions; Correction, Commun. Pure App!. Math. 23, 1-38 and 459-509 (1970); 24, 279-288 (1971). [11] Taylor, M., Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.
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[14] Adams, R.A., Sobolev Spaces, Academic Press, New York, 1975. [15] Atiyah, M.F., and Singer, I.M., The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69, 422-433 (1963). [16] Beals, R., A general calculus of pseudo-differential operators, Duke Math J. 42, 1-42 (1975). [17] Beurling, A., Quasianalyticity and general distributions, Lectures 4 and
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[20] Boutet de Monvel, L., Operateurs pseudo-diffdrentiels analytiques et opdrateurs d'ordre infini. Ann. Inst. Fourier Grenoble 22(3) (1972).
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[22] Bros, J., and Iagolnitzer, D., Support essentiel et structure analytique des distributions, Seam. Goulaouic-Lions-Schwartz 1974-1975, exposes no. XVI and XVIII, Ec. Polytechnique, Paris. [23] Calderdn, A.P., Uniqueness in the Cauchy problem for partial differential equations. Amer. J. Math. 80, 16-36 (1958). [24] Cafderdn, A.P., Boundary value problems for elliptic equations, Outlines of the Joint Soviet-American Symposium on Partial Differential Equations, pp. 303-304. Novosibirsk, 1963.
[25] Calderdn, A.P., and Vaillancourt, R., On the boundedness of pseudodifferential operators. J. Math. Soc. Japan 23, 374-378 (1971). [26] Calderdn, A.P., and Vaillancourt, R., A class of bounded pseudodifferential operators, Proc. Nat. Acad. Sci. U.S.A. 69, 1185-1187 (1972). [27] Calderdn, A.P., and Zygmund, A., Singular integral operators and differential equations, Amer. J. Math. 79, 901-921 (1957). [28] Cauchy, A., Cours d'Analyse de 1'Ecole Royale Polytechnique, lere partie: Analyse Alge brique, Debrue, Paris, 1821.
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[29] Dirac, P.A.M., The physical interpretation of the quantum dynamics, Proc. R. Soc. London Ser. (A) 113, 621-641 (1926-1927). [30] Duistermaat, J.J., Fourier Integral Operators, Lecture notes at the Courant Inst. Math. Sci., New York, 1973. [31] Duistermaat, J.J., and Honnander, L., Fourier integral operators II, Acta Math. 128, 183-269 (1972). [32] Egorov Yu, V. The canonical transformations of pseudodifferential operators, Uspekhi Mat. Nauk XXIV-5(149), 235-236 (1969). [33] Fefferman, C., and Phong, D.H., On positivity of pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 75, 4673-4674 (1978). [34] Folland, G.B., Lectures on Partial Differential Equations at the Tata Inst. of Bombay, Springer-Verlag, Berlin, 1983. [35] Fourier, J., Thgorie Analytique de la Chaleur, M6moire lu devant I'Acad. Sci. Paris (1807/11), edited by Didot, Paris, 1822. [36] Friedrichs, K., On the differentiability of the solutions of linear elliptic differential equations, Comm. Pure App!. Math. 6, 299-326 (1953). [37] Gelfand, I.M., and Silov, G.E., Generalized Functions I and 2, Academic Press, New York, 1964/1968. [38] Hadamard, J., Le Probleme de Cauchy et les Equations aux Derivees Partielles Hyperboliques, Hermann, Paris, 1932. [39] Heaviside, 0., On operators in mathematical physics, Proc. R. Soc. London 52, 504-529 (1893) and 54, 105-143 (1894). [40] Honnander, L., Differential opeators of principal type, Math. Ann. 140, 124-146 (1960). [41] Hormander, L., Pseudo-differential operators, Comm. Pure App!. Math. 18, 501-517 (1965). [42] H6rmander, L., Pseudo-differential operators and nonelliptic boundary value problems, Ann. Math. 83, 129-209 (1966). [43] HOrmander, L., Pseudo-differential operators and hypoelliptic equations, Amer. Math. Soc. Symp. on Singular Integrals, 138-183 (1966).
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Index of Notation
Multiindices and geometry of R"
a! = (al!)...(a"!),
For a = (al,..., an) E Z+, Ial =
0
otherwise.
xn)1/Z,x"=xT...x"-,BR
{xElR';jxl
Forj=l,...,n,a,=a/ax,,D2=-i03=(t/i)(a/ax,),a"=a;'...
ann = 01"1/8x1' ...axR^, D° = (-i)IOIa Spaces of functions defined on W1 S: Schwartz space, i.e., space of COO functions cP such that all the norms
E R and Ia+/31 < k}
kPIk =
for k E Z+ are finite. Co : space of test functions, i.e., of C°° functions with compact support. When E Co and supp cP C S1, we also write cP E Co (S2). P: space of C°G functions with polynomial growths at infinity: we write /' E P° if /, is continuous and (1 + IxI2)-NV,(x) is bounded for some
NEZ+; we write V) EPif a°iI'EP°for all aEZ+. LP: Lebesgue space, i.e., space of measurable functions u such that the norm NormLP (u) =
(f1ux1'dx 103
Index of Notation
104
if I
J
u(x)v(,r) dx
if uv E L', if u E L2.
IIuII1i = (u. IC) = J Iu(x)I2dx Spaces of distributions defined on R"
S': space of temperate distributions (topological dual of S), i.e., space of semi-linear forms on S 3 ,; - (u. ) E C such that I (a,
)I
for all
ES
and some CEll andNEZ+. D': space of distributions (topological dual of C°`); we also use the notation D'(S2) for the dual of Co (( ). E': space of distributions with compact support (topological dual of C"); we also use the notation E'(Q) for the dual of C"(SZ). H': Sobolev space of exponent s E P, i.e., subspace of S' formed by distributions u such that ASir E L2 where AS(S) = (1 + They are Hilbert spaces when equipped with the norms
IIuII' = (2ir)-"IIA'hIIi1= (27r)"
1(1 +
We also use H-" = U5H5 and H" = f1SH'. Other spaces on IR"
S"': space of symbols of order in, i.e., of C' functions a(x, ) defined on IR" x R" such that the functions All3l-' c7Qa are bounded for all a.,3 E Z" , where A113I-' is the function (I +Il;I2)(1,SI-"')/2. We also use S-" _ S' = and S ( ) _ {a E C" (St x W'); yea E S'" for all ,^ E Co (i2)}. For S,".. F) and Sa(F), see page 76. A"': space of amplitudes of order in, i.e., of C" functions a(x) defined on 11P" such that all the norms IIIaIIIk = sup{I(1 +
IxI2)-m"2d'ra(x)I;x
E R" and Ial < k}
(in III 111k in is implicitly fixed) for k E 7L+ are finite.
Index of Notation
105
space of pseudodifferential operators of order m as defined on page 50. We also use %F-OC = n,,41, (smoothing operators), *" = U,,, %' and W (1l) (resp. 'comp("),'Y"'(I')) for operators with symbols in Sla(Sl) (resp. Scomp(r), Sia(r))Miscellaneous
A unit test function is a nonnegative C" function supp V C tai
satisfying
J2(x)dx=l.
and
Operations on distributions: for u E S' and ' E S
(u, V) _ (u,,) (symmetry)
Ax) = 0(-x) p(x) = f
(Fourier transformation)
d
(D' u, y') = (u, D" p) (differentiation) (Vu, gyp) = (u,
gyp) (multiplication by a && E P)
(u, 0 = (u, ) (conjugation)
°a*
(Tyu,,P) = (u, r_ yp) (translation by a y E R") supp u = support of u, see page 16 sing supp u = singular support of u, see page 16 WFu = wave front set of u, see page 77
r_yyJ(x) = y:(x - y)
Operations on symbols: for a and b E Sc
(x, ) =
(27r)_ f e-i(y,77)a(x
- y,
71) dy di (adjunction)
r!)b(x - y, ) dy d? (composition) a#b(x, = (27r) -n f Char a = characteristic set of a, see page 76-77
Index
adjoins, adjunction, 37, 48 amplitudes, 33 a priori estimates, 70-71
symbols, operators, 42-43, 55, 62, 71, 76 energy
conservation of, 86 estimates, 80 bicharacteristic curve, 79 binomial coefficients, formula, 3
Cauchy problem, 79, 85 characteristic set, 76-77 classic symbols, 32 composition, compound, 37, 48 conic, 76 cotangent bundle, 60-61
differential operator, symbol, 22, 30, 32, 39, 43, 50, 60, 61-62
differentiation of distributions, 13-14 distributions, 12-23 in an open set, 15-17 products of, 14-15, 26, 92 real-valued, 24 restrictions of, 92-93 temperate, 12 with compact support, 16-17
elliptic estimate, 62
finite propagation speed, 87 Fourier integral operators, 67 Fourier transform(ation), 9-13 Friedrichs's lemma, 65
GArding's inequality, 56
Hamiltonian vector field, 78 Holder spaces, 19, 25-26 holomorphic functions. 23-24 hypoellipticity, 28, 55, 64, 65
inversion formula, 11, 13
Klein-Gordon equation, 93
Lebesgue spaces, 7-8 Leibniz's formula, 4 local property, 50, 61-62
107
108
Index
local solvability, locally solvable, 28, 64, 70, 89-91
symbol, 29-30 pseudolocal property, 51, 55
microlocalization, 64, 76 microlocal property, 51, 76, 78 multiindex, 2
quasi-elliptic, quasiprincipal symbols,
45-46 restrictions of distributions, 92-93
noncharacteristic, 76 S,11.1,, calculus, 45-46, 64, 89
operators, 47-61
Schur's lemma, 54, 63
differential, 22, 30, 50, 60, 61-62 elliptic, 55, 62, 71
Schwartz space, 5
Fourier integral, 67 of principal type, 70-71 principally normal, 70-71 properly supported, 61 pseudodifferential, 50 smoothing, 50, 55 wave, 83
singular support. 16 smoothing operator, 50, 55 Sobolev spaces, 18 subellipticity, 65-66 support, 6, 16 symbols, 28-43 differential, 30, 32, 39, 43
oscillatory integrals, 32-37, 44
Paley-Wiener-Schwartz theorem, 17, 25 Parseval's formula. 11, 13 partitions of unity, 7 Peetre's inequality, 21 Poisson brackets, 44, 70 polyhomogenous symbols. 32 polynomial growths at infinity, 5-6 principally normal, 70-71 principal symbol, 43, 70 principal type, 70-71 products of distributions, 14-15, 26. 92 projection theorem, 77 propagation of singularities, 78-79 properly supported, 61 pseudodifferential operator, 50
semi-linear form, 8
elliptic, 42-43, 55, 62, 76 idempotent, nilpotent, 45 of the adjoint, compound operator. 37, 48 polyhomogenous (or classic), 32 principal, 43 (pseudodifferential), 29-30
quasi-elliptic, quasi-principal, 45-46 Taylor's formula, 2 test functions, 6 unit test function, 7
wave equation, operator, 83 wave front set, 77, 91
7158 ISBN 0-8493-7158-9