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.)I-:::::, ).-n/2
as>.-+
+oo,
whenever a(O) =F 0. In this chapter we shall study some natural generalizations of these results, where the integrals now will take their values in LP spaces. Specifically, we shall consider operators of the form T.xf(x) = [
JR..
ei.\cf>(x,y) a(x, y) f(y) dy,
>.
> 0,
where now a is a smooth cutoff function and
..)
f(y) dyt
~ Cll/llp,
we see that (2.1.3) implies the Hausdorff-Young inequality:
llfllp• ~ Cllfllp·
57
2.1. Non-degenerate Oscillatory Integral Operators Before proving the theorem it is illustrative to restate the non-degeneracy condition in an equivalent form. Let ct/> = {(x,t/>~(x,y),y,-tf>~(x,y))}
c
T*IRn
X
T*IRn
be the canonical relation associated to the non-homogeneous phase function tf>. Then by the remark at the end of Section 0.5, Ct/> is Lagrangian with respect to the symplectic form~ A dx- dTJ A dy. Moreover, the condition (2.1.1) is equivalent to the condition that the two projections ct/>
./
'\.
(x,t/>~(x,y)) E T*IRn
(y, -t/>~(x, y)) E T*IRn
be local diffeomorphisms (i.e., have surjective differentials). In results to follow we shall encounter variations on this geometric condition. It of course means that
IVx[tf>(x,y)- t/>(x,z)]l ~ IY- zl,
IY- zl small.
(2.1.1')
This is what we shall~ in the proof of Theorem 2.1.1 and it of course just follows from the fact that
8 2 tf>(x,y)) 2 Vx[tf>(x,y)-t/>(x,z)] = ( ax;aYk (y-z)+O(Iy-zl ). Proof of Theorem 2.1.1: B_v using a smooth partition of unity we can decompose a(x,y) into a finite number of pieces each of which has the property that (2.1.1 1) holds on its support. So there is no loss of generality in assuming that
IVx[tf>(x,y)- t/>(x,z)] I~ ely- zl on supp a,
(2.1.4)
for some c > 0. To use this we notice that
IIT.x/11~ = where
K,x(y,z) =
1
Jf
K,x(y,z)f(y)f(z)dydz,
ei.X(tf>(x,y)-t/>(x,z)]
a(x,y)a(x,z)dx.
R"
However, (2.1.4) and Lemma 0.4.7 imply that
IK_x(y,z)l $ CN(1 + -XIy- zi)-N 'VN.
(2.1.5)
58
2. Non-homogeneous Oscillatory Integral Operators Consequently, by Young's inequality, the operator with kernel K>. sends £ 2 into itself with norm O(A-n). This along with (2.1.5) yields
IIT>./11~ $CA-n 11/11~,
I
as desired.
2.2. Oscillatory Integral Operators Related to the Restricted Theorem AB we pointed out in the introduction, there are very important situations where the non-degeneracy hypothesis of Theorem 2.1.1 is not fulfilled. For instance, if tjJ(x, y) = lx - Yl then the mixed Hessian of tjJ cannot have full rank n since for fixed x the image of y -+ ¢>~ ( x, y) is sn- 1 (and hence this map can not be a submersion). In fact, one can check that for this phase function the differentials of the projections from Ct/> to T*IRn have corank 1 everywhere. Nonetheless, since the image of y-+ t/J~(x,y) has non-vanishing Gaussian curvature, we shall see that the oscillatory integrals T>. associated to this phase function map .LP(IRn)-+ Lq(IRn) with norm O(A-nfq) for certain p and q. Moreover, we shall actually prove a stronger result which says that certain oscillatory integral operators sending functions of n - 1 variables to functions of n variables satisfy the same type of estimates. AB before, these oscillatory integrals will be of the form T>.f(z) =
J
ei>.tf>(z,y) a(z, y) f(y) dy,
(2.2.1)
except, now, a E COO(IRn x IRn- 1) and tjJ is real and C 00 in a neighborhood of supp a. Thus, in what follows, z shall always denote a vector in IRn and y one in IRn- 1 . The canonical relation associated to tjJ is now a subset of T*IRn x T*IRn- 1 . The hypotheses in the oscillatory theorem will be based on the properties of the projections of Ct/> into T*IRn- 1 and the fibers of T*IRn. First, the non-degeneracy hypothesis is that rankdliT•Rn-1
=2(n- 1},
(2.2.2)
if JIT•Rn-1 : C4> -+ T*IRn- 1 ia the natural projection. Thus, (2.2.2) says that the differentials of this projection must have full rank everywhere,
2. 2. Opemtors Related to the Restricted Theorem
59
or, to put it another way, the mixed Hessian always has maximal rank, that is, 1)2q,
rank (ay;l)zk)
= n- 1.
(2.2.2')
Assumption (2.2.2) of course is an analogue of the hypothesis in Theorem 2.1.1 and it is enough to guarantee that T.\: V(IR.n- 1 )-+ Lq(IR.n) with norm O(,x-(n- 1)/q) if q ~ 2 and p ~ q'. To get better results where the norm is O(.x-nfq) a curvature hypothesis is needed. To state it, we first notice that, since Ct/> = {(z, forn (2) 1 $ s < ~ for n = 2.
~ 3;
Notice that the exponents in the Corollary are conjugate to those in Theorem 2.2.1. Keeping this in mind, let us see how (2.2.6) follows from (2.2.5). We first notice that there is no loss of generality in assuming that
s= for some C
00
(1Rn- 1 )
{(y,h(y))}
function h. If we then let
<jJ(z,y) = (z, (y,h(y))} it follows that
n-
(R")
11911 L 21"•U> , " (R")
we conclude that (2.2.9) holds if and only if (2.2.10) An advantage of this reduction is that, unlike (2.2.9), both sides involve functions of the same number of variables. In proving this inequality we may assume that z E R.n splits into variables z = (x, t) E R.n-l x R. such that
det(ax~~k (x,t;y)) #0
(2.2.11)
onsuppa.
With this in mind, we define the frozen operators
rt f(x)
=
L. -ei~tP(x.t;y)a(x. 1
t; y) f(y) dy.
Then
TAT~g(x, t) = /_: r((rl~)* [g( ·, t')] (x) dt'.
(2.2.12)
Using the argument in our observation that the n-dimensional HardyLittlewood-Sobolev inequality follows from the one-dimensional version, we see that (2.2.10) would follow from Proposition 0.3.6 and the estimate
_ 2(n+l)
p-
n+
.
(2.2.10')
2.2. Operators Related to the Restricted Theorem
63
In fa.Ct, this inequality, (2.2.12), and Minkowski's integral inequality give
IIT~Ttg( • I t)IILP1 {Rn-1) n(n-1)
$C>..- n+ 1
100 e 1) _ 1t-t'l-1+;-;r l!g(·,t')IILP(R"-1)dt'.
00
Raising this inequality to the p' power, integrating with respect to t, and applying the one-dimensional ij:ardy-Littlewood-Sobolev inequality leads to
IIT~T~giiLP'(Rn)
(100 -oo 1100 _ 00 1t- t'l-1+ e ;-;r llg( ·, t')IILP(R"-1) dt' lp'dt)~ 1/p' n(n-1) (100 ) = C')..- n(n-1) $ C')..- n+ 1 llg( t)II~"(Rn-1) df n+ 1 IIYIILP(Rn)•
$ C>..- !!..1.!!..=!1 n+ 1
1 )
1/p
-OO
•!
as desired. So we are left with proving (2.2.10'). We shall do this by interpolating between £ 2 -+ L 2 and £ 1 -+ L estimates. The first is easy since (2.2.11) and the non-degenerate oscillatory integral theorem give ~ n-1 1 " 2 !ITt IIIP(R"-1) ::; c>..-11/IIP(R"-1)·
00
Since the adjoint operator satisfies the same bounds, we conclude that (2.2.13) So far we have only used the non-degeneracy hypothesis. To prove the L estimate we must now use the curvature hypothesis. To do this, we first notice that the kernel of T((Tt )* is
£ 1 -+
00
K~t' (x, x')
= {
}Rn-1
ei~(tl>(x,t;y)-1/>(x',t';y)] a(x, t; y) a(x', t'; y) dy.
We may assume that a has small support and we claim that
!K~t'(x,x')!
$ C(>..!(x,t)- (x',t')l)-"; 1
(2.2.14)
To prove this, we use Taylor's formula to write
t/J(x,t;y)- t/J(x',t';y) = (Vx,tt/J(x,t;y), ((x,t)- (x',t')))
-+ O(l(x,t)- (x',t')l 2). Thus, if (x, t)- (x', t') belongs to a small conic neighborhood of the unit vectors ±v(x, t) in (2.2.4'), we can apply the stationary phase formula (1.1.19) to get (2.2.14) for this case, assuming, as we may, that (x', t')
2. Non-homogeneous Oscillatory Integral Operators
64
is close to (x, t). On the other hand, if (x, t) - (x', t') is outside of this conic neighborhood, it follows from the definition of v(x, t) that
IVy[(x, t;y)- (x', t';y)] I~ cl(x, t)- (x',t')l,
for some c > 0,
provided again that (x', t') is close to (x, t). So in this case, Lemma 0.4.7 implies that a stronger estimate holds, where in the right side of (2.2.14) we may replace (n - 1) /2 by any power. The estimate (2.2.14) of course implies that n-1 n-1 II Tt (Tt') *I 11 Loo(Rn-1) ~ c>..-2 llfii£1(R"-1)· It-t ~-~
~
1
2
(2.2.15)
n-1 1 n-1 d . Smce 2(n+1) = 2" n+l an
). - "~";11) =
(>.. -(n-1)) =~: (>..- n;1) n~1'
It - t'l- 1+( ;-;,)
=
(It - t'l- "; 1 ) "~ 1 1
the missing inequality, (2.2.10'), follows from interpolating between (2.2.13) and (2.2.15). I
Proof of Theorem 2.2.1, Part (2): Here n = 2 and we must show that forq=3p1 and 1 ~p.. - 2 /q 11/IILP(R)·
(2.2.16)
To take advantage of the fact that q > 4, we write
(T~f(z) ) 2
=IIei~[tfJ(z,y)+tfJ(z,y')]
a(z, y)a(z, y') f(y)f(y') dydy'. (2.2.17)
We would like to use the non-degenerate oscillatory integral theorem to estimate the Lqf 2 (JR 2 ) norm of this quantity, However, the mixed Hessian of the phase function ~(z; y, y')
= ¢(z, y) + 4>(z, y')
has determinant
~ "'"1 y l '+'z2y
1-"'
11 ( 11 11 ( r/J~ ( z, y1 ) - '+'Z1Y' A.11 ( z, y1 )"' "'"1 Y' - '+'Z1Y z, y )"' '+'Z2Y '+'Z2Y z, y ) I (2 •2• 18) '+'z2y' 1
and so the assumptions are not verified as the determinant vanishes on the diagonal y = y'. On the other hand, the Carleson-Sjolin assumption (2.2.8) implies that
8 iP Idet (aza(y, y')) I~ ely- y'l 2
(2.2.19)
2.9. Riesz Means in llln
65
for some c > 0 if !y-y'l is small. In fact, at the diagonal, they' derivative of (2.2.18) equals the determinant in (2.2.8). There is no loss of generality in assuming that (2.2.19) holds whenever the above integrand is nonzero. To exploit (2.2.19) as well as the fact that the integrand in (2.2.17) is symmetric in (y, y'), it is convenient to make the change of variables
u= (y-y',y+y'). Then since
!dujdyl
= 2, it follows that
a2~ )I ~ c!utl· I (8z8u det
(2.2.19')
Also, since ~(z;u) is an even function of the diagonal variable u1, it must be a C 00 function of u~. So we make the change of variables
v = (~u~,u2)· Then since
!dv/dul =!uti, it follows that ldet
(:::JI ~c.
(2.2.19")
We can now apply Corollary 2.1.2. If r' = q/2 and if r is conjugate to it follows that
r',
IIT~JIIiq(R2)
=
II(T~/) 2 11Lr'(R2)
=
11//
f(y)f(y') !dv/d(y,y')l-l dviL
ei'P(z,v)
~ c>.-2/r' (//lt(y)f(y') ldv/d(y,y')l-llr dvy/r ~ c>.-4/q
(If
!f(yW lf(y'W IY- y'l-l+a dydy') l/r'
with a: = 2 - r. Since a:=
[p/r] -1- [(p/r)'rl,
(2.2.16) follows from applying the Hardy-Littlewood-Sobolev inequality.
I 2.3. Riesz Means in llln Let q({) be homogeneous of degree one, C 00 ' and nonnegative in an\ 0, with n ~ 2. For 6 ~ 0, we define the Riesz means of a given function by
S~f(x) =
(21r)-n
f
}Rn
ei(x,e) (1-
q({/>.)) 6 j({) d{. +
(2.3.1)
2. Non-homogeneous Oscillatory Integral Operators
66
tt = t
for t > 0 and zero otherwise. Iff E S (and thus j E S) it follows from Fourier's inversion theorem that S~f(x) -+ f(x) for every x as >. -+ +oo. In this section we shall consider the convergence of the Riesz means of LP functions. To apply the oscillatory integral theorems of the previous section, we shall assume that the "cospheres" associated to q, Here
6
E
= {~: q(~) = 1},
(2.3.2)
have non-vanishing Gaussian curvature. Note that, since q is smooth and V q ¥- 0, E is a C 00 hypersurfa.ce. The assumption regarding the curvature of E is equivalent to (
a2q )
rank a~ja~k
(2.3.3)
= n- 1. The most important case is when q(~) = 1~1· For a given 1 ~ p
~
oo we define the critical index for LP(JR.n):
6(p)
= max{nl~- ~1- ~.o}.
Note that 6(p) > 0 p a necessary condition for
S~f
-+
f
(2.3.4)
fl. [2n/(n + 1), 2nf(n- 1)). It is known that , in V
when
f
E V, p # 2,'
is that 6 > 6(p). When 6(p) = 0 this is a theorem of C. Fefferman. The other cases follow from the fact that the kernel of S~ is in LP only when 6 > 6(p) for 1 ~ p ~ 2nf(n + 1). (This can be seen from Lemma 2.3.3 below.)
Sf.
Theorem 2.3.1: Let 8 6 = Then if the cospheres E associated to q(~) have non-vanishing Gaussian curvature, and (1) n ~ 3 and p E [1, 2 ~:l)J U [ 2 ~~l), oo), or (2) n = 2 and 1 ~ p ~ oo,
it follows that
1186 IIILP(Rn) ~ Cp,6llfiiLP(Rn)
when 6 > 6(p).
(2.3.5)
Corollary 2.3.2: If p is as in Theorem 2.3.1 then
S~f
-+
f
in V(JR.n),
when f E LP(IR.n) and 6 > 6(p). The proof of the corollary is easy. First (2.3.5) implies that the means S~ are uniformly bounded on LP when 6 > 6(p). Next, given f E LP and
2.9. Riesz Means in Rn
67
e > 0 there is a g E S such that II/-gllv < e, and hence IIS~f- S~gllv = O(e). Since both g and g are inS, Fourier's inversion theorem and the uniform boundedness of S~ : V' --+ V' imply that S~g --+ g in V'. This implies the assertion, since, by Minkowski's inequality, one sees that IIS~f- fllv = 0(3e) when A is large. The proof of Theorem 2.3.1 requires knowledge of the convolution kernel of the operator S 6 :
This kernel will be a sum of two terms, each involving an oscillatory factor and an amplitude. To describe the phases, note that our assumptions imply that given X E an \ 0, there are exactly two points 6(x),{2(x) E E such that the differential of the map
vanishes when e = e;(x). In fact, e;(x) are just the two points in E with normal x. We now define (2.3.6) Since e;(x) is homogeneous of degree zero and smooth in an\ 0, it follows that '1/J; is also smooth and it is of course homogeneous of degree one. Note that in the model case where q = 1e1, we would have '1/J; = ±lxl. Lemma 2.3.3: The kernel of S 6 can be written as al(x)ei1/Jl(x)
6
K (x) =
(1 + lxl)
~ 6 2
+
+
a2(x)ei1/12(x) ~ 6' (1 + lxl) 2 +
(2.3.7)
where the a; are bounded below near infinity and satisfy
I(:xrx a;(x)l :::; Calxl-lal
'Va.
(2.3.8)
The proof of the lemma is a straightforward application of the stationary phase method. First, let TJ E C~(an) be a function which is constant on dilates of E and equals one near E but vanishes near the origin. Then the difference of K 6 and
is in S, since the Fourier !_ransform of the difference is in C~. Therefore we need only show that K 6 can be written as in (2.3.7). ·
2. Non-homogeneous Oscillatory Integral Operotors
68
If we recall that Theorem 1.2.1 says that the inverse Fourier transform of surface measure on the cosphere satisfies
-
(27r)
n
{ "( ) b1(x)ei1/ll(x) JE e' x,w du(w) = (1 + lxl)(n-1)/2
~(x)ei1/12(x)
+ (1 + lxl)(n-1)/2'
for functions bj satisfying (2.3.8), then it follows that two terms. The first is
I
K6 is the sum of
b1(px)eiP'Pl(x) 1 6 d (1 + lpxl)(n-1)/271(P)( - p)+ p = ei1/ll(x)
I
i
bt((p + 1)x) ij(p + 1) eiP'Pl(x) dp, (1 + (p + 1)lxl) (n- 1)/ 2 +
where ij is 71 times a smooth function coming from the polar coordinates. Since lt/11(x)l ~ clxl for some c > 0, and since the Fourier transform of is homogeneous of degree -6 - 1 and smooth away from the origin, it follows that the last integral is of the form a1(x)/(1 + lxl)(n+l)/ 2+6 with a1 satisfying (2.3.8). Since the other term has the same form, we are done. To apply Lemma 2.3.3 we shall use a scaling argument and the following.
Pt
Lemma 2.3.4: Ifa(x,y) is in C[f and suppa(x,y) [~,2]} then
C
{(x,y): !x-y! E
f ei>.,P;(x-y)a(x,y)f(y)dyll L•(Rn) ~CqA-n/qllfiiLP(Rn)• II }Rn
(2.3.9)
if q = ~:!;}p' and (1) 1 ~ p ~ 2 for n ~ 3; (2) 1 ~ p < 4 for n = 2.
Furthermore, for a given tPj the constants depend only on the size of finitely many derivatives of a. To verify this result we need to check that tPj(x, y) = tPj(x-y) satisfies the n x n Carleson-Sjolin condition in Corollary 2.2.3. This is easy since, by construction,
Vxt/lj(x,y) = ej(X- y). This implies that for every xo, Sx0 = {VxtPj(xo,y)} = E, and, hence, the curvature condition holds. Since the Gauss map sn- 1 --+ E is a local diffeomorphism, it follows that, for fixed y, the differential of x--+ ~ .( ,._.,) t= E has constant rank n-1. So the non-degeneracy condition is
2.3. Riesz Means in R.n
69
also satisfied. In view of these facts, Lemma 2.3.4 follows from Corollary 2.2.3. Notice that p < q in (2.3.9). So if we use Holder's inequality and the fact that a vanishes for y outside of a compact set, we get
{ ei>..P;(x-y) a(x, y) f(y) dyll Lq(Rn) ~ Cq A-n/q llfiiLQ(Rn)· II }Rn To apply this, we fix {3 E Ccf(IR.) supported in {8: satisfying }:~00 {3(2-l s) = 1, 8 > 0. We then define
Kf(x) = {3(2-1lxi)K6 (x),
8
(2.3.9')
E [~,2]} and
l = 1, 2, 3, ... ,
and Kg(x) = K 6 (x)- L:i Kf(x). We claim that if q is as in (2.3.91), then we have the following estimate for convolution with these dyadic kernels:
(2.3.10) The estimate for l = 0 is easy since Kg is bounded and supported in the ball of radius 2. Since this implies that Kg is in £ 1 , Young's inequality implies that convolution with Kg is bounded. To prove the estimate for l > 0 we note that the Lq -+ Lq operator norm of convolution with Kf equals the noJin of convolution with the dilated kernels
Kf(x) = AnKf(Ax),
A= 21•
But this function equals n-1 " a 1 (Ax)ei>..p 1 (x) n-1 " .,.1, ( ) A-2 -v {3(! I} = A-2 -v (">. ) '"Y'1 X x (A -1 + lxl)(n+l)/2+6 a1 , x e
plus a similar term involving tP2· Since (2.3.8) 1(/xt'a1(A,x}j ~Co. for every a, (2.3.91) implies that
implies that
IIKf * /llq ~ Cq An; 1 -~- 6 11/llq•
i
But ~ = 6(q) and so (2.3.10) follows. Finally, by summing a geometric series we conclude that (2.3.10) implies that for q as above
IIS6 /llq ~ Cq,oll/llq, 6 > 6(q). This proves (2.3.5) for q ~ 2(n + 1)/(n- 1),n ~ 3, and q > 4 for n = 2. The remaining cases follow from duality and interpolating with the trivial inequality IIS6 /112 ~ 11/112· If we drop the assumption that E has non-vanishing Gaussian curvature, it becomes much harder to analyze the I.l' mapping properties
70
2. Non-homogeneous Oscillatory Integral Operators of S 6 . This is because the stationary phase methods used before break down, and, except in special cases, it is impossible to compute K 6 . Nonetheless, it is not hard to prove the following. Theorem 2.3.5: Assume only that q(e) is homogeneous of degree one and both C 00 and nonnegative in an\ 0. Then, if S 6 is the Riesz mean associated to q, it follows that (2.3.11)
Proof: The operator we are trying to estimate is
s6 f(x) = (2tr)-n }Rn f ei(x,e) (1- q(e))t i<e) ~· If {3 is as above and we now let
sf f(x) = (2tr)-n
1
ei(x.e) /3(2l(1-
q(e)))(1- q(e)) 6 i<e)~. l
= 1, 2, ...•
00
sgf(x)=S6 J(x)- LSff(x), 1 #
I
then sg : £ 1 - £ 1 since its kernel is in we could show that, for any e > 0,
II Sf /1!1
s.
Hence we would be done if
~ Ce 2-(o- n;l-e]lll/1!1·
(2.3.11 1)
To prove this we now let Kf be the kernel of this operator. Then, for every N,
Kf(x) = (2tr)-nlxi-2N
I
ei(x,e) (
-~)N {f3(2l(1-q(e)))(1-q(e))6} ~·
Since ( -~)N {/3(2l(1 - q(e)))(1- q(e)) 6} that
= 0(22Nl T
6l),
it follows
IKf(x)l ~eN 2- 6l!xf2li- 2N, and so, for large N and fixed e > 0,
{
Jlxl>2O.
-oo
It then must follow that
L
'I'!. I= rt. (
!; )
if >. E [2k, 2k+l].
li-kl:5ko+2
Using this we get 00
IT! f(x)l 4 S:
L k=-oo
sup
I'I'!.f(x)l 4
AE[2•,2•+1)
Based on this, we claim that (2.4.1') follows from the localized esti-
mate sup I'Tff(x)lll S: C6ll/ll4, II AE(2•,2Hl) 4
k E Z.
(2.4.4)
This claim is not difficult to verify. In fact, since, by Theorem 0.2.10,
74
2. Non-homogeneous Oscillatory Integral Opemtors
II(L: lfkl 2 ) 112 114 :::; Cll/114, we see that (2.4.4) and Minkowski's inequality yield
l
supl7'ff(x)l 4 dx::; .bO
f: I f: I lfkl
sup
k=-oo
lrf(
>.E[2•,2"+1]
:::; C
L !;)(x)l li-kl:5ko+2
I
L: 00
dx
4 dx
k=-oo
:::; c (
4
lfk1 2 )
14
2•
dx:::;
c'
I
ltl 4 dx.
k=-oo
By dilation invariance, (2.4.4) must follow from the special case where k = 0. Furthermore, if we again use the fact that the difference between Tf.f(x) and S1J(x) is pointwise dominated by the Hardy-Littlewood maximal function and recall (2.4.3), we deduce that this in turn would be a consequence of
II
sup IS!f(x)lll :::; AE[1,2]
4
C.sll/114.
if suppj c {e: lei E [2-ko- 2 ,2ko+ 2)}. (2.4.111 )
This completes the first reduction. We now turn to the main step in the proof which involves the use of the half-wave operators eitQ defined in (2.4.2). To use them we need to know about the Fourier transform of the distribution T~ = lri 6X(-oo,O] E S'(R.). By results in Section 0.1, the Fourier transform must be homogeneous of degree -6 - 1 and C 00 away from the origin. This is all that will be used in the proof; however, we record that the Fourier transform is actually the distribution
where
Using this we first notice that
2.4,. Kakeya Maximal Functions, Maximal Riesz Means in .R2 75 Based on this we make the decomposition 00
sit
= si,o I + 2: si,k /, k=1
where, if f3 is as above, then for k = 1, 2, ...
si,k I = (21r)-1c6 A-6
I:
e-iAt /3(2-kltl)
(t + io)-6-1 eitQ I dt. (2.4.5)
These are Fourier multiplier operators, of course, and the multiplier,
mi k• behaves like 2- 6kp(2k(A- q(,))), with p a fixed Ccr'(IR) function. In ~articular, mi k becomes an increasingly singular function around A· E, but, on the' other hand, its L 00 norm is decreasing like 2-k6 as k
-+
+oo. Taking this into account, we naturally expect that for any
E>O sup ISikf(x)lll $ Ce2-( 6 -e)k 11/114, II AE[1,2) ' 4
for
f
as in (2.4.111 ). (2.4.6)
By summing a geometric series this of course implies (2.4.1") .• The inequality is trivial when k = 0. In fact, the Fourier' transform of the distribution [1 - 'Ek:: 1 /3(2-kt)] (t + i0)- 6- 1 must be smooth since the latter is in E'. Consequently, 0 must be a C 00 multiplier.
mi,
Therefore, since we are assuming that / has fixed compact support, the special case where k = 0 in (2.4.6) follows from the Hardy-Littlewood • maximal theorem. To estimate the terms involving k > 0 we shall use the following elementary result.
Lemma 2.4.2: Suppose that F is C 1 (JR). Then if p > 1 and 1/p +
1/r/ = 1
s~p IF(A)jP $
IF(O)jP +
P(
I
IF(A)IP
dA) 1/p' ·
(I
IF' (A)jP
dA) 1/P
To prove the lemma one just writes
IF_(A)jP = IF(O)jP
+loA ~IF(s)IP ds = IF(O)IP +p loA IFip-1·F' ds
and then uses HOlder's inequality to estimate the last term. To apply the lemma in the special case where p =. 2, we now fix p E Ccr'(IR) satisfying suppp C [~,4] and p = 1 on [1,2]. Then, using
76
2. Non-homogeneous Oscillatory Integral Operators (2.4.5) and Holder's inequality, we conclude that the left side of (2.4.6) is majorized by
I (/1 j x I (/I j -Jx
p(A)e-i.Xt.B(2-kltl)(t + i0)-6-1eitQ f dt 12
dA) 1/2 L
(p(A)e-i.Xt).B(2-kltl)(t + i0)-6-1eitQ f dt 12
dA) 1/2 L
Plancherel's theorem implies that this expression is controlled by
r I 2 dt ) 11211 I (lltie[2•-1,2k+1J e' !I 1t12+26 r I 2 dt )11211 I( Jlt1E[2•-1,2k+l) x e' !I 1t126 "tQ
4
.tQ
4
Thus (2.4.6) would be a consequence of the estimate
I( r
I "tQ 12
JltiE[2•,2k+l) e' f
dt ) 112 ltil+e
~ Cell/114,
~
4
suppj C {': 1!1 E [1,2]},
E
> 0.
By taking complex conjugates one sees that we need only estimate the expression involving t integration over [2k, 2k+l]. Moreover, by a change of scale argument, one sees that this in turn is equivalent to proving the following result. ' Proposition 2.4.3: For which
E
> 0 and r > 1 there is a constant Ce for
(2.4.7) Remark.
In Chapter 6 we shall see that the operators
1·
are bounded on L 4 (R.2 ) if and only if u ~ On the other hand, (2.4.7) implies that for any u > 0 this expression is in £ 4 (£2 ([1, 2])).
2.4. Kakeya Maximal Functions, Maximal Riesz Means in lll2 77 Let us introduce some notation that will be used in the proof of the proposition. First, if p and {3 are as above and if we set :Frf(x, t)
= p(t) f ei{x,e) eitq({) /3(1el/r) /(e) d{, JR2
then it is clear that (2.4.7) would be a consequence of
II(/ I:Frf(x, t)1 2 dtf 12 IIL'(R2 )~ ee Te llfiiL'(R2)·
(2.4.7')
The first step in the proof of this inequality is to decompose the square function inside the L 4 norm for fixed x. To do this we write
where
:F~f(x, t) = p(t) f
JR2
ei{x,e)eitq(e>a(r- 112 q(e)-
j) /3(1el/r) /(e) d{
for some function a: E e 0. Since this implies that for such (e, 17), m~J and all of its derivatives are O((r + l(e,f1)1}-N), the estimate (2.4.10) follows as before. In view of (2.4.10), the desired inequality would follow from showing that (2.4.11)
The key observation behind this inequality is that, if 'Ill has small enough support and if we let
then the ~ourier transform of (x, t) -+ ~,j f(x, t) vanishes outside of A~,j. To exploit this we shall use the following geometric lemma.
Lemma 2.4.5: lfe > 0 is as in the definition of A~,j' there is a uniform constant C so that, if j and j' are fixed,
•
Here A~,; + A~, ,j' denotes the algebraic sum of the two sets, and if we identify R.2 and c in the usual way, arg e~ is the argument of the unit vector e~. We postpone the proof of this result. It is based on the fact that our assumption that the cospheres associated to q(e) have non-vanishing Gaussian curvature implies that the cones {(e, q(e))} c R.3 \0 have one non-vanishing principal curvature. This allows the overlap to be finite rather than just O(rl/2). To apply this result, by symmetry, we may clearly restrict the v summation in (2.4.11) to indices as in Lemma 2.4.5. It then follows that the fourth power of the resulting expression can be estimated in the
2.4. Kakeya Maximal Functions, Maximal Riesz Means in IR2 81 following way using Plancherel's theorem and the overlap lemma:
=I L:!L:i:·j I i:',j' ,,
2
dxdt
j,j' v,v'
~I L:IL:(xA:.;+A:,.)(e, 11)(.r:J !)" * (i:',j' /)"1 2 ded11 j,j' v,v'
~ere IL:L:!i:·iti:',i't! 2 dxdt j,j' v,v'
Consequently, if we use (2.4.10) again, we conclude that we would be done if we could show that
II(~ IF:•j /1 2) 112 I L4(R ~ Clogr llfii£4(R2)· 3)
(2.4.12)
V,]
This completes the orthogonality arguments. The proof of (2.4.12) will be based on estimating the L2(R.3 ) -+ L 2(R.2) norm of a "Kakeya maximal function" which involves averages over rays on light cones associated to q(e). To use the Kakeya maximal estimates to follow, we shall need to make use of a variant of Littlewood-Paley theory which involves a decomposition of the Fourier transform into functions supported in a lattice of cubes. To describe what is needed, we first recall that the e-support of the symbols o:~,j of the operators :F;•i are sets Q 11,j all of which are comparable to cubes of side-length r 112 . In particular, each intersects at most a fixed number of cubes in a r 112 lattice of R.2. With this in mind, we let a be as above and set
fm(e)
= o:(r-l/26- m1)o:(r-l/ 2e2- m2)f(e),
m
= (mt,m2) E 'l'}.
Thus EmEZ2 /m =f. In addition, if for a given (v,j), we let Iv,j c Z 2 be those m for which :F;•i fm is not identically zero, it follows from the above discussion that Card (I11,j) ~ C,
(2.4.13)
82
2. Non-homogeneous Oscillatory Integral Operators with C being a uniform constant. Also since the sets Q11,j have finite overlap there is an absolute constant C such that Card {(v,j): mE I 11J} :5 C 'Vm E Z2.
(2.4.14)
We now turn to (2.4.12). Let
f ei(x-y,{) eitq({) o:~,j (t, e) cl{ JR2
K!;J (x, t; y) =
be the kernel of :F;•i. In a moment we shall see that
f IK~J(x,t;y)ldy :5 C JR2
(2.4.15)
uniformly. Thus, the Schwarz inequality and (2.4.13) give
1.1"!/-,j f(x,t)1 2 :5 C
/1 2:: f 2::
fm(Y)I 2 1K!;,j(x,t;y)ldy
mEI~.;
:5 C'
l/m(Y)I 21K!;..i(x, t;y)l dy.
mEI~.;
If we now use (2.4.14) we see that, for a given g(x, t),
1/ L; 1.1"!/-,j J2::
f(x, t)l 2 g(x, t) dxdtl
II,J
:5 C
m
l/m(Y)I 2 su~{ "•3
/rJ{IK~·j
(x, t; y)llg(x, t)l dx dt} dy.
Since the square of the left side of (2.4.12) is dominated by the supremum over allllgll2 = 1 of the last quantity, by applying the Schwarz inequality, we see that we would be done if we could prove
(L, •:JI J/ IK;J (
y
x, t; )I g(x, t) dxdt
I' dy)
112
:5 C llogrl 312 llgii£2(R3)•
(2.4.16)
as well as the following.
Lemma 2.4.6: For 2 :5 p :5 oo,
II ( 2:: l/mi 2 ) 112 IILP(R2) :5 C II/IILP(R2)· m
Proof of Lemma 2.4.6: When p = 2 the inequality holds because of Plancherel's theorem. If we apply a vector-valued version of the M. Riesz interpolation theorem (which follows from the same proof) we conclude
!LI. Kakeya Maximal Functions, Maximal Riesz Means in JR2 83 that it suffices to prove the inequality for p = oo. By dilation invariance we may take T = 1. Then, if Q = [-1r, 1r] 2 and if a is the inverse Fourier transform of a,
(L
lfm(0)1 2) 112
mEZ2
:5
L (L nEZ2 mEZ2
11
f(y- 21rn)a(21rn1- yl)a(21rn 2 - y2)e-i(m,y) dyi 2
Q
Y
12
But the terms in the absolute values of the last expression are the Fourier coefficients of the function there. So Parseval's formula says that the last expression is
211"
L (lif(y- 21rn) a(21rn1- Y1) a(21rn2- Y2)1 2 dy) 112 nEZ2
Q
:::; Cllflloo .
L (lia(21rn1 - Y1) a(21rn2 - Y2) 12 dy) 112 nEZ2
Since
aE S
Q
the last sum converges and this finishes the proof.
I
To prove (2.4.15) and (2.4.16) let us introduce some notation. Given a direction (cos 8, sin 8) E S 1 we let 'Y8 C R.3 be the ray defined by
'Y8 = { (x, t):
X+ tq{(cos8,sin8) =
0}.
(2.4.17)
Clearly these light rays depend smoothly on 8 and our assumption that rank ( a[;l~c) = 1 implies that 'Y8 -=f. 'Y8' if 8 -=f. 81 • We also remark that the union of all the rays is the dual cone to {(e,q(e))} c R.3 \0. The estimate about the kernels which we require is the following lemma.
Lemma 2.4. 7: If e~ are the unit vectors occurring in the definition of F;J and if (cos 811 , sin 811 ) = (~, then given N there is an absolute constant C N for which
IK~,j (x, t; y)l :::; eN r( 1 + r112 dist ((x- y, t), 'Yo..)) -N.
(2.4.18)
Note that (2.4.18) implies that, for fixed y, the kernels are essentially supported in a tubular neighborhood of width r- 1/ 2 around 'Y8~ + (y, 0). With this in mind one gets (2.4.15).
84
2. Non-homogeneous Oscillatory Integral Opemtors Proof of Lemma 2.4.7: To prove (2.4.18}, we recall that o:~J has e-support contained in a sector r~.~ of IR2 \0 of angle O(r- 112 ). Thus, since q( is homogeneous of degree zero,
This implies that for some c > 0
provided that dist ( (x - y, t}, 'Yo.,) is larger than a fixed constant times r- 1/2. By applying Lemma 0.4.7 we get (2.4.18}, since l(&t'o:~'jl $
C0 r-lal/ 2 and l{supp~o:~'j}l $Cr.
I
Using (2.4.18}, one sees that (2.4.16} follows from the following Kakeya maximal theorem.
Proposition 2.4.8: Let 'YO be defined by (2.4.17}. If, for a given 0 < 6 < ~, we let
'Ro = { (x, t) E IR2 x [0, 1] : dist ( (x, t), 'YO) < 6}, it follows that
Proof of Proposition 2.4.8: First we fix a E Cox>(R.2 ) satisfying 0 $ a and alsop E Cox>(JR) vanishing outside a small neighborhood of(}= 0. We set
where X[o, 1] denotes the characteristic function of [0, 1]. Then it suffices to prove that Aog(y)
(211"}-211 ei((y-x,~)+t(q((cosO,sinO),~)] ao(8; t, e)g(x, t) dedxdt = (211")-211 ei((y.~)+t(q((cosO,sinO),~)] ao(8; t, e)9(e, t) dedt
=
R3
R2
R R2
2.4. Kakeya Maximal Functions, Maximal Riesz Means in IR2 85 satisfies
IA9g(y)lll L2(R2) $ C llog6I 3/ 2 11YII£2(R3)· ll sup 9
(2.4.19)
Here "-" denotes the partial Fourier transform with respect to x. Just as before, to apply Lemma 2.4.2, we need to make a couple of reductions. First, if we define dyadic operators
A9"g(y) = (21!")-2
II
ei[(y,~)+t(q((cos9,sin9),~)] X
where
.B(Iel/r) ao(8; t, erq·(e, t) dedt,
.B is as above, then it suffices to prove that IA9g(y)lll ::; Clogr IIYII£2(R3)• ll sup 9 £2(R2 )
T
> 2.
(2.4.19'}
E A~"+~. where Cis a fixed constant l.E(1,2]
IS~,kf(x)lll :5 Ce,6 2-(6-6(p)-e)k 11/llp, P
for k = 1, 2, ... , f as in (2.4.1 11 ).
(2.4.61)
However, by using the definition of t5(p) we see that, by interpolating with the L 4 estimate, this would follow if we could prove the inequality for the special cases of p = 2 and p = oo. The inequality for p = 2 trivially holds by repeating the arguments used for L 4 since a stronger version of (2.4. 7) holds:
II (121eitQ /12 dtf'2112 :511/112. due to the fact that eitQ maps £ 2 to itself with norm 1. To prove the £ 00 estimate, one notices that the proof of Lemma 2.3.3 shows that for>. E (1, 2J the kernels of the operators S~ k are dominated by ( n-1 Ce2- 6-,--e)k(1 + lxl)-n-e, n = 2, I
which of course implies (2.4.6') for p = oo.
I
2..4. Kakeya Maximal Functions, Maximal Riesz Means in JR2 89 To conclude this section let us see how the arguments which were used to prove the maximal theorems involving singular multipliers can also be used to prove maximal theorems for operators with singular kernels. Specifically, as above let E = { x E R.2 : q(x) = 1}. Then we have the following "circular maximal theorem."
Theorem 2.4.9: If we assume that q is as above and if we assume that E has non-vanishing curvature, then for p > 2
(2.4.25) Here dn denotes Lebesgue measure on E.
Remark. This result is sharp in the sense that (2.4.25) can never hold for p $ 2. To see this there is no loss of generality in assuming that (-1,0) E E and that the normal there is (-1,0). If we take fv to be the characteristic function of the ball of radius times lx21-l/Pilog lx2!1-I, it follows that fv E LP for p > 1. But the nonzero curvature of E implies that for p ~ 2
!
h
fv(x- ty) dn
=+oo
for x =(xi. t),
lx1l
< 1.
On the other hand, a limiting argument shows that for p > 2 the maximal function in (2.4.25) extends to a bounded operator on LP(R.2) even though functions in this space are only defined almost everywhere. To prove the maximal theorem, we notice that Theorem 1.2.1 implies that the Fourier transform of dn is the sum of two terms each of which is of the form a(,)eiq(e)
(1 + lel)l/2, where ±q(') has the same properties as q and
Thus if we abuse notation a bit and replace
q by q, and then set
Qf(x t) = (21T)-2/ei(x,e)eitq(e) a(t') , (1 + ltel)l/2
j(')~
,
2. Non-homogeneous Oscillatory Integml Opemtors
90
then (2.4.25) would follow from
~~~~giQ/(x,t)IIILP(R2 ) ~ Cpii/IILP(R2)•
P > 2.
To prove this we define the dyadic operators
Then since the supremum over t of the absolute value of the difference beoo
tween g f(x, t) and }: Q2" f(x, t) is dominated by the Hardy-Littlewood k=l
maximal function of f, it suffices to prove that for
T
>1
As before, since these are dyadic operators, one can use Littlewood-Paley theory to see that the inequality holds if and only if II sup IQT f(x, t)lll tE(1,2) P
~ Cp T-e(p) 11/llp.
2
< p < oo.
(2.4.26)
As with Riesz means, the proof is based on establishing the inequality for the special case of p = 4 and then interpolating with easy £ 2 and £ 00 estimates. The main estimate then turns out to be II sup IQd(x, t)ill tE(1,2) 4
~ Ce T-l/B+e 11/114 Ve > 0.
(2.4.26')
If we let p E C3"((l, 2)) then we can apply Lemma 2.4.2 in the case where p = 4 to see that II supt lp(t)Qd(x, t)1111 is dominated by
a(te) /A(t) ricll3 II/ ei(x,~) eitq(~) f3(1cl/r) ._ (1 + t!ei)l/2 ._ ....._ L4 (R2 x(I,2)) x II/ ei(x.~) eitq(~) f3(1el/r) a(te) d ( p(t)a(te) ) } II {. lq(e)p(t) (1+tlel)l/2 + dt (1+tlel)l/2 f(e)cte L'(R2x(l,2J>. A
x
Thus, since q(e) ~ron supp.B(Iel/r), if we now let
Fd(x, t) = p(t)
f ei(x,~) eitq(~).B(Iel/r)
a(t,e) J(e) de,
!L4. Kakeya Maximal Functions, Maximal Riesz Means in R.2 91 where we 88Sume 1(8t)i(~ta(t,e)l::; C;0 (1 + lel)-lol, then it suffices to prove the estimate (2.4.27) This of course should be similar to (2.4.7'). In fact, if n is the function occurring in the proof of this inequality and if we now set
:F~f(x, t) = p(t)
j
ei(x,{) eitq({) .B(Iel/r) n(r_ 1, 2q(e) -
i) a(t, e> /(e) cte,
then the difference between this operator and F~/(x, t) defined in (2.4.8) has L 4 (R.2)-+ L 4 (R.3 ) norm O(r-N) for any N. The important thing about F~f(x, t) is that its partial Fourier transform with respect tot vanishes for TJ fl. I~= [r 112j-2rl/2,r 1/2j+2rl/2j. Note that, as j and j' vary over the ~ r 112 indices for which :F~ =F 0, XIt+It' (TJ) ::; Cr 112. Thus, Plancherel's theorem and Schwarz's in-
k
J,3
equality yield
i:
~~ ~~f(x, t) 14 dt = 3
i:
14
~~f(x, t)F{ f(x, t) 12 dt
3.3
= (27r)- 1/ _ :
14x
~
1t+It'
(TJ}(F~ff*(J-t' !)1 2 dt
3·3
::; Cr 112 / _ : 41 (F~ff*(ij' !)1 2 dt 3.3
Therefore, since the difference between :F~ and ~~ has rapidly decreasing norm, we conclude that
II:FT IIIL4 (R3)
::; Cr 118 ll (
L I:F~/1 2 ) 112 IIL'(R3) +eN T-N IIIIIL (R 4
2 )·
j
Finally, since the proof of (2.4. 7") applies to the slightly different operators in this context we can estimate the right side and get (2.4.26'). To finish the proof of (2.4.26) we see, by interpolating with the £ 4 estimate, that this inequality would follow from
II tE(l,2) sup lgd(x,t)lll :=;CIIfllv. P
p=2orob.
92
2. Non-homogeneous Oscillatory Integral Operators The inequality for p = 2 follows from applying Lemma 2.4.2 and using the £ 2 boundedness of eitQ. The inequality for p = oo follows from the fact that the kernels off-+ Qd( ·, t) are uniformly in L 1 (JR 2 ).
Notes Theorem 2.1.1 is due to Hormander [6]. Theorem 2.2.1 is due to Carleson and Sjolin [1] and Hormander [1] in the two-dimensional case and to Stein [4] in the higher-dimensional case. We have given a slightly different proof of Stein's oscillatory integral theorem which uses an argument in Journe, Soffer, and Sogge [1]. See also Oberlin [1]. Recently, Bourgain [3] has shown that this higher-dimensional oscillatory integral theorem cannot be improved in the sense that there are phase functions satisfying the Carleson-Sjolin condition for which (2.2.5) does not hold for p > 2. On the other hand, it is not known whether the range of exponents in Corollary 2.2.3 is sharp. The two-dimensional restriction theorem is due to Fefferman and Stein (Fefferman [1]) and Zygmund [2]. The £ 2 restriction theorem is due to Stein and Tomas (Tomas [1]). Bourgain [2] has recently improved Corollary 2.2.2 slightly in higher dimensions. T):vl maximal theorem for Riesz means in two dimensions is due to Ca.rbery [1], although the proof given here is slightly different, as it is based on the alternate proof of the circular maximal theorem of Bourgain [1] given in Mockenhaupt, Seeger, and Sogge [1].
Chapter 3 Pseudo-differential Operators
The rest of this course will mainly be concerned with "variable coefficient Fourier analysis" -that is, finding natural variable coefficient versions of the restriction theorem, and so forth. One of our ultimate goals will be to extend these results to the setting of eigenfunction expansions given by the spectral decomposition of a self-adjoint pseudo-differential operator. To state the results, however, and to develop the necessary tools for their study, we need to go over some of the main elements in the theory of pseudo-differential operators. These will be given in Section 1 and our presentation will be a bit sketchy but essentially self-contained. For a more thorough treatment, we refer the reader to the books of Hormander [7], Taylor [2], and Treves [1]. In Section 2 we present the equivalence of phase function theorem for pseudo-differential operators. This will play an important role in the parametrix construction for the (variable coefficient) half-wave operator. Finally, in Section 3, we present background needed for the study of Fourier analysis on manifolds, such as basic facts about the spectral function. We also present a theorem of Seeley on powers of elliptic differential operators which allows one to reduce questions about the Fourier analysis of higher order elliptic operators to questions about first order operators.
3.1. Some Basics We start out by defining pseudo-differential operators on an. We say that a function P(x,e) in C 00 (an X an) is a symbol of orderm 'or more
3. Pseudo-differential Operators
94
succinctly P(x,e) E
sm, if, for all multi-indices a, {3, (3.1.1)
To a given symbol we associate the operator
P(x, D)u(x) = (21r)-n
= (21T)-n
ff f
ei(x-y,{) P(x, e) u(y) d(.dy
ei{x,{) P(x, e) u(e) d(..
By the second formula it is clear that P(x, D)u is well-defined and C 00 0 has symbol 1 • • • e~n. when u E S; also notice that D 0 = ( = We shall say that an operator P : COO -+ C 00 is a pseudo-differential operator of order m if it equals P(x, D), for some P(x, e) E sm. Finally, an operator R which is in s-oo = nmsm is called a smoothing operator since all derivatives of its kernel are 0((1 + lx- yi)-N) for any N, and, hence, R: S' -+ C 00 (1Rn). It is not hard to see that Puis well-defined when u is a distribution. Note that the distribution kernel of P(x, D) is the oscillatory integral
t Jx )
(21r)-n
f
ei(x-y,{) P(x, e) d(. = lim (21r)-n e--+0
eo er
f
ei{x-y,{) p(ee)P(x, e) d(.,
(3.1.2) where p E COO equals one near the origin. By the results in Section 0.5, this definition does not depend on the particular choice of p. Moreover, away from the diagonal, { (x, y) : x = y}, the kernel of P is C 00 , and all of its derivatives are O(lx-yi-N) when lx-yl is larger than a fixed positive constant. Taking this into account one can see that pseudo-differential operators are pseudo-local: If u is C 00 in an open set 0, then so is Pu. Thus, they do not increase the "singular support" of u. However, unlike differential operators, they are usually not local-that is, it is not usually true that supp Pu C supp u. A chief result is that pseudo-differential operators are closed under composition. Theorem 3.1.1: Suppose that P(x, e) E sm and Q(x, e) E SP.. Then P(x, D) o Q(x, D) is a pseudo-differential operator having a symbol P o Q E sm+p. given by
(3.1.3)
3.1. Some Basics
95
By this we mean that p
0
Q-
~~De p
L
.
(!r'Q
E
sm+J1-N
'VN.
(3.1.3')
loi./21T)n
JJ
ei.\((x-z),(11-Z)) P(x, >.17)Q(z, ~) d17dz, (3.1.5)
where we have set >. = 1~1 and ~ = ~/ >.. Note that the phase function appearing in the last oscillatory integral,
~ = ((x- z), (17- ~)), satisfies Th~,
(3.1.6) V 71 ,z~ = (x- z,~ -17). by Lemma 0.4.7, if p(s) E C~(IR) equals 1 near 0 and vanishes
when lsi > ~' we see that, modulo a function which is smooth and rapidly decreasing in ~, P o Q equals (>./21T)n
JJei.X{(x-z),(11-Z)) p(lx- zl)p(l -1171)P(x, >.17)Q(z, ~) d17dz.
We can estimate this using the method of stationary phase. First, notice that (3.1.6) implies that the unique stationary point of ~ is (17,z) = (~,x). The Hessian of (17,z)-+ ~is the 2n x 2n matrix ( 0
-1
-1) 0
(I= n x n identity matrix),
which has determinant 1. Thus, since ~ vanishes at the stationary point and since the integration is over JR.2n, we see that formula (1.1.20) implies that, modulo a symbol in sm+~£- 1 , (Po Q)(x,~) equals (>./21T)- 2n/ 2 • (>./21T)n P(x, >.~) Q(x, ~) = P(x, ~) Q(x, ~).
Thus. we have nroved (3.1.3') for N =
1
9. Pseudo-differential Operators
96
The proof of the formula for the general case is' similar, except that one needs to use Taylor's formula: P(x, TJ) Q(z, ~)
=
[~, (:~ rk P(x, ~)(TJ- ~) 0]
L O~lai,I.BI 0.
(3.2.2)
Clearly rp = (x- y, e} satisfies (3.2.2), and we have seen that in this case Prp is a pseudo-differential operator of order m. Our main result is that this is true for other phase functions as well. Theorem 3.2.1: Suppose that rp is as above and that
cp(x, y, 0 = (x - y, e} + O(lx- Yl 2 1el).
(3.2.3)
Then, if P(x, y, e) E sm, Prp is a pseudo-differential operator of order m, and, moreover, if we set P(x,e) = P(x, x,e), it follows that Prp-P(x, D) is a pseudo-differential operator of order (m- 1). Conversely, given a pseudo-differential operator P there is an operator Prp such that P- Prp is a smoothing operator. "
By (3.2.3) we of course mean that
for every n. Before turning to the proof, let us first see that this result implies that compactly supported pseudo-differential operators are invariant under changes of coordinates. Specifically, suppose that n and nit are open subsets of R.n and that K. : n -+ nit is a diffeomorphism; then one can define a map sending functions in n to functions in nit by setting X
En.
Our next result says that there is also a push-forward map for pseudodifferential operators. Corollary 3.2.2: Let tt, n, and nit be as above. Then if P(x, e) E sm vanishes for X not belonging to a compact subset of n, there is a 2 We have placed this condition only to allow our assumptions on rp to be of a local nature in what follows.
3. Pseudo-differential Operators
102
pseudo-differential opemtor Prt(x, D) which is compactly supported in Ott such that, modulo smoothing opemtors, y
Furthermore, if ~'(x)
= (a~;ax)
= ~(x).
(3.2.4)
is the Jacobian matrix,
Plt(~(x), 2(x,z)} L=x
a2 }z=x• modulo a symbol of order m. Also, since the first term on the right side of (3.2.9) is linear, it follows from our choice of 4> that {Dr;ei~4>2 }z=x = D~(icp(x,y,t;,)) when Ia I = 2. Consequently, if we let
Q1(x,y,t;,) =
I:
~~ (:erQ(x, Vxcp)D~icp E s~-'- 1 ,
lal=2 then (3.2.11) gives the following result.
(3.2.12)
3. Pseudo-differential Operators
106
Theorem 3.2.3: Assume that P(x, y, e) E sm and that cp(x, y, e) is as above. Then if Q(x, D) is a pseudo-differential operator of order p. and Q1 is defined by (3.2.12), e-icpQ(x, D) [Peicp] =
L lo!.;E;, j=l 00
I=
(3.3.2)
LE;, j=l
where E; : £ 2 -+ L 2 are the projection operators that project onto the one-dimensional eigenspace E; with eigenvalue >.;. By (3.3.1) the eigenvalues are all positive, and, since +oo is the only limit point, we may assume that they are ordered so that
>.1 $ >.2 S: · · · . Since the eigenspaces are mutually orthogonal (3.3.2) gives 00
ll/lli2(M) =I: IIE;/IIi2(M)• j=l
Let {ej ( x)} be the orthonormal basis associated to the spectral decomposition. Then of course
E;f(x) = e;(x)
JM f(y) ej(Y) dy.
9.3. Self-adjoint Elliptic Pseudo-differential Operators
109
We claim that e;(x) E C 00 (M). To see this, we notice that one can use Theorems 3.1.5 and 3.1.6 to prove that for every k = 1, 2, ... there is an inequality (3.3.3)
This inequality and the Sobolev embedding theorem imply that e;(x) is C 00 since it belongs to every Sobolev space L~. It also shows that the spectrum of any C 00 ( M) function is rapidly decreasing, that is,
>.f
I/
VN
u(y) e;(Y) dyl-+ 0
if u E C 00 (M).
(3.3.4)
In the next chapter, we shall investigate the distribution of the eigenvalues of P, or, more specifically, the behavior of the function
N(>.) = #{j : >.; :::; >.}. Let us see now that this function is tempered. To do this we let
L
S>.l(x) =
E;l(x)
>.; $>. be the projection operator onto U>.;$>.£;. Then (3.3.3} implies'
IIS>.IIIL: :::; c>.klllll£2· By the Sobolev embedding theorem we see from this that if IE U>.;$>.£; then
lllllvx• :::; CIIIIIL[n/2+11 .n/ 2+111111£2, 2
where [ ·] denotes the greatest-integer function. However, if we define the spectral function
L
S>.(x,y) =
e;(x)e;(y),
>.; $>. then since this is the kernel of S >.,
I/
S>,(x,y)l(y)dyl = IS>.I(x)l:::; c>.ni 2+111111L2·
Since this inequality holds for all that
I
E L2 , by duality, we can conclude
( / IS>.(x,y)l2dyf/2:::; c>.n/2+1, which by the above gives the pointwise estimate x,yEM.
9. Pseudo-differential Operators
110 Finally, since
N(>.) =
JM S>,(x,x)dx,
it follows that this function is tempered. Later on we shall improve these estimates and see that N(>.) = c>.n + O(>.n- 1 ) when P satisfies some natural assumptions. So far we have been discussing first order operators; however, the following result reduces the study of properties of the spectrum etc. of elliptic self-adjoint operators of arbitrary order to the first order case.
Theorem 3.3.1: Let P E 'W~(M) be self-adjoint and positive with m > 0. Then the operator P 11m defined by the spectral theorem is in w~ 1 . Its principal symbol is (p(x, e)) 11m, if p(x, e) is the principal symbol of P. Let us sketch the proof. In local coordinates we choose
where x E C 00 vanishes near the origin but equals one near infinity. If we let Q1 = (Q1 + Qi)/2 then Q1 is self-adjoint, and it is also classical by Theorem 3.1.3. In addition, Theorem 3.1.1 implies that (QI)m- P is in wcl'- 1 . By the arguments of Section 3.1 we can recursively choose self-adjoint classical pseudo-differential operators Q; of order 2 - j such that P- (Q1 + · · · + QN)m is in wcl-N for every N. Thus, if we let Q E W~ be a representative of the formal series E Q;, we conclude that P- Qm is smoothing. Since each Q; is self-adjoint, Q equals its adjoint up to a smoothing operator, and so, after possibly adding such an operator, we can assume that the Q .constructed is self-adjoint. Since it is elliptic and first order (3.3.1) implies that it has at most finitely many non-positive eigenvalues, and, therefore, after possibly modifying it on a finite-dimensional space, we can assume that Q is positive as well. To summarize, we have seen that there is a positive first order self-adjoint elliptic Q E W~ such that
P-Qm=R, where R is smoothing. We claim that Q- P 11m is smoothing as well. To see this let 'Y C C be the contour shown:
3.3. Self-adjoint Elliptic Pseudo-differential Operators
Then, by Cauchy's formula and (3.3.2),
p-1/"'1 = ~ ~
211"Z
1
z-1/m (z- P)-1 dz,
'Y
and
=~ 2n
1
z- 1/m(z- P+ R)- 1 dz.
'Y
Therefore,
Q-1- p-1/m =
-1·1 ;11"~ i
211"Z
=
z-1/m [(z- p + R)-1- (z- P)-1] dz
'Y
z- 1/m [(z- P + R)- 1R(z- P)- 1] dz.
However, since R is smoothing one can see that the operator inside the brackets is smoothing and that the integral converges and defines a smoothing operator R1. But this implies the desired result, since Q _ p1/m = -QR1 P 11m is smoothing.
111
112
3. Pseudo-differential Operators
Notes The theory of pseudo-differential operators goes back to Hormander [2) and Kohn and Nirenberg [1) and it has as its roots earlier work of Calderon and Zygmund [1) (see also Calderon [1) and Seeley[1)). The equivalence of phase function theorem presented in Section 3.2 is taken from Hormander [4) and the important Theorem 3.3.1 is from Seeley [2).
Chapter 4 The Half-wave Operator and Functions of Pseudo-differential Operators
Let M be a compact manifold, and suppose that Pis in \11~1 (M) with positive principal symbol p(x, e). Then if P is self-adjoint, as before, let N(>.) denote the number of eigenvalues of P which are ~ >.. The main result of this chapter is the sharp Weyl formula,
where c denotes the "volume" of M, that is, c = (211")-n /r { cl{dx. l{(x,e)ET· M :p(x,e):'5:1}
The proof of the sharp Weyl formula begins with the observation that N(>.) =
JM S>.(x,x)dx,
where S>.(x, y) is the kernel of the summation operator S>.f = L>.,_ ·. E;f (see Section 3.3 for the notation). This operator is an example of a function of P since if we let m(r) = m>.(r) be the characteristic function of the interval ( -oo, >.], then 00
S>.f = m(P)f =
L m(>.;)E;f· j=l
4. The Half-wave Opemtor
114
If we let m(t) be the Fourier transform of m, then Fourier's inversion formula gives
S,f ~
(2~)- 1
= (211")-1
1: L:
ffl(t)[t,•"''E;t] dl m(t)eitP I dt.
The operator eitP is the solution operator to the Cauchy problem for the half-wave operator (i8/fJt + P), and since we shall see that we can compute the kernel of this operator very precisely when It! is small, it is natural to also study
S>.l = (211")- 1
L:
p(t) m(t) eitP I dt
when p is a bump function with small support. If p equals 1 near the origin then a Tauberian argument involving sharp £ 00 estimates for £ 2-normalized eigenfunctions will show that the difference between the kernels of S>. and S>. is O(.~n- 1 ). Finally, we shall be able to compute the kernel of the "local operator" S>. and see that S>.(x,x) = c(x)An + O(An- 1 ), -
#
for the appropriate constant c(x). This along with the estimate for the remainder yields the sharp Weyl formula. Variations on this argument will be used throughout much of the monograph, and at the end of the chapter we shall see "that if m( A) E S~-', then m(P) is a pseudo-differential operator of order p. on M whose principal symbol is m(p(x,e)).
4.1. The Half-wave Operator In this section we shall construct a parametrix Q(t) for the Cauchy problem: (i8/fJt
+ P) u(x, t) =
0,
u(x,O)
= l(x).
( 4.1.1)
Since 00
u(x, t) =
L eit>.; E;l, j=1
we shall denote the operator sending 1 to u(x,t) by eitP. The main result here is that eitP can be represented by a Fourier integral. More
4.1. The Half-wave Opemtor
115
specifically, in local coordinates, we shall find that, for small times t, modulo a smoothing operator, eitP equals Q(t)l = (211")-n
JJei~(x,y,e)eitp(y,e) q(t, x, y, e) l(y) ~dy,
(4.1.2)
where rp is the type of phase function studied in Section 3.2 and q E S 0 , that is,
Using stationary phase, we shall observe that the singularities of the kernel of eitP are very close to the diagonal when t is near 0, and it is precisely this fact that will allow us to reduce global problems, such as proving the sharp Weyl formula or obtaining sharp estimates for the size of eigenfunctions, to localized versions which lend themselves directly to the techniques developed before. Let us now turn to the details. We shall first work locally constructing the parametrix for functions with small support, and then, at the end, using a partition of unity, glue together the pieces. In local coordinates on a patch n c M, the self-adjoint elliptic operator P E \11~1 is (modulo an integral operator with C 00 kernel) of the form P(x, D) where 00
P(x, e)
"' L
PI-j(X, e),
j=O
with the P; being homogeneous of degree j and P1 = p(x, e) the principal symbol. To be able to apply the results of the last section, we assume that the density dx on M agrees with Lebesgue measure in the local coordinates. This can always be achieved after possibly contracting n. Let us fix a relatively compact open subset w of n and try to construct an operator of the form (4.1.2) so that Q(t)l is a parametrix for eitP I whenever I E C3"(w) and t is small. In order to apply Theorems 3.2.1 and 3.2.3, we shall want rp to be in 8 1 and also satisfy (4.1.3)
while q E SO must have small enough support around the diagonal {(x,y): x = y} so that
IVecpl ;:::: cix - Yi
on supp q.
( 4.1.4)
4. The Half-wave Operator
116
The first step in achieving this for small time t is to construct rp. If Q(t) is to be an approximate solution to (4.1.2), then (iaj&t + P)Q(t) must be smoothing, and this will be the case if is in
s-oo,
(4.1.5)
where we have set
4'(t, x, y, e> = cp(x, y, e> + tp(y, e). But, by Theorem 3.2.3, the quantity in (4.1.5) equals [p(y,e)- p(x, Vxrp)] · q +lower order terms. Thus, it is natural to require that rp solves the eikonal equation
p(x, Vxcp) = p(y,e),
lx - Yl
small.
(4.1.6)
To see that a solution verifying (4.1.3) always exists, we shall need a fundamental result from the theory of Hamilton-Jacobi equations, whose proof will be postponed until the end of this section. Lemma 4.1.1: Letp be a realC00 function in a neighborhood of(O,T/) E R.n X R.n such that p(O, 71) = 0,
a
~ p(O, 71)
"# 0,
and suppose that 1/J is a real-valued C 00 function in R.n-l satisfying
a axJ_'1/J(O) = '1i'
j = 1, ... ,n -1.
Then there is a neighborhood of the origin and a unique real-valued solution ¢ E C 00 of the equation
p(x, Vx¢) = 0 satisfying
¢(x1 , 0) = '1/J(x'),
Vx¢(0)
= '1·
Here x' = (xt, ... , Xn-d·
Since the principal symbol is real and homogeneous of degree one, we need only apply the lemma to see that (4.1.6) can be solved when lei = 1; for then, if we extend rp to be homogeneous of degree one, (4.1.6) will be satisfied for all The resulting phase function need not be smooth ate= 0; however, this is irrelevant since the contribution in
e.
4.1. The Half-wave Operotor
117
the integral (4.1.2) coming from small' is smooth. With this in mind, if we fix the parameters y E w and ', then Lemma 4.1.1 implies that there is a unique function rp = rp(x, y, ')solving the nonlinear equation (4.1.6) that satisfies the boundary conditions
rp(x,y,,)
=0
when (x- y,,)
=0
and Vxrp
='
when x
= y, (4.1.3')
when xis close toy. Since (4.1.3') clearly implies (4.1.3), rp has the right properties. Having chosen rp, we need to impose a condition on the symbol q so that when t = 0, Q(t)- I is smoothing if I is the identity operator. To do this we recall that Theorem 3.2.1 implies that there is a symbol
vanishing outside a small enough neighborhood of the diagonal so that rp is defined there and
(211")-n
II
ei.] denotes the characteristic function of the interval ( -oo, >.], then S>., = (211")- 1
I:
(X(-oo,>.J)"(t)eit? dt.
The Fourier transform of the characteristic function of ( -oo, 0] is the distribution
i(t+i0)- 1 =i lim - 1-. e--+0+ t + Ze
=1r6o(t)+i~, t
(4.2.3)
and so (4.2.4)
4·
126
The Half-wave Operator
Note that (t + i0)- 1 is only singular at t = 0; therefore, since Theorem 4.1.2 allows us to compute the kernel of eitP very precisely when ltl < c, it might be reasonable to compareS>. with
S>. = (211")- 1
I:
p(t) i(t + i0)- 1 e-i>.t eitP dt,
(4.2.5)
if p is the Fourier transform of some p E S and satisfies
ltl < c/2,
p(t) = 1,
ltl >c.
and p(t) = 0,
If we let X( -oo,>.] = X( -oo,>.] * p, note that the "approximate summation operators" are given by 00
(4.2.51)
S>.f = LX(-oo,>.J(A;)E;f. 1
Let us now use Theorem 4.1.2 to compute S>.(x,x). We assume that coordinates are chosen so that, around x EM, dx agrees with Lebesgue measure. If this is the case then (4.2.5) implies that
S>.(x,x) = (21r)-n- 1
+ (211")- 1
jj p(t)i(t+i0)- 1 q(t,x,x,e)eit(p(x,e)->.)~dt
I:
p(t)i(t + i0)- 1 R(t, x, x)e-i>.t dt, (4.2.6)
where q and R are as in Theorem 4.1.2. However, since R is C 00 , Corollary 0.1.15 implies that the last term in (4.2.6) is 0(1). To prove that the first term on the right side of (4.2.6) is equal to c(x)An +O(An- 1 ), note that Taylor's formula implies that
q(t, x, x, e)= q(O, x, x, e)+ tr(t, x, x, e) where r E SO. However, it only contributes O(An- 1 ) to the error since (4.2.3) gives
JJp(t)(t + io)-
1 tr(t, x, e) eit(p(x,e)->.)
=
jj p(t) r(t, x,e)
=
J
eit(p(x,e)->.)
r(A- p(x,e),x,e) ~.
dt~
dt~
127
4.2. The Sharp Weyl Formula where, with an abuse of notation, f( ·, x, ~)denotes the Fourier transform of p( · )r( · , x, ~). But the last integral is clearly dominated by
(4.2.7) (The last estimate is easy to prove after recalling that ~ -+ p(x, ~) is positive and homogeneous of degree one in IR.n \ 0.) By putting together these estimates, we see that we have shown that
S>.(x,x)
= (21r)-n- 1 X
If
p(t)i(t + i0)- 1
q(O, X 1 x, ~) eit(p(z,()->.) d{dt + O(.~n- 1 ).
(4.2.8) However, (4.1.9) implies that the main term here is
(211")-n-1 =
II
p(t)i(t + i0)-1 eit(p(z,()->.) d{dt
(211")-n
j
(x(-oo,O]
= c(x)>..n + (211")-n
*P)(p(x,~)- >..)cl{
J
(x(-oo,O]
* (p -1)v)(p(x,~)- >..)cl{. (4.2.9)
To estimate the last term, note that (p-1)v(s) = -Jst/J(s), where ~(t) = (1- p(t))fit. Since t/J is a bounded function which is rapidly decreasing at infinity, and since, by (4.2.3), (x(-oo,O]
* (p-1)v)(s) = -t/J(s),
one can use (4.2.7) to see that the last term in (4.2.9) is O(>..n- 1). Finally, since, by (4.1.9), q(O,x,x,~)-1 = 0(1~1- 1 ), it is clear that this argument also gives
and this means that we have proved the desired estimate
Therefore, we would be done if we could show that
IS>.(x,x)- S>.(x,x)l::::; c>..n- 1,
xEM.
(4.2.10)
4. The Half-wave Operator
128
To prove this, we note that (4.2.4) and (4.2.5) imply that the Fourier transform of the function A-+ S.>.(x,x)- S.>,(x,x)
vanishes when It! Tauberian lemma.
< e/2. To exploit this we shall use the following
Lemma 4.2.3: Let g(A) be a piecewise continuous tempered function of JR. Assume that for A > 0
lg(A + s)- g(A)I::; C(1 + A)a,
O<s::;l.
(4.2.11)
Then, if g(t) = 0, when iti ::; 1, one must have (4.2.12)
Remark. Clearly, the assumption that g(t) vanish for small t is needed. For, if g(A) = amAm + · · · + ao is a polynomial of degree m, then (4.2.11) holds for a= m- 1, while clearly (4.2.12) cannot hold if am =F 0. On the other hand, in this case g is, in some sense, concentrated at the origin, since sing supp g = {0}. # • Proof of Lemma 4.2.3: Let {>.+I
G(A)
= l.>.
g(s) ds. I
Then G is absolutely continuous and, except on possibly a set of measure zero, G' exists and satisfies
IG'(A)I
= lg(A + 1)- g(A)I ::; C(1 + A)a.
It is also clear that the Fourier transform of G vanishes in [-1,1]. Next since (4.2.11) and the triangle inequality imply that
it follows that (4.2.12) would follow from the estimate
!,
To prove this last inequality let TJ E S satisfy TJ(t) = 0, when It! < and TJ(t) = 1, for It! > 1, and let t/J be defined by ~(t) = (it)- 1TJ(t). Then, since t/J is bounded and rapidly decreasing at infinity, it is easy to
129
4.2. The Sharp Weyl Formula check that G' * 1/J =G. Consequently, the estimate for G' gives
IG(A)I = I(G' * 1/J)(A)I
J
~ C(1 + A)a ~
I1/J(s)l(1 + lsl)a ds
C(1 + A}a,
in view of the rapid decrease of 1/J.
I
If we apply the lemma to g(A) = S.>,(x,x)- S.>.(x,x}, then, since we have already observed that g vanishes near 0, we need only prove the following two estimates:
IS.>.+s(x,x}- S.>,(x,x)l ~ C(1 + A)n- 1 ,
0
< s ~ 1, (4.2.13)
IS.>.+ 8 (X, x) - S.>,(x, x)l
~
C(1 +At- 1 ,
O<s~l.
(4.2.14} However, we claim that these inequalities are a corollary of the following estimates for the L 00 norm of eigenfunctions. Lemma 4.2.4: Let X>. be the spectml projection opemtor X>.!=
L
(4.2.15}
E;f.
.>.;E(.>.,.>.+I]
Then, for A ~ 0,
llnfiiL""(M) ~ C(1 + A)(n- 1)/2 11/IIP(M)·
(4.2.16}
To see why (4.2.16} implies (4.2.13} and (4.2.14), note that (4.2.16} holds if and only if the kernel of the spectral projection operator satisfies sup
f ln(x,y)l 2 dy~C(1+A)n- 1 •
(4.2.17}
xeMJM
However, since
n(x,y} =
L
e;(x} e;(y),
.>.;E(.>.,.>.+1]
where {e;(x)} is an orthonormal basis associated to the spectral decomposition, we see that
L
jln(x,y)l 2 dy=
le;(x}l 2 •
.>.;E(.>.,.>.+I]
Consequently, since
S.>.+ 8 (x,x}- S.>.(x,x) =
L .>.;E(.>.,.>.+s]
le;(x}l 2 ,
130
4- The Half-wave Operator it is obvious that (4.2.17) implies (4.2.13). On the other hand, to see that (4.2.14) follows as well, note that B>.+s- B>. = (211")- 1
J
p(t) i(t + iO)-l ( e-ist-
1) eit(P->.) dt.
Thus, since (e-ist- 1) vanishes at t = 0, one can use (4.2.3) and argue as above to see that, for every N, IS>.+s(x,x)S>.(x,x)l:::;
eN ""' L..,(1 + lA- Ajl}- N le;(x)l 2 . j
But since the estimates for X>. imply that this term is 0( An- 1 ) as well, we need only to prove (4.2.16) to finish the proof of the Weyl formula. Proof of Lemma 4.2.4: The first step in trying to apply the above arguments is to notice that it suffices to prove the dual version of (4.2.16):
lln/II£2(M) :::; C(1 + A)(n- 1)/ 2 11/llu(M)·
(4.2.18)
Next, to exploit Theorem 4.1.2, we notice that it suffices to prove the analogous inequality for certain "approximate spectral projection operators." Namely, for a fixed XES, satisfying x(O) > 0, X~ 0, and x(t) = 0 unless ltl :::; e, we define X>.!= LX(Aj -A)E;f·
(4.2.151)
j
(Such a function always exists, since if the function p in (4.2.5) is real, then x(A) = (p(A/4)) 2 has the right properties.) It is useful to have the £ 2 norm on the left side, since orthogonality and the fact that x(O) "I 0 imply that (4.2.18) would be a consequence of the analogous estimates for the approximate operators:
llx>./112 :::; C(1 + A)(n- 1)/211/111·
(4.2.18')
Moreover, the above arguments also apply here since x( t) = 0 when ltl ~ e and X>. = (211")-1
L:
x(t) e-it>. eitP dt.
Next, notice that if X>.(x, y) is the kernel for x>.. then X>.(x, y) =
L x(Aj -A) e;(x) e;(y). j
(4.2.19)
.pJ. Smooth Functions of Pseudo-differential Operators
131
Hence, it is easy to see from the Schwarz inequality that the L1 (M)-+ L2 (M) operator norm of X.\ satisfies
llx.\II~Ll(M}P(M)) $:~~I lx.\(x, y)l 2 dx = sup 2)x(A; -
(4.2.20)
A)) 2 1e;(Y)I 2
yEM j
$llxllv"'(R} · sup X.\(y,y). yEM
In the last inequality we have used the fact that
x ;::: 0. Finally, if we let
c(t, x, e) = X(t)q(t, x, x, e), where q E S0 is the symbol in the parametrix for eitP, then (4.2.19) and Theorem 4.1.2 give lx.\(x, x)l
$1Ln $eN
c(A- p(x, e), x, e) del+
I
11.:
x(t)e-it.\ R(t,x, x) dtl
(1 + lA- p(x, em-N de+ 0(1)
$ C(1 + A)n- 1
I
(by (4.2.7)).
I
Combining this with (4.2.20) completes the proof.
Remark. Lemma 4.2.4 is a generalization of the (£ 1 , £ 2 ) restriction theorem for the Fourier transform in Rn. In fact, duality and a scaling argument show that the latter is equivalent to the uniform estimates
•
f ei(x,() f(e)dell $ C(1 + A)(n- 1}/2 11/IIP(Rn)· 11 (211")-n ji(IE(.\,.\+1) L""(Rn} The next chapter will be devoted to proving a generalization of the full restriction theorem under the assumption that the principal symbol p(x, e) satisfies certain natural curvature conditions. These estimates will allow us to extend the Tauberian argument used above to handle other situations, such as proving estimates for Riesz means on M.
4.3. Smooth Functions of Pseudo-differential Operators Let m E C 00 (R) belong to the symbol class SP., that is, assume that (4.3.1)
4·
132
The Half-wave Opemtor
Then, using the spectral decomposition of P, we can define an operator m(P) that sends C 00 (M) to COO(M) by 00
L m(>.;) E;f,
m(P)f =
(4.3.2)
j=l
if Pis as above. Using the ideas in the proof of the sharp Weyl formula, we shall see that m(P) is actually a pseudo-differential operator.
Theorem 4.3.1: Let P E \11~1 (M) be elliptic and self-adjoint with respect to a positive C 00 density dx. Then, if m E Sf.l., m(P) is a pseudodifferential operator of order p., and the principal symbol of m(P) is
m(p(x,,)). As before, one can see that the same result holds for pseudo-differential operators of arbitrary positive order as well. Also, since Theorem 3.1.6 says that zero order pseudo-differential operators are bounded on l.Jl, we also have the following.
Corollary 4.3.2: Let P be as above. Then if m E llm(P)fiiLP(M) ~ Cp llfiiLP(M)•
SO,
1 < P < oo.
(4.3.3)
Proof of Theorem 4.3.1: We assume that local coordinates are chosen as in Theorem 4.1.2, and, as in the proof of Theorem 4.2.1, we fix p E S(IR) satisfying p(t) = 1, ltl ~ e/2, and p(t) = 0, ltl >E. Then, using the half-wave operator, we make the decomposition
/_:fJ(t) m(t)
m(P) = (27r)- 1 = m(P)
eitP dt
L:(l-
+ (27r)- 1
fJ(t)) m(t) eitP dt
+ r(P),
where, if 1/J is defined by
,P =
1-
m(>.) = (m * p)(>.)
p, and
r(>.) = (m * 1/J)(>.).
Since m satisfies (4.3.1), m(t) is C 00 away from t = 0, and rapidly decreasing at oo; thus r(>.) E S. Since the kernel of r(P) equals 00
r(P)(x, y) =
L r(>.;) e;(x) e;(y), 1
one can, therefore, use the crude estimates from Section 3.3 for the size of the derivatives of the eigenfunctions e;(x) together with the Weyl formula to see that r(P)(x,y) is C 00 •
Notes
133
On the other hand, Theorem 4.1.2 implies that, in our local coordinate system, the kernel of m(P) equals
(21r)-n- 1
11
p(t) m(t)q(t, x, y, {) eitp(y.{) eicp(x.y,{) dedt
+ (211")- 1
I
p(t) m(t) R(t, x, y) dt.
Since R is C 00 and since p(t)m(t) E e'(JR), the second term must be in C 00 (M x M). Thus, we would be done if we could show that
(21r)-n- 1
11
p(t) m(t) q(t, x. y, {) eitp(y.{) eicp(x,y.{) dedt
(4.3.4)
is the kernel of a pseudo-differential operator with principal symbol m(p(x,{)). To see this we note that, as before, we can write
q(t, x, y, {) = q(O, x, y, {) + tr(t, x, y, {),
so.
where r E Since m is singular at the origin, we would, therefore, expect that the main term in (4.3.4) would be
(211")-n- 1
II
p(t) m(t) q(O, x, y, Q,eitp(y,{) eicp(x,y,{) dedt
= (211")-n
I
m(p(y, {)) q(O, x, y, {) eicp(x,y,{) cte.
Since, q(O,x,x, {)-1 E s- 1 , Theorem 3.2.1 implies that this is the kernel of a pseudo-differential operator having principal symbol m(p( X,{)), and since m-m E S, we conclude that this kernel has the desired form. Thus, we would be done if we could show that
(21r)-n- 1
11
p(t) m(t) tr(t, x, y, {) eitp eicp dedt
is the kernel of a pseudo-differential operator of order :5 p. - 1. But this too follows from Theorem 3.2.1 after checking that
(211")- 1
L:
p(t) m(t) tr(t, x, y, {) eitp(y,{) dt E s~'- 1 •
I
Notes The parametrix construction for the half-wave operator is taken from Hormander [4]. This was a modification of the related construction of Lax [1] which used generating functions. We have used Lax's construction at the end of Section 4.1 to construct parametrices for strictly
134
4·
The Half-wave Opemtor
hyperbolic differential operators. As the reader can tell, Lax's approach is somewhat more elementary, but, since the phase functions in the Lax construction need not be linear in t, it is harder to use in the study of eigenvalues and eigenfunctions. The proof of the sharp Weyl formula is from Hormander (4], except that here the Tauberian argument uses the bounds for the spectral projection operators in Lemma 4.2.4 which are due to Sogge (2] and Christ and Sogge (1]. The Tauberian arguments which were used in the proof go back to Avakumovic [1] and Levitan [1], where they were used to prove the sharp Weyl formula for second order elliptic self-adjoint differential operators. The material from Section 4.3 is due to Strichartz (2] and Taylor (1].
Chapter 5
LP Estimates of Eigenfunctions
In Chapter 2 we saw that if E C R.n is a compact C 00 hypersurfa.ce with non-vanishing curvature then
1< .f(x) = (27r)-n {
ei(x,e) /(e)~,
he:q(E)E[.>.,>.+l)}
then taking e = 1/ >. and applying a scaling argument shows that the last inequality is equivalent to the uniform estimate
IIX>./IIL2(Rn) ~ C(1 + A)c5(p) 11/IILP(Rn)>
1 ~ P ~ 2 ~-tl) l
with 6(p) = nl~- ~~-~Notice that, for this range of exponents, 6(p) agrees with the critical index for lliesz summation (see Section 2.3). The operators X>. of course are the translation-invariant analogues of the spectral projection operators which were introduced in the proof of the sharp Weyl formula. The main goal of this ch~pter is to show
136
5. LP Estimates of Eigenfunctions that these operators satisfy the same bounds as their Euclidean versions under the assumption that the cospheres associated to the principal symbols Ex= {e: p(x,e) = 1} c T;M\0 have everywhere non-vanishing curvature. Since 6(1) = (n- 1)/2 this is the natural extension of the estimate (4.2.18). After establishing this "discrete L 2 restriction theorem" we shall give a few applications. First, we shall show that the special case having to do with spherical harmonics can be used to give a simple proof of a sharp unique continuation theorem for the Laplacian in an. Then we shall see how the Tauberian arguments that were used in the proof of the sharp Weyl formula can be adapted to help prove sharp multiplier theorems for functions of pseud{)odifferential operators. Specifically, we shall see that estimates for Riesz means and the Hormander multiplier theorem carry over to this setting.
5.1. The Discrete L 2 Restriction Theorem Let M be a C 00 compact manifold of dimension n ~ 2. We assume that P = P(x, D) E w~1 (M) is self-adjoint, with principal symbol p(x, e) positive on T* M \ 0. Then if n are the spectral projection operaJ;o~s defined in (4.2.15), we have the following result.
Theorem 5.1.1: Assume that, for each x E M, the cospheres {e : p( X' e) = 1} c T; M \ 0 have everywhere non-vanishing curvature. Then if 6(p) = nl~- ~~- ~ and A> 0 1< < 2(n+l) _p_ n+ '
(5.1.1)
llnfii£2(M) ~ C(1 + A)(n-l)( 2-p)/4p llfiiLP(M)•
2(n+l) < < 2. --n+3 _p-
(5.1.2) Furthermore, these estimates are sharp. If we use Theorem 3.3.1, then we see that the dual versions of these inequalities yield the following estimates for the "size" of eigenfunctions on compact Riemannian manifolds.
Corollary 5.1.2: Let t1 9 be the Laplace-Beltrami operator on a compact C 00 Riemannian manifold (M,g). Then, if {A;} are the eigenvalues of
5.1. The Discrete L 2 Restriction Theorem -6.9 , and if one defines projection operators R>.f = Ly:X;e[>.,>.H] E;f, one has the sharp estimates IIR>.fllLv(M) $ C(1 + A)c5(q) llfiiL2(M)• IIR>.fllLv(M) $ C(1 +A) (n - 1}(2 -q')/4q llfiiL2(M)• 1
2< . : L 2 - L 2 with norm one, the M. lliesz interpolation theorem implies that we need only prove the special case where p = 2(n + 1)/(n + 3); that is, it suffices to show that when p = 2 ~-tl>. (5.1.3) To prove this we shall use the idea from the proof of Lemma 4.2.4 of proving an equivalent version of this inequality which involves operators whose kernels can be computed very explicitly. The operators X>. used in the proof of Lemma 4.2.4 have kernels which are badly behaved in the diagonal {x = y}. So, in the present context, it is convenient to modify their definitions slightly. To do this we let E > 0 be as in Theorem 4.1.2. Thus, in suitable local coordinate systems, eitP has a parametrix of the form (4.1.16) as long as It! <E. Then for 0 <Eo< E to be specified later we fix 0 -=f. X E S(IR) satisfying x(t) := 0 if t ~ (eo/2,eo), and define approximate projection operators x>.f
= x(P- .X)f = L:x(.X;- .X)E;f. j
Then, as before, orthogonality arguments show that the uniform bounds (5.1.3) hold if and only if llx>.fllL2(M> $ c(1 + .x)o(p) llfiiLP(M>·
p = 2 ~ti>,
.x > o. (5.1.3')
137
5. LP Estimates of Eigenfunctions
138
Let Q(t) and R(t) be as in Theorem 4.1.2. Then
X>.f =
..!... I 2~
Q(t)fe-it>. x(t) dt +..!...I R(t)fe-it>. x(t) dt. 2~
Since R( t) has a C kernel, the last term has a kernel which is 0( >.- N) for any N and hence it satisfies much better bounds than those in (5.1.3'). Therefore, if we work in local coordinates so that dx agrees with Lebesgue measure, it suffices to show that 00
T>.f(x)
= (2~)-n-1 I f f ei[. x x(t) q(t, x, y, e) f(y) ctedtdy
satisfies the bounds in (5.1.31) if cp and q are as in (4.1.16). Notice that since c- 1 1x- Yi :5 IVecp(x,y,e)l :5 Clxconstant C, it follows that on the support of the integrand
IVe[cp(x,y,e)+tp(y,e)]l #0 if
Yi
for some
lx-yl f/. [C0 1eo,Coeo]
for some constant Co. Therefore, by Theorem 0.5.1, if we let a(t, x, y, e) = 71(x, y)x(t)q(t, x, y, e), where '7 E C 00 equals 1 when lx - Yl E [C01eo, Coco] and 0 when lx - Yl f/. [(2Co)- 1eo, 2Coeo], it follows that the difference between T>.f and
T>.f(x) =
(2~)-n- 1 I I f ei[. a(t, x, y, e) f(y) dedtdy
has a kernel with norm 0(>.-N) for any N. On account of these reductions, we would be done if we could show that (5.1.3")
The proof of this follows immediately from the estimates for non-homogeneous oscillatory integrals and the following result. Lemma 5.1.3: Let a(t, x, y, e) E S0 be as above and set
K>.(x,y) =
JJ
ei[.ta(t,x,y,e)dedt,
Then, if eo is chosen small enough we can write n-1 K>.(x, y) = >. ----r a>.(X, y) ei>.,P(x,y),
>. > 1.
(5.1.4)
where t/J and a>. have the following properties. First, the phase function t/J is real and C 00 and satisfies the n x n Carleson-Sjolin condition on suppK>.. In addition, a>. E C 00 with uniform bounds
l~,ya>.(x,y)l :5 Co.
139
5.1. The Discrete L 2 Restriction Theorem If we apply Corollary 2.2.3 we see from the lemma that the oscillatory integral operator with kernel K>. is bounded from £ 2 norm
-+
L
~
with
Since the n x n Carleson-Sjolin condition is symmetric, the adjoint ofT>. must also satisfy this condition. So, by duality, we see that the operators n-1 2(n+1) T>. are bounded from L 7i+3 to L 2 with norm O(.X 2(n+l) ). But 6(p) = 2 ~+11 ) when p = 2 ~tl> and hence the proof of Theorem 5.1.1 would
I
be complete once Lemma 5.1.3 has been established.
Proof of Lemma 5.1.3: The proof will have two steps. First we shall show that K>. is of the form (5.1.4), and then we shall show that the phase function satisfies the n x n Carleson-Sjolin condition. The first thing we notice is that, if p(y, e) fl. (.X/2, 2-X],
I
eit(p(y,{)->.la(t, x, y, e) dt = O((lp(y, e) I+ .x)-N)
"
for any N. But p(y, e) ~ 1e1. So if {3 E C8"(1Rn \ O)"equals one when 1e1 E (c- 1, C], with C sufficiently large, the difference between K>. and
II
ei[rp(x.y,{)+tp(y.{)) e-i>.t f3(e/ .X) a(t, x, y, e) d{dt
(5.1.5)
is C 00 , with all derivatives being O(.X-N) for any N. On account of this, it suffices to show that we can write (5.1.5) as in (5.1.4). But if we set
\ll(x, y; t, e) = rp(x, y, e)+ t(p(y, e)
- 1)
and
then, after making a change of variables, we can rewrite (5.1.5) as (5.1.51) Notice that a>. is compactly supported and that all of its derivatives are bounded independently of .X. So we can use stationary phase to evaluate this integral if the Hessian with respect to the n + 1 variables (e, t) is non-degenerate on suppa>._. By symmetry it suffices to show that this is
1
140
5. LP Estimates of Eigenfunctions the case when e lies on the en axis. If we write
e = (e2 •... 'en) then
0
0
Ifen To see that this matrix is non-degenerate we first notice that, by homogeneity and our assumption that e lie on the en axis, we have p{Jy, e) = p(y, e)/en· So the non-degeneracy of the Hessian is equivalent to (5.1.6)
I
,
But our curvature hypothesis implies that o2 pfoe' 2 is non-singular and (4.1.3) implies that o2cpfoe' 2 consists of elements which are O(lx-yl 2 ) = O(e~) on suppa~. So, if eo in the statement of the lemma is sufficiently small, (5.1.6) holds and consequently det o2 'ifl fo(e, t) 2 =F 0 as desired. Furthermore, since cp satisfies (4.1.3) one can modify the argument before the proof of Lemma 2.3.3 to see that, if eo is sufficiently small, then, on suppa~, (e, t) -+ 'if! has a unique stationary point for every fixed (x, y). Hence Corollary 1.1.8 implies that (5.1.51) can be written as in (5.1.4). What remains to be checked is that the phase function satisfies the n x n Carleson-Sjolin condition. That is, we must show that, for every fixed x,
Sx = {Vx,P(x,y): a~(x,y) =F 0} c
T;x
(5.1.7)
is a C 00 hypersurface with everywhere non-vanishing Gaussian curvature, and also that
a2,p
rank oxf)y
=n- 1
on suppa~.
(5.1.8)
Since the adjoint ofT~ has a similar form, the proof of (5.1.7) also gives that, for each y, Sy
= {Vy,P(x,y): a~(x,y) =F 0} c T;x
has everywhere non-vanishing Gaussian curvature. This, together with (5.1.7) and (5.1.8), implies that the n x n Carleson-Sjolin condition is satisfied. To prove (5.1.7) and (5.1.8), let us first compute ,P. We note that V~,t'ifl = ( IPe(x,y,e)
+ tpe(y,e),p(y,e) -1).
5.1. The Discrete L 2 Restriction Theorem
141
Thus, if (t(x, y), e(x, y)) is the solution to the equations
{
cp~(x, y, e)+ tp~(y, e) = 0, p(y,e) = 1,
(5.1.9)
it follows from Corollary 1.1.8 that our phase function must be given by
1/J(x,y) = w(x,y;t(x,y),e(x,y)) = cp(x,y,e(x,y)).
(5.1.10)
To check that the curvature condition is satisfied, notice that
Vxt/J(x,y)
= cp~(x,y,e(x,y)) + ae~~y) cp~(x,y,e(x,y)).
However, the first half of (5.1.9) implies that
ae(x,y) ax Pt.'( y,e (x,y )) ax cp£.'( x,y,e (x,y )) = -t (x,y )ae(x,y)
a {p(y,e(x,y))}. = -t(x,y) ax Since the second half of (5.1.9) implies that the last expression vanishes identically we conclude that
Vxt/J(x,y) = cp~(x,y,e(x,y)). But this along with the eikonal equation (4.1.6) implies that
p(x, Vxt/J(x,y)) = p(y,e(x,y)). So if we use the second half of (5.1.9) again, we conclude that the hypersurfaces in (5.1.7) are just the cospheres Ex= {e: p(x,e) = 1}, which have non-vanishing curvature by assumption. To verify (5.1.8) we fix y and let p(e) = p(y, e). Then if ~(x- y) is determined by the analogue of (5.1.9) and (5.1.10) where cp is replaced by the Euclidean phase function (x- y, e) and p(y, e) by p(e) we have already seen (in the proof of Lemma 2.3.4) that rank a 2 ~(x-y)jaxay::::: n -1. But 1/J(x, y) = ~(x- y) + O(lx- y! 2 ) as x- y which implies that a2 .,pjaxay = a2 ~(x- y)jaxay + 0(1). From this and the fact that ~is homogeneous of degree one, we conclude that rank a2 .,p1axay ~ n - 1 on suppa~ if eo is small enough. However, since we have just seen that y- Vxt/J(x,y) is contained in the hypersurface Ex, the rank can be at most n- 1 and so we get (5.1.8). I
Remark. The proof of Lemma 5.1.3 shows that the operators x~ inherit their structure from the wave group eitP. In fact, if C c T*(M x IR) \0 x T* M \0 is the canonical relation associated to the wave
5. LP Estimates of Eigenfunctions
142
group, which is given in (4.1.21), it follows that the canonical relation associated to t/J satisfies
C,p
c {(x,e,y,TJ): (x,t,e,p(x,e),y,TJ) e c, for some 0 < t < e and (x,e) such that p(x,e) = 1}. (5.1.11)
To see this, notice that, for x close toy, there is a unique positive small timet such that (x,t,y) is in the singular support of the kernel of eitP. But, if we call this time t(x, y), it follows from Euler's homogeneity relations, cp = {cp(,e},p = {p(,e}, along with (5.1.9) and (5.1.10) that .,P(x, y) = -t(x, y). We should also point out that this implies that, when P = ~, t/J is just minus the Riemannian distance between x and y.
Proof of sharpness: We first prove that for 1 ~ p
~
2
lim sup sup A-n(~-~)+~ llx>./112 > c >.--++oo /ELP(M) II/IlP - 1
(5.1.12)
for some c > 0. This of course implies that the estimate (5.1.1) must be sharp. To prove (5.1.12) we fix {3 E S(R.) satisfying /3(1) "I 0 and /3(r) = 0 if T ¢ [-e,e], where e is as in Theorem 4.1.2. We then fix xo EM and define
f>.(x)
= Lf3(Aj/A)e;(xo)e;(x). j
Notice that this is the kernel of {3(P/A) evaluated at (xo,x). So, if we argue as in Section 4.3, we conclude that, given N, there must be an absolute constant CN such that 1/>.(x)l ~ CNAn(1 +A dist (x,xo))-N. Consequently, 11/>.llp ~ CbAn-nfp. On the other hand, since /3(1) "I 0, we conclude that there must be a co > 0 such that, if A is large enough,
lln/>.11~ =
L
I/3(Aj/A)I 2 Ie;(xo)l 2 lle;(x)ll~
>.; E [>.,).+ 1)
~CO
L
>.;E[>.,).+l)
le;(xo)l 2·
143
5.1. The Discrete L 2 Restriction Theorem However,
~
h
L
lej(x)l 2 dx
~ N(>.. + 1) -
N(>..).
M >.;E(>.,.Hl]
(5.1.121)
Combining this with the lower bounds for the norm of f>. yields sup lln/ll2
II/IlP
/ELP
for some c
~c>..-n+n/p{N(>..+ 1 )-N(>..)}l/2
> 0. However, since N(>..)
~
>..n,
limsup>..-(n-l){N(>.. + 1)- N(>..)} >.-++oo
> 0,
and hence (5.1.12) follows from (5.1.121). The proof that the other estimate in Theorem 5.1.1 is sharp is more difficult and requires a special choice of local coordinates in M. Before specifying this, let us first notice that it suffices to show that, for large enough>.., sup llx>./ll2 /ELP
> c>..(n-t)(1p-i>
II/IlP -
'
1 :5 P :5 2,
(5.1.13)
if X>. are the approximate spectral projection operators occurring in the proof of Theorem 5.1.1. To prove this lower bound we shall use the last remark. That is we shall use the fact that -'1/J(x, y) = t(x, y), where, if xis close but not equal toy, t(x,y) is the unique small positive number such that JIM(It(x,y)(Y,TJ) = x for some TJ E T; M \ 0. Here 41t : T* M \ 0 --+ T* M \ 0 is the flow out for time t along the Hamilton vector field associated to p(x, e), and II M : T* M \ 0 --+ M is the natural projection operator. Keeping this in mind, there is a natural local coordinate system vanishing at given Yo E M which is adapted to '1/J(x, y). This is just given by ~(z)=x
e
if1IM4llxi(YO,e)=z
withe=x/lxl-
(5.1.14)
Here denotes the coordinates in T;0 M \ 0 which are given by an initial choice of local coordinates around Yo (see Section 0.4). Note that, if P = FfS:;, the local coordinates would just be geodesic normal coordinates around Yo and so one should think of (5.1.14) as the na.tural "polar coordinates" associated to p(x, e). They are of course well defined for z
5. LP Estimates of Eigenfunctions
144
near Yo and C 00 (away from possibly z =Yo) since, for a given initial choice of local coordinates vanishing at YO,
and since, by assumption, Pee has maximal rank n- 1. The reason that these coordinates are useful for proving (5.1.13) is that, in these coordinates, '1/J(x, 0) = Ixi- Moreover, using the semigroup property of ~t, one sees that
'1/J(x,y)
=-(XI-
YI) and - Vxt/l(x,y)
= Vyt/l(x,y) = (1,0),
if x = (xi,O),y = (yt.O), and 0
.
II
ei~((,P(x,y)-y1)-(,P(x,y)-y1)) x
o:~(y) o:~(jj) a~(x, y) a~(x, jj) dydydx.
Notice that, by (5.1.151), the term in the exponential is O(e~) on the support of the integrand. Using this and the fact that the a~ belong to a bounded subset of C 00 shows that, for small enough e1, the last expression is bounded from below by a fixed positive constant times
e~nAn-l(A-(n- 1 )1 2 ) 2
{
lrh
la~(x,O)I 2 dx-Ce~n+liO~I,
where Cis a fixed constant. However, since q(O,x,x,e) = 1 + O(lel- 1), it follows from the proof of Lemma 5.1.3 that, if eo is small and A large, then there must be positive constants such that {
lrh
la~(x,O)I 2 dx ~ ciO~I ~ c' A-(n- 1)/2.
Thus, if A is large and if e1 is small, we reach the desired concltision that #
llx~hll2 ~ CQA-(n- 1)/ 4 ,
some
co> 0.
Finally, since llhiiP::; CA-(n-1)/2p, we conclude that (5.1.13) must hold.
I
Application: Unique continuation for the Laplacian A special case of Corollary 5.1.2 concerns spherical harmonics. If M = 1, n ~ 3, and A 9 = As is the usual Laplacian on sn- 1 the eigenvalues of the conformal Laplacian, -As + [(n - 2)/2] 2, are {(k + (n- 2)/2) 2}, k = 0, 1, 2, .... The eigenspace corresponding to the kth eigenvalue is called the space of spherical harmonics of degree k and it has dimension ~ kn- 2 for large k. If we let Hk be the projection onto this eigenspace, then Corollary 5.1.2, duality, and the fact that Hk = Hk o Hk give the bounds
sn-
IIHk/IILP'(Sn-1) :S C(1 + k) 1- 2/n 11/IILP(Sn-1)• Notice that for this value of p we have 1/p-1/p'
P=
n~2·
(5.1.16)
= 2/n, so the exponents
5. LP Estimates of Eigenfunctions
146
in (5.1.16) correspond to the dual exponents in the classical Sobolev inequality for the Laplacian in an: lluiiLP'(R") ~ C 116-uiiLP(R")•
P = n~2•
u E Ccf(an).
We claim that (5.1.16) along with the Hardy-Littlewood-Sobolev inequality yield a weighted version of this. Namely, if p and p1 are as above there is a uniform constant C for which lllxi-TuiiLP'(R") ~ Clllxi-T 6-uiiLP(R")• ifuEC«f(an\0) and dist(r,Z+~)=~.
(5.1.17)
The condition on T is of course related to the spectrum of the conformal Laplacian on sn-l. Before proving (5.1.17) let us see how it yields the following unique continuation theorem. Theorem 5.1.4: Let n ~ 3 and let X be a connected open subset of an containing the origin. Suppose that D 0 u E 0 c{X) for ial ~ 2 where p = 2nf(n + 2). Assume further that
Lf
I
{5.1.18)
for some potential V E L':!c2 {X). Then if u vanishes of infinite order at the origin in the LP mean, that is, if for all N {
Jlxi2
00
~)1 + k)- 2 /n{l + lk + ~- rl·lt- si)-N = O{lt- sl-1+ 2/n). k=O
Hence we get IITd(t, · )IILP'(dw) :$ C
i:
It- sl-1+ 2/n llf(s, · )IILP(dw) ds,
which leads to (5.1.21) after an application of the Hardy-LittlewoodSobolev inequality (i.e., Proposition 0.3.6) as 1/p- 1/p' = 2/n. 1
5.2. Estimates for Riesz Means In this section we shall study the Riesz means of index to P(x,D):
S~f(x) =
L {1- >.;/>.) 6E;f.
/j ~
0 associated (5.2.1)
.>.;$.>.
As in IR.n, these operators can never be uniformly bounded in V(M), p 'f:. 2, when /j :$ 6(p) if 6(p) is the critical index 6(p)
= max{nl~- ~~- ~,0}.
(5.2.2)
On the other hand, we shall prove the following positive result which extends Euclidean estimates in Section 2.3.
Theorem 5.2.1: Let P(x, D) E w~1 (M) be positive and self-adjoint. Then there is a uniform constant C5 such that for all>.> 0 (5.2.3) IfwealsoassumethatthecospheresassociatedtoP,'Ex = {e :p(x,e) = 1}
5. LP Estimates of Eigenfunctions
150
c
T; M \ 0 have non-vanishing Gaussian curvature, then for p E [1, 2(n + 1)/(n + 3)] U [2(n + 1)/(n- 1), oo] and A> 0
IIS~fiiLP(M) ~ cp,6llfiiLP(M)•
6 > 6(p).
(5.2.4)
As a consequence of this result, we get that, for p < oo as in the theorem, S~f--+ f in the l.Jl topology if 6 > 6(p). Notice that, for the exponents in both (5.2.3) and (5.2.4), 6{p) = ~- ~ ~· Hence, by Lemma 4.2.4, Theorem 5.1.1, and duality, both inequalities are a consequence of the following.
nl
1-
Proposition 5.2.2: Suppose that P(x, D) E \11~ 1 is positive and selfadjoint. Suppose also that for a given 1 ~ p < 2 there is a uniform constant C such that
llnfii£2(M) ~ C(1 + A)n(1/p-1/2)-1/211fiiLP(M)•
A~ 0.
(5.2.5)
Then (5.2.4) holds.
Proof of Proposition 5.2.2: Since the Fourier transform ofT~ is c6(t + iO) - 6- 1 , we can write
S~f =
(21r)-1c6A-6
I
eitP fe-it>.(t
+ i0)-6-1 dt.
To use the parametrix for eitP we let e be as in Theorem 4.1.2 and fix p E C~(IR) which equals 1 for ltl < e/2 and 0 for ltl >e. We then set 6
-6
...Fi
S>.f = S>.f + .«).f,
where
S~f =
(21r)-1c6A-6
I
eitP fe-it>.p(t)(t
We would be done if we could show that, for 6
+ i0)-6-1 dt.
> 6(p),
-6
IIS>.fiiP ~ C6llfiiP•
(5.2.6)
~ C6llfllp·
(5.2.7)
II.RifiiP
We first estimate the remainder term and in fact show that it satisfies a much stronger estimate:
IIR~fll2 ~ C(1 + A)[n(1/p-1/2)-1/2J-611fllp·
{5.2.7')
This is not difficult. We note that, for 6 ~ 0, the Fourier transform of ( 1 - p( t)) (t + iO) - 6- 1 is bounded and rapidly decreasing at infinity. This means that = A- 6 r6(A- Aj)Ej/ j
Rif
L
151
5.2. Estimates for Riesz Means for some function r6 satisfying lr6{.X)I :5 CN{l + 1-XI)-N for all N. Hence, for large N, 00
II.Ri/11~ :5 cN-X- 26 L:(t +I-X- ki)-NIIxkfll~ k=1 00
:5 c'_x-26 L(1 +I-X- ki)-N (1 + k)2(n(1/p-1/2)-1/2]11/ll~ k=1
:5 c" (1 +.X) -2[6-n(1/p-1/2)+1/2]11/ll~. as desired. To estimate the main term we decompose S~ as in the proof of Theorem 2.4.1. To this end, we fix /1 E C~(IR \ 0) satisfying E~-oo .B(2ks) = 1, s '# 0. We then write -6 -6 ~ -6 S>.f = s>.,of + L....J s>.,kf, k~1
where for k = 1, 2, ...
stkf = {21r)- 1c6.x- 6
I
eitP fe-it>./1(-XTkt) p(t)(t + i0)- 6- 1 dt.
=
Note that S~ kf 0 if 2k is larger than a fixed multiple of .X. We claim that (5.2.5) and the finite propagation speed of the singularities of ( eitP)(x, y) can be used to prove that, for any e > 0, (5.2.8) By summing a geometric series this leads to (5.2.6). Actually the estimate fork= 0 just follows from Theorem 4.3.1. This is because where mi 0 (r) is the convolution of {1- r/.X)t with .x- 1,.,(r/.X) if 1J is
'
00
the inverse Fourier transform of {1-
E
/1{2-kr)) E C~(IR.). Hence, for
k=1
anyN,
I(:Tr
mi,o(r)l :5 Ca,N_x-lal {1 +r/A)-N.
Thus, if p > 1 the estimate is a special case of (4.3.3). Since pseudodifferential operators of order -1 are bounded on £ 1, the case of p = 1 follows via Theorem 4.3.1 and the easy fact that the pseudo-differential operators with symbol mi_ 0 (p(x,~)) are uniformly bounded on £ 1.
5. LP Estimates of Eigenfunctions
152
To handle the terms with k
s~lkf =
(2?r)- 1c6A - 6
I
+ (2?r)- 1c6A - 6
~
1 in (5.2.8) we write
Q(t)fe-it>.{3(A2-kt) p(t)(t + i0)_ 6_ 1 dt
I
R(t)fe-it>.{3(ATkt) p(t)(t + i0)- 6- 1 dt
1
where Q(t) is the parametrix for eitP. Since the kernel of R(t) is C 00 one sees that the kernel of the second operator is 0(2-kN) for any N. Hence it suffices to show that the integral operator with kernel
Htk(x~ y) =
(2?r)-n-1c6A -6
II
ei(rp(x~y~e)+t(p(y~e)->.)]
x q(tl x, y, e> f3(ATkt> p(t> (t + io)- 6 - 1 dedt
satisfies the bounds in (5.2.8). A main step in the proof of this is to show that, if E > 0 is fixed, then for every N
r
Jlx-yl>>.-12k(l+•) #
IH~ k(x,y)l dy, I
"rJlx-yl>>.-12•)) ~ CTk(6-n( 1/p- 1/ 2)+1/ 2lllfiiLP(M)• (5.2.8') if B(xo, r) denotes the ball ofradius r around xo with respect to a fixed smooth metric. But, if we use HOlder's inequ~ity, we can dominate the left side by {2k /At( 1/p- 1/ 2) IIS~Ikfii£2(M)·
Next, we observe that S~ kf = eN 2-k6 (1
-6
mi k(A -
+ l2kTI AI) -N f~r any N.
P)/ where
lienee (5.2.5) gives
2
IIS>.Ik/IIL2(M) 00
~ CT2k6
L(1 + 2k A-11A _ ii)-N (1 + j)2(n(1/p-1/2)-1/2)11/ll~ j=O
~
C' T2k6 A2(n(1/p-1/2)-1/2) (A/ 2k) 11/11~.
5.3. More General Multiplier Theorems
153
So, if we combine this with the last estimate we conclude that the left side of (5.2.8') is always majorized by 2-k6 (2k / >..)n(1/p-1/2) >._n(1/p-1/2)-1/2(>../2k)1/211/llp
= Tk[6-n(1/p-1/2)+1/2) 11/llp, as desired.
To finish matters and prove (5.2.9) we let atk( ·, x, y, ')denote the inverse Fourier transform oft---+ c0 >.. - 6q(t, x, y, 0.8(>..2-kt)p(t)(t+i0)- 6- 1. Then one sees that
ID~DJai,k(r,x,y,,)l :$ Ca-yNTk 6 (2kf>..)a(1+12krf>..I)-N (1+1,1)-hl. (5.2.10) But
Htk(x,y)
= (211")-n
I
ei..,x,y,,)~,
and so (5.2.10) (and the fact that IVepl ~ cix- Yl on the support of the symbol) gives the bounds
IH~ k(x,y)l :$ CNTk>..n i>..Tk(x- Y)I-N, I
I
which of course yield (5.2.9).
Remark. One could also prove (5.2.4) in a more constructive way by combining arguments from the proofs of Theorems 5.1.1 and 2.3.1. In fact, the stationary phase arguments that were used in the proof of Lemma 5.1.3 show that the kernel of S~ is 0(>..-N) outside of a fixed neighborhood of the diagonal, while near the diagonal it is of the form
s~ (X, y) where
t/J;
= >.. (n-1)/2-6 ( ei.X,Pl(x,y) ai,1 (x, y) + e-i.X1/J2(y,x)Qt2(X, y))
I
is as in Lemma 5.1.3 and a 6 ·( x, y )I< ID x,ya_x, _ Ca (d"IS t (x, y ))-(n+l)/2-6-lal . 3
Since the phase functions satisfy the n x n Carleson-Sjolin condition one can therefore break up the kernel dyadically near the diagonal and argue as in the proof of Theorem 2.3.1 to obtain (5.2.4).
5.3. More General Multiplier Theorems Suppose that mE £CX>(JR). Fix .BE C~( (1/2, 2)) satisfying L:~oo ,8(2ir) = 1, T > 0, and suppose also that sup>..- 1 -X>O
1-oo l>..aD~(.B(r/>..)m(r))l 00
2 dr < oo,
0 :$a:$ s,
(5.3.1)
5. LP Estimates of Eigenfunctions
154
where sis an integer> n/2. Then if p(e) is homogeneous of degree one, positive, and C 00 in Rn \0 it follows that m(p(e)) satisfies the hypotheses in the Hormander multiplier theorem, Theorem 0.2.6. Therefore (m(p(D))f)(x) = (211")-n
f ei(x,~)m(p(e)) i(e) de
}Rn
is a bounded operator on LP(Rn) for all1 < p < oo. The purpose of this section is to extend this result to the setting of compact manifolds of dimension n ~ 2.
Theorem 5.3.1: Let m E L00 (R) satisfy (5.3.1). Then if P(x, D) E \11~1 (M) is positive and self-adjoint it follows that
llm(P)fiiLP(M) :::; Cp llfiiLP(M)•
1 < P < oo.
(5.3.2)
Proof: Since the complex conjugate of m satisfies the same hypotheses we need only prove (5.3.2) for exponents 1 < p :::; 2. This will allow us to exploit orthogonality and also reduce (5.3.2) to showing that m(P) is weak-type (1,1): JL{x: lm(P)f(x)l > o:} :::; Co:- 1 11flh·
(5.3.2')
Here JL(E) denotes the dx measure of E C M. Since m(P) is bounded on £ 2 , (5.3.21) implies (5.3.2) by the Marcinkiewicz interpolation theorem. The proof of the weak-type estimate will involve a splitting of m(P) into two pieces: a main piece to which the Euclidean arguments apply, plus a remainder which can be shown to satisfy much better bounds than are needed using the estimates for the spectral projection operators. Specifically, if p E CQ"(R) is as in the proof of Theorem 5.2.1, we write
m(P)
= m(P) + r(P),
where
m(P) = (m * p) (P) =
_.!:.._I eitP p(t) m(t) dt. 211"
To estimate the remainder we define for>.= 2i ,j = 1, 2, ... , m~(r) = {3(rf>.)m(r). Then we put
r~(P) = 2~ and notice that ro(P)
I
eitP(1- p(t)) m~(t) dt,
= r(P) -
E r 2 (P) t
is a bounded and rapidly
k~1
decreasing function of P. Hence ro(P) is bounded from £ 1 to any £P space. Therefore, we would have
llr(P)fll2 :::; C llfll1,
(5.3.3)
155
5.3. More General Multiplier Theorems which is much stronger than the analogue of (5.3.21), if we could show that (5.3.3')
To prove this we use the L 1 operators to get
-+
L2 bounds for the spectral projection
00
llr.x(P)/11~ ::;
L
llr.x(P)Xkfll~
k=O 00
::; C
L TE(k,k+l) sup lr.x(r)l(1 + k)n-
1
11/11~.
k=O
Hence (5.3.3') would be a consequence of 00
L: k=O
sup
lr.x(r)l2(1 + k)n-1::; c.xn-2s.
(5.3.4)
TE(k,k+l)
We claim that this follows from our assumption (5.3.1). The first thing to notice, though, is that since m,x(r) = 0 forT fl_ (A/2,2Aj both m,x(r) and r,x(r) are 0((1 + lrl + 1-XI)-N) for any N if T fl. [.X/4,4-X]. Hence .£S:3.4) would follow from
L:
sup
lr.x(r)l2::; c.x1-2s.
(5.3.41)
kE(.\/4,4.\) TE(k,k+l) But if we use the fundamental theorem of calculus and the CauchySchwarz inequality, we find that we can dominate this by I
jlr.x(r)l 2 dr+ jlr~(r)l 2 dr=
2~ jlm.x(t)(1-p(t))1 2 dt +
2~ j 1tm.x(t)1 21(1- p(t))l 2 dt.
Recall that p = 1 for ltl < e/2, so a change of variables shows that this is majorized by
_!_,x -1-2s 211"
j ltsm.x (tf .X)I2 dt
= _x-1-2s jlv:(.xm,x(.Xr)) 12 d; = _x1-2s. {.x-t jl.xsv:(f3(rf.X)m(r))l2dr}. By (5.3.1) the expression inside the braces is bounded independently of A, giving us (5.3.4').
5. LP Estimates of Eigenfunctions
156
Since we have established (5.3.3), it suffices to show that m(P) m(P)- r(P) is weak-type (1,1}:
JL{x: lm(P)I(x)l >a}~ Ca- 1 111111·
=
(5.3.5)
But if we argue as before, this would follow from showing that the integral operator with kernel
K(x, y)
= (211")-n- 1 = (211")-n
If ei[rp(x,y,~)+tp(y.~)]
1eirp(x,y.~)
p(t) m(t) q(t, x, y, ') ~dt
m(p(y, ,), x, y, ,) ~
(5.3.6)
is weak-type (1,1}. Here we have abused notation a bit by letting m(r,x,y,,) denote the inverse Fourier transform oft ---+ p(t)m(r) X q(t, x, y, ,), that is,
m(r,x,y,O = ([p( · )q( · ,x,y,OJv * m)(r).
(5.3.7)
Notice that, if we define m_x(r,x,y,,) in a similar manner, then this function is C 00 and, moreover, (5.3.1) and the fact that q E SO imply
L
sup
l!Di'De(m_x(p(y,A,),x,y,A'})I 2 ~.,>.e), x, yf >.,>.e) m.x (p(y/ >., >.,.,), x, yf >., >.,.,) dx ded,.,.
However, if we use (5.3.11) we see that, for any N, this is dominated by
2: JJJ >.n(1 + >.ie- ,.,1) -NIDJ m.x (p(y/ >.,>.e), x, yf >.,>.e) 1
1
O:Sh;I:SN
x
IDJ m.x (p(y/ >., >.,.,), x, yf >.,>.,.,)I ded17dx. 2
Therefore, if we apply the Schwarz inequality and (5.3.8), we get (5.3.10), since the integrand in the last expression is compactly supported in x. To finish the proof we must show that when Ia: I= s
jl(x- y)aK,x(x,y)l 2 dx
~ ~-
But IVePI ~ clx- Yi on suppq. Therefore, the above arguments show that the integral is dominated by
L JJJ >.n(1 + >.ie- '71) -NIDil Dt (m,x(p(yf>., ~),x, yf>., >.e)) I O:Sia;I:Ss O:Sh;I:SN
x 1Di2 D~ 2 (m.x (p(y/ >., >.,.,), x, yf >., >.17)) lded17dx.
As before, this along with (5.3.8) and an application of the Schwarz I inequality implies the missing inequality.
Notes Theorem 5.1.1 is due to Sogge [2], Christ and Sogge [1], and Seeger and Sogge [2]. The argument showing that the estimates in this theorem are sharp is a variable coefficient version of an argument of Knapp (see Tomas [1]) which applied to the £ 2 restriction theorem for the Fourier transform in an' and the Riemannian version of this argument was given in Sogge [4]. See also Stanton and Weinstein [1]. Davies [2] has shown that the estimates in Corollary 5.1.2 need not hold if the metric is not
Notes assumed to be C 00 • Even sharp L 00 estimates are not known when the for 0 < a < 2. However, using heat kernel metric is assumed to be techniques (see Davies [1), [2)), one can show that, for L 00 uniformly elliptic metrics, IIXAII(£2,£"") = O(.~n/ 2 ). A reasonable conjecture might metrics, with 0 .II(£2,£oo) = O(A(n-a)/2 ). be that for Using the Hadamard parametrix and the proof of Lemma 4.2.4, one can show that the bounds are 0(A(n-l)(2 ) for C 1•1 metrics. It would also be interesting to find the appropriate extension of Theorem 5.1.1 to compact manifolds with boundary. In two dimensions, Grieser [1) has shown that if P = ~and if the boundary is geodesically concave (i.e., diffractive), then the estimates in Theorem 5.1.1 hold; on the other hand, (5.1.1) can only hold for a smaller range of exponents for manifolds with convex (i.e., gliding) boundary. The strong unique continuation theorem for the Laplacian is due to Jerison and Kenig [1), and independently to Sawyer [1) in three dimensions. The simplified proof we have used, though, is from Jerison [1). For related arguments which show how the L 2 restriction theorem for the Fourier transform in Rn and related oscillatory integral theorems can be used to prove uniqueness theorems and embedding theorems see Hormander [8), Kenig, Ruiz, and Sogge [1), Sogge [5), and Wolff [1], [2). The estimates for Riesz means are due to Sogge [3) and Christ and Sogge [1). The best prior results were due to Hormander [3) and Berard [1). The extension of the Hormander multiplier theorem to the setting of compact manifolds was proved in Seeger and Sogge [1).
ca
ca
159
Chapter 6 Fourier Integral Operators
We start out with a rapid and somewhat sketchy introduction to Fourier integral operators, emphasizing the role of stationary phase and only presenting material that will be needed later. In Section 2 we give the standard proof of the L 2 boundedness of Fourier integral operators whose canonical relations are locally a canonical graph and we state and prove a special case of the composition theorem in which one of the operators is assumed to be of this form. The same proof of course shows that this theorem holds under the weaker assumption that C1 x C2 intersects {(x, y, TJ, y, TJ, z, () : (x, E T* X\ 0, (y, TJ) E T*Y \0, (z, () E T* Z \ 0} transversally, although it is a little harder to check here that the phase function arising in the proof of the composition theorem is non-degenerate. The next thing we do is to prove the pointwise and LP regularity theorems for Fourier integral operators and show that these are sharp if the operators are conormal with largest possible singular supports. Although this theorem came first, its proof uses the decomposition used in the proof of the maximal theorems for Riesz means and the circular maximal theorem given in Section 2.4. In the last section we apply the estimates for Fourier integral operators to give a proof of Stein's spherical maximal theorem and its variable coefficient generalizations involving the assumption of rotational curvature. In anticipation of the last chapter, we point out how this assumption is inadequate for variable coefficient maximal theorems in the plane.
e,
e)
6.1. Lagrangian Distributions
161
6.1. Lagrangian Distributions In Section 0.5 we studied certain types of homogeneous oscillatory integrals whose wave front sets turned out to be Lagrangian submanifolds of the cotangent bundle. In this section we return to the study of such distributions, this time taking a somewhat more global point of view. We start out with a few definitions. First of all, the Besov space 00Hu(R.n) is defined to be the space of all u E S'(R.n) for which u E L~c (R.n) and, moreover,
llullooH
"
(Rn) = ( {
l1e19
lft(,)l 2 ~) 112 f
+sup( j?:_O
J2i$Jei$2Hl
j2uiu(,)j 2 ~) 112 < oo.
(6.1.1)
If X is a C 00 manifold of dimension n we can extend this definition by using local coordinates. We define 00H~c(X) to be all u E V'(X) for which (t/Ju) 01\';-l is always in 00Hu(R.n) whenever n c X is a coordinate patch with coordinates ~t and t/J E Clf(O). Next, if A C T* X\ 0 is a C 00 closed conic (immersed) Lagrangian submanifold, we define the space of all I;..airangian distributions of order m which are associated to A, 1m( X, A), as follows. We say that u E Im(X,A) if N
IT P;u
E 00H~~-n;4 (X),
(6.1.2)
j=l
whenever P; E '111~ 1 (X) are properly supported pseudo-differential opertors1 whose principal symbols P;(x,,) vanish on A. The reason for the strange convention concerning the order will become apparent if one considers pseudo-differential operators. Specifically, if a(x, ') E sm(R.n) then it follows from Theorem 0.5.1 that the Schwartz kernel of a(x, D),
u(x,y) = {21r)-n
Jei(x-y,e)a(x,,)~,
satisfies WF(u) C A= {(x,x,,,-,)}. Moreover, Theorem 6.1.4 below 1 P is said to be properly supported if, given a compact set K C X, there is always a compact set K' C X such that supp u C K ~ supp Pu C K' and u 0 inK' ~ Pu. = 0 inK. The reader can check that any pseudo-differential operator can be written as the sum of a properly supported pseudo-differential operator plus a smoothing error. Notice also that if X is compact then every pseudo-differential operator is properly supported.
=
6. Fourier Integral Operators
162
shows that (6.1.2) is exactly the right normalization so that the order of the Lagrangian distribution u is the same as the order of the associated pseudo-differential operator, that is, u E rn(R.n x IRn, A). Another remark is that ifu E /m(X, A) then WF(u) CA. To see this, we pick (x 0 , 'o) fl. A and let r be a small conic neighborhood of (xo, 'o) satisfying f n A = 0. Then, if P; E w~1 ,j = 1, ... , N, have principal symbols supported inside r, (6.1.2) must hold. From this one deduces WF(u) = 0 which gives us the claim. To prepare for the main result of this section, the equivalence of phase function theorem for Lagrangian distributions, we need a few preliminary results. The first one concerns the Fourier transform of u E /m(R.n, A) when A takes a special form. We saw in Section 0.5 that if H(') E C 00 (1Rn \ 0) is real and homogeneous of degree one, then A= {(H'(,), ')} is a conic Lagrangian submanifold of T*IRn \0. In this setting we have:
nr
Proposition 6.1.1: lfu E I~mp (IRn, A) with A of the form {(H'(,), ')} then for 1,1 ~ 1, u(') = e-iH(e)v(') with v E sm-n/4(1Rn). Proof: Let p E Clf(IRn) equal !me near the origin and let h = pHo, with Ho = (1- p)H. Then Hrr- hE S and so it suffices to show that
v(') = eih(e) u(') E sm-n/4. Set h; = oh/8,;. Then, by construction, h;(D) is properly supported since this operator is convolution with the inverse Fourier transform of h;(') which is compactly supported. Hence, since the principal symbol of Dk(x; - h;(D)) vanishes on A, we get from (6.1.2) that
v.BIJ(x;- h;(D))a;u E 00H_m-n/4 if la:l
= I.BI-
This means that for R > 1
{
1R/2::;iei::;2R
l,.a
IJ {-D.h ·('))a;u(,)l2 cl{ :S CaR2(m+n/4), 3 3 la:l =
I.BI,
or, equivalently,
{
1R/2::;iei9R
1'12lai1Dav(,)l2 cl{ :S CaR2(m+n/4).
By rescaling, we see that vn(') = v(RI,.)/ Rm-n/4 satisfy the uniform estimates
6.1. Lagmngian Distributions By the Sobolev embedding theorem, this implies that ID0 vR(e)l when lei= 1, or, equivalently, ID0 vl ~ Ca(1 + len-m+n/ 4 -lal.
163 ~
Ca
1
We need one other result for the proof of the equivalence of phase function theorem. Recall that in Section 0.5 we saw that every Lagrangian section ofT*R.n is locally the gradient of a C 00 function. A similar result holds for homogeneous Lagrangian submanifolds. Proposition 6.1.2: Let 'YO = (xo, eo) E A C T* X\ 0, with A being a C 00 conic Lagrangian manifold. Then local coordinates vanishing at xo can be chosen such that
e
( 1) A 3 (X' e) -+ is a local diffeomorphism, (2) and there is a unique real homogeneous HE C 00 such that, near (xo, eo),
A= {(H'(e), en. Let us first assume that (1) holds and then see that this implies (2). This is easy, for, near 'Yo, A = {(if>(e), en for some l/> which is homogeneous of degree zero and C 00 near eo. We saw in the proof of Proposition o.5.4 that the canonical one form w = E e;dx; must vanish identically on A. This means that if 4>; denotes the jth coordinate,
L:e; dlf>;(e)
= o.
Or, if we set H(e) = Ee;l/>;(e), then
dH(e) = L:4>;(e)de;. that is, l/>;(e) = aH(e)Jae;. giving us (2). To prove that local coordinates can be chosen so that ( 1) holds we need an elementary result from the theory of symplectic vector spaces whose proof will be given in an appendix. Lemma 6.1.3: lfVo and V1 are two Lagrangian subspaces ofT*R.n one can always find a third Lagrangian subspace V which is transverse to both Vo and V1. Recall that two C 00 submanifolds Y, Z of a C 00 manifold X are said to intersect transversally at xo E Y n Z if Tx 0 X = Tx 0 Y + Tx 0 Z. Proof of (1): We first choose local coordinates y so that 'YO = (0, cl} with E'l = (1, 0, ... , 0). The tangent plane to A at 'Yo, Vo, must be a Lagrangian plane. If it is transverse to the plane W = {(y, cl)} = {(y, dyl)}, we can take y as our coordinates since the transversali ty of Vo and W is equivalent to (1). If not, we use Lemma 6.1.3 to pick a Lagrangian plane V which is transverse to both Vo and V1 = {(0, en.
164
6. Fourier Integral Operators The transversality of V and V1 means that V is a section passing through 'Yo and hence V = {(y,d(yl + Q(y)))} for some real quadratic form Q. If we now take x1 = Yl + Q(y),x; = Y;,j = 2, ... ,n, as our new coordinates, it follows that, in these coordinates, the tangent plane at 'Yo, Vo, and V = {(x,dxi)} are transverse, giving us (1). I We now come to the main result. Recall that the homogeneous phase function ¢(x, 9) is said to be non-degenerate if d¢ '# 0 and when ¢ = 0 the N differentials d(o¢fo9;) are linearly independent. We saw before that this implies that :Ecf> = {(x, 9) : ¢ (x, 9) = 0} is a C 00 submanifold of X x (aN\ 0) and that A= {(x, ¢~(x, 9)) : (x, 9) E :Ecf>} is Lagrangian.
0
0
Theorem 6.1.4: Let¢ be a non-degenerate phase function in an open conic neighborhood of (xo, lio) E an X (aN\ 0). Then, if a E S~£(an X aN), p. = m+ n/4- N /2, is supported in a sufficiently small conic neighborhood r of(xo,9o) itfollows that
u(x) = (211")-(n+ 2N)/4 {
JRN
eicf>(x,(J) a(x, 9) d9
(6.1.3)
is in Im(an,A) with A as above. If we also assume that coordinates are chosen so that A= {(H'(,), ,)}, then
eiH(e)u(')- (21r)nf 4a(x, 9)ldet ¢"1-l/2e¥ sgncf>" E sm-n/ 4 - l (6.1.4) for 1,1 > 1 near'o = ¢~(xo,9o), where (x,9) is the solution of¢ (x,9) = 0, ¢~(x, 9) = ,, and
0
"'" = (
"+'
rP~x ¢~(} ) "'" "+'(Jx
"'"(}(} "+'
.
Conversely, every u E Im(an, A) with W F( u) contained in a small neighborhood of (xo, 'o) can be written as (6.1.3) modulo C 00 • One should notice that this result contains the equivalence of phase function theorem for pseudo-differential operators, Theorem 3.2.1, since the phase function rp(x, y, ') there and the Euclidean phase function (x -y,,} both parameterize the trivial Lagrangian {(x,x,,, _,)}.Similarly, if u E I~mp (an x an, A), with A the trivial Lagrangian, it follows that u(x, y) is the kernel of a pseudo-diffE'ft~tial operator of order m. So this clarifies the remark made earlier that the order of the distribution kernel of a pseudo-differential operator agrees with the order of the pseudo-differential operator. Using (6.1.4) we can define ellipticity. We say that u E F(X, A) is elliptic if, when coordinates are chosen so that A = {( H' ('), ')}, the absolute value of either of the terms on the left side of (6.1.4) is bounded
165
6.1. Lagrangian Distributions from below by l'lm-n/ 4 for large '· Since det t/J11 is homogeneous of degree -(N- n) this just means that a(x,8) is bounded from below by l8lm+n/ 4-N/2 when 181 is large and (x, 8) E :Eel>.
Proof of Theorem 6.1.4: We may assume that a(x, 8) vanishes when x is outside of a compact set and that coordinates have been chosen so that A= {(H'(,), ,)}. We shall then use stationary phase to evaluate eiH(e>u(') = (21r)-(n+2N)/4
JJ i[cf>(x,B)-(x,e)+H(e)J a(x, 8) d8dx. (6.1.5)
The first thing we must check is that the Hessian (with respect to the variables of integration) of the phase function is non-degenerate, that is, det t/J" '# 0. But this follows from the fact that the maps
Eel>
3
(x, 8) - (x, t/J~(x, 8))
E
A and A 3 (x, ') - '
are both diffeomorphisms. Since t~J9 = 0 on Ecf>, this means that the map r 3 (x, 8) - (t/J~. t~J{J) has surjective differential on :Eel> and hence in r, if this set is small enough. But this is just the statement that det t/J" '# 0. Since the stationary points, depending on the parameter ', are nondegenerate, we can assu~e, ..after possibly contracting r, that, for every 'near there is a unique stationary point. Since we may also assume that t/J~ '# 0 in r it follows that the difference between (6.1.5) and
,o,
(21r)-(n+2N)/4
JJei[cf>(x,B)-(x,e)+H(e)lp(8/lel) a(x, 8) d8dx
(6.1.5')
is rapidly decreasing if .B E CQ"(IRn \ 0) equals one for 181 E re-I C] with C sufficiently large-in particular, large enough so that .8(8/IW = 1 at a stationary point. Next, if we set >. = lei and w = '/1,1, we can rewrite (6.1.51) as I
(21r) -(n+2N)/4 >. N
JJei.X[cf>(x,B)-(x,w)+H(w)] .8(8) a(x, >.8) d8dx.
Notice that the integration is over a fixed compact set and that, at a stationary point, we have t/Jo = O,x = H'(w), and hence, by Euler's homogeneity relations,
tfJ(x,8) = (tfJ{J(x,8),8} = 0 and
(x,w} = (H1(w),w} = H(w).
Thus one reaches the conclusion that the phase function always vanishes at the stationary point. Therefore, the stationary phase formula (1.1.20) tells us that the difference between (6.1.51) and
(21r)n1 4 >.(N-n)/2 a(x, >.O)Idet t/J11 (x, 8) ~-l/ 2 e¥ sgncf>"
(6.1.6)
6. Fourier Integral Operators
166
is in sm-n/4- 1I since (m + n/4- N/2) + (N- n)/2 dettf>" is homogeneous of degree -(N- n) and so
= m- n/4.
But
A(N-n)/21det 4>" (x, 8)1-1/2 = ldet 4>" (x, .X8)1-1/2.
Hence, (6.1.6) is the second term in (6.1.4), which gives us the first half of the theorem. To prove the converse, we use Propositions 6.1.1 and 6.1.2 to see that it suffices to consider u E Im(!Rn I A) having the property that v = ueiH E sm-n/4 is supported in a small conic neighborhood of eo. We then let ~(x, 8) = 8tj>f8x and put
ao(x,8)
= (21r)-n/ 4 vo~(x,8)ldettf>"l 1 / 2 e-7 8 gn¢" E sm+(n-2N)/4.
If we then define uo by the analogue of (6.1.3) where a(x, 8) is replaced by ao(x, 8), it follows that u-uo E Jm- 1. Continuing, we construct Uj from
aj(x,8) E sm+(n- 2N)/ 4-i so that u-(uo+·· ·+uj) E lm-i- 1(1Rn,A). If we then pick a"" Eai, it then follows that (6.1.4) holds mod C 00 •
I Let us end this discussion by making a few miscellaneous remarks concerning the many ways that one can write Lagrangian distributions. First, if u is given by an oscillatory integral as in (6.1.3), one can add as many 8 variables as one wishes and get a similar expression involving the new variables. In fact, if Q(8NH• ... , 8N') is a non-degenerate quadratic form in N' - N variables, one checks that the phase function ;[,(8) = 4>(8) +Q(8N+I•····8N,)/I81, defined in the region I(8NH·····8N')I < 181, is also non-degenerate and parameterizes A. So by_ Theorem 6.1.4 we can express u by an oscillatory integral involving 4> and a symbol a(x,B) E sm+n/4-N'/2. A more interesting construction involves the reduction of theta variables. As a preliminary, we need an observation about the rank of the Hessian of the phase function with respect to the theta variables. Specifically, let /IA : A 3 (x, e) ---+ X and /IE~ : Eq, 3 (x, 8) ---+ X denote projection onto the base variable, and let K. : Eq, 3 (x, 8) ---+ (x, t/>~(x, 8)) E A. Then, if K.(xo, 8o) = (xo, eo), we can compare the rank of t/>~o(xo, 8o) with the rank of d/IA(xo,eo). In fact, since K. is locally a diffeomorphism and Eq, is n-dimensional-both because we are assuming that 4> is non-degenerate-we have the formula dim
Kerd/IE~ =
n- rankd/IA·
(6.1.7)
But if a tangent vector is in Kerd/IE~ it must necessarily be of the form v = Ei vj8/88j. And it must also satisfy Ei vj8 2¢jaoiaok = 0,
6.1. Lagrangian Distributions
167
= 1, ... , N, since this is a necessary and sufficient condition for v to be a tangent vector to E4>. Since the dimension of such tangent vectors at (xo, 8o) is N- rank 4>~9 (xo, Oo), (6.1.7) implies the following:
k
Proposition 6.1.5: If (xo, 'o) E A we have
4> is non-degenerate and (xo, t/>~(xo, 8o))
N- rankt/>~9 (xo,8o) ~ n- rankdllA(xo,,o).
=
(6.1.8)
Let us now see that this result implies that we can reduce the number of theta variables ifrankdllA is large. Let r = rankt/>~9 (xo,8o). We may assume that tf>~, 9 ,(xo,8o) is invertible where 81 = (8t, ... ,8N-r) and 811 = (8N-r+b ... ,ON)· By the implicit function theorem, near (xo,,o), there is a unique solution 811 = g(x, 81) to the equation tf>~,(x, 81, 811 ) = 0. Clearly g must be smooth and homogeneous of degree one in 8'. We then lt:t 8 = (81, 8''- g(x, 81)) and define~ by ~(x, 0) = tf>(x, 0). Since'#:: = 0 if and only if 011 = 0 we conclude that ~"- = 0 and 411: x,9"
011 =
9',9"
9" = 0 when
0. Therefore, if we set
1/J(x,O')
= tf>(x,81,g(x,81)) = ~(x,O',o),
we get
·'·" 'l'x9' = ;." xB' IB"=O
I
;." B"=O" and ·'·" '1'9'9' = "'8•8• These conditions imply that 1/J is non-degenerate. Furthermore, the first condition, along with Euler's homogeneity relations and the fact that ~11- = 0 when 011 = 0, gives that 1/J~ = 4>~ when 011 = 0. Consequently, x9" 1/J also parameterizes the Lagrangian A near (xo, 'o). Therefore, if u(x) is given by (6.1.3), with a(x, 0) having small enough support, there must be a symbol bE sm+n/4 -(N-r)/2 such that, modulo C 00 , 'I'
u(x) = ( 21r)-[n+2(N-r)]/4 { ei'I/J(x,9') b(x, 8') d8'. }RN-r
(6.1.9)
Notice that the order of the symbol here has increased by r/2 from the order of the symbol a(x, 8) in (6.1.3) to compensate for the fact that, in (6.1.9), the integration involves r fewer variables. From this we see that if rankdllA = r and if u E I~mp(X, A) then we can write u as a finite sum of oscillatory integrals of the form (6.1.9) modulo C 00 • If dllA has constant rank then we say that u E Im(x, A) is conormal since A is the conormal bundle of Y = llA(A), which must be an r-dimensional smooth submanifold of X by the constant rank theorem.
6. Fourier Integral Operators
168
A concrete example which illustrates the remarks about the reduction of theta variables concerns the Fourier integral operators eitP when the cospheres associated to the principal symbol of P, {e: p(x,e) = 1} c T; X, have nonvanishing Gaussian curvature. Under this assumption, we saw in the proof of Lemma 5.1.3 that the phase functions ~t(x,y,e) = rp(x,y,e) + tp(y,e) satisfy rank82 ~t!aeiaek n. -1 for small nonzero times t. This means that, modulo a smoothing error, for such times one can write
=
(eitP f)(x) = {
JM
1oo ei'I/J,(x,y,9) at(x, y, 8) f(y) d8dy -oo
for some at (x, y, 8) E s(n-I )/2 . The symbol and phase function depend smoothly on the time parameter when t ranges over compact subintervals of [-co,c]\0; however, not as t-+ 0 since eitPit=O is the identity operator and hence can not be expressed by oscillatory integrals involving one theta variable. In the special case where P = ~ one can use (4.1.21) to see that, fort> 0, t/Jt(x,y,8) must be a nonvanishing function of (x, t, y) times 8(ltl- dist (x, y)), where dist (x, y) denotes the distance with respect to the Riemannian metric g. The solution to the Cauchy problem for 82 jat 2 -l:l. 9 of-course can be written in a similar form, involving two oscillatory integrals both having this same phase function.
6.2. Regularity Properties Here we shall study the mapping properties•of a special class of Fourier integral operators. If X and Y are C 00 manifolds, then we shall say that an integral operator :F with kernel :F(x,y) E Im(x x Y,A) is a Fourier integral operator of orderm if A c {(x!y,e, 17) E T*(X x Y) \0: i' 0, 1J i' 0}. We shall usually write things, though, in terms of the associated canonical relation,
e
C = {(x,
e, y, 17) : (x, y, e, -17) E A} C (T* X\ 0) x (T*Y \ 0),
(6.2.1)
and from now on use the notation :FE Im(x, Y;C). The reason that it is more natural to express things in terms of the canonical relation, rather than the Lagrangian associated to the distribution kernel, will become more apparent in the composition formula below and the formulation of various hypotheses concerning the operators. Notice that, since A is Lagrangian with respect to the symplectic from ox +oy, the minus sign in (6.2.1) implies that C must be Lagrangian with respect to the symplectic form ox - oy = E df.j 1\ dxj - E d1Jk 1\ dyk in (T* X\ 0) X (T*Y \ o).
6.2. Regularity Properties
169
In this 5{ ction we shall study the mapping properties of Fourier integral operato 'S whose canonical relation is locally the graph of a canonical transformation-or locally a canonical graph for short. By this we mean that if "YO = (xo, Yo, 110) E C then there must by a symplectomorphism x defined near (yo, 110) so that, near ')'o, C is of the form
eo,
{(x,e,y,7J): (x,e)
= x(y,7J)}.
(6.2.2)
Notice that this forces dim X = dim Y. In addition, this condition is equivalent to the condition that either of the natural projections C -+ T* X\ 0 or C -+ T*Y \0 (and hence both) are local diffeomorphisms. Clearly, if C is locally a canonical graph then the projections are local diffeomorphisms. To see the converse we notice that if, say, C -+ T*Y \ 0 is a local diffeomorphism, then, near -yo, we can use (y, 71) as coordinates for C and hence the canonical relation must locally be of the form (6.2.2). The fact that X must then be canonical is a consequence of the fact that ox - oy vanishes identically on C which forces oy = x• (u x). Let us also, for future use, express this condition in terms of phase functions which locally parameterize C. If ¢(x, y, 8) is a non-degenerate phase function parameterizing A, then, by (6.2.1), C = {(x, ¢~(x, y, 8), y, -¢~(x, y, 8)) : (x, y, 8) E E}·
We claim that C being locally a canonical graph is equivalent to the condition that
¢" det ( ¢~:
¢" ) ¢~: "f:. 0 on E.
To see this, we notice that this is the Jacobian of (y, 8)
(6.2.3) -+
(¢~, ¢~). So
if (6.2.3) holds, then we can solve the equations r/J~(x, y, 8) = 0,
e= ¢~(x, y, 8)
with respect to (y, 8) and hence use (x, e) as local coordinates on E. Thus, (6.2.3) is equivalent to the condition that the projection C -+ T* X \ 0 is a local diffeomorphism, which establishes the claim. Having characterized the hypothesis, let us turn to the main result.
Theorem 6.2.1: Let X andY ben-dimensional C 00 manifolds and let FE JTR(X, Y;C), with C being locally the graph of a canonical transformation. Then
(1) F: L~omp(Y) -+ L~0c(X) if m ~ 0, (2) F: L~mp(Y)-+ Lfoc(X) if 1 < p < oo and m ~ -(n -1)1~- ~I, (3) F: LiPcomp(Y, o:) -+ LiPloc(X, o:) if m ~ -(n- 1)/2.
170
6. Fourier Integral Operators Furthermore, none of these result.s can be improved if :F is elliptic and if corank dllx x y = 1 somewhere, where llx x y : C -+ X x Y is the natural projection operator.
Notice that the non-degeneracy hypothesis that C be locally a canonical graph is the homogeneous version of the non-degeneracy hypothesis in the non-degenerate oscillatory integral theorem, Theorem 2.1.1. In both cases, the non-degeneracy hypothesis is equivalent to the condition that the projection from the canonical relation toT* X be non-singular. Also, by the discussion at the end of the previous section, the last condition in the theorem means that sing supp :F( x, y) contains a C 00 hypersurface. We shall first prove the most important part, the £ 2 estimate of Eskin and Hormander. 1b do this we shall need the following composition theorem. Theorem 6.2.2: Let :F E J~mp(X, Y; Cl) and g E IComp(Y, Z; C2). Then, if C1 is locally the graph of a canonical transformation, :Fog E pn+~£(X, Z;C) where
C = C1 oC2
= {(x,e,z,(): (x,e,Y,'1) E C1 and (y, 71, z, () E C2
for some (y, 71) E T*Y \ 0}.
Also, if both :F and g are elliptic, then so is :F o g.
By taking adjoints (see below), one sees of course that the same result holds if we instead assume that C2 is locally a canonical graph.
Remark. In Chapter 4 we saw that, for small t, the canonical relation of the operator eitP is
c = {(x,t,e,r,y,'1): (x,e) = 4't(Y,'1), r = p(x,e)}, where cllt is the canonical transformation defined by flowing along the Hamilton vector field associated to p(x, e) for time t. However, Theorem 6.2.2 shows that if this holds for small time then it must hold for all times since ei(t+s)P = eitP o eisP has canonical relation
c
0
Cs
= {(x, t, e, T, y, 11) : (x, e) = o { (y, 71, z, () :
4't(Y, '1),
T
= p(x, e)}
(y, 71) = 4'8 (z, ()}
= {(x, t, e, T, y, 71) : (x, e)
= 4't+s(Y, 71),
T
= p(x, e)}.
Hence if C has the stated form for t in [-e, c] then it must also have this form fort in [-e,e] + [-e,e] = [-3e,3e], and by iterating this one concludes that C is as above for all time.
171
6.2. Regularity Properties If we assume the composition theorem for the moment, it is easy to prove part (1) of Theorem 6.2.1. After perhaps breaking up the operator we may assume that :FE ~omp(X, Y;C) with Cas in (6.2.2). But then :F* E ~mp(Y,X;C•) where
c• = {(y,7J,x,e): (x,e,y,7J) E C} = {(y,1J,x,e):
(x,e)
= x(y,7J)}.
c•
Hence, C o must be the trivial relation {(y, 1J, y, 17)} and therefore TheoreiJ15 6.2.2 and 6.1.4 imply that':F* :F must be a pseudo-differential operator of order 0. So the L 2 boundedness of pseudo-differential operators of order 0 gives
j 1Ful 2dx = j r Fu u dy $ 11.r•Full2llull2 $ Cllull~, proving ( 1). Notice that Theorem 6.2.2 implies that PF E Jm+jj(X, Y;C) if Pis a pseudo-differential operator of order JL on X. Using this and Theorem 6.2.1 one obtains the following regularity theorem.
Corollary 6.2.3: Let .r E (1) F : L~mp(Y)
-+
I
~(X, Y; C)
be as in Theorem 6.2.1. Then
Lfoc m-o p (X), if 1 < p < t
00
and ap =
(n- 1)11/P- 1/21, (2) F: LiPcomp(Y, a)-+ LiPloc(X, a- a 00 ), with aoo = (n- 1)/2. As before, all of these results are sharp if .r is elliptic and corank diixxY = 1 somewhere. Notice that the Corollary says that, compared to the L 2 estimates, • one in general loses (n- 1)11/p- 1/21 derivatives in LP and (n- 1)/2 derivatives in the pointwise sense.
Proof of Theorem 6.2.2: We may assume that Ct is parameterized by a non-degenerate phase function t/J(x, y, 9) which is defined in a conic region of (X x Y) x (IRN1 \ 0) and that C2 is parameterized by a non-degenerate phase function cp(y, z, u) defined in a conic region of (Y x Z) x (IRN2 \ 0). It then follows that Ct o C2
= {(x, t/J~, z, -cp~) : t~J{J(x, y, 9) = 0, cp~(y, z, u) = 0, t/J~(x, y, 9) = -cp~(y, z, u)}.
Equivalently, if we set
e = ((191 2 + lul 2)112y,9,u), and define the homogeneous phase function ~(x,
z, 8) = tjJ(x, y, 9)
+ cp(y, z, u),
6. Fourier Integral Operators
172 we have
Ct o C2
= {(x, ell~, z, -ell~) : (x, z, 9) E E4> }.
Notice that ell is defined in a conic region of (X x Z) x (IRN \ 0), where N = N1 + N2 + n, with n = dim Y = dim X. We claim that ell is nondegenerate. This will show that Ct o C2 is a smooth canonical relation which is Lagrangian with respect to ux- uz. To prove the claim we must show that the differentials d(oellfo9j), j = 1, ... , N, are linearly independent on E4>. One can rephrase this as the requirement that the N X ( N + n +dim Z) matrix IJ2 ell I aea( e, X' z) have full rank N here. This in turn just means that
(
,1,11 + """' 'l'yy ryy ,1,11 'I'(Jy (f'l/1
Tt7y
,1,11 'l'y(J ,1,11 'I'(J(J 0
rya
"""'
0 (f'l/1
Tt7t7
,1,11 'l'yx
ryz )
(f'l/1
,1,11
0
0
(f'l/1
'I'(Jz
(6.2.4)
Tt7Z
has full rank when 4J9 = 0, cp~ = 0, and l/J~ + cp~ = 0. But we have already seen that Ct being locally a canonical graph is equivalent to the condition that the (Nt + n) x (Nt + n) submatrix I
be non-singular when degenerate ~orces
4J9
= 0. In addition, the fact that cp is non-
( (f'l/1
(f'l/1 (f'l/1 ) rayraaraz
to have full rank N2. By combining these two facts, and noting the form of the matrix, one sees that (6.2.4) must have rank Nt + n + N2 = N, giving us the claim. It is now a simple matter to finish the proof. Using a partition of unity, we may assume that a1(x,y,8) E sm-Nd 2+n/ 2 and a2(y,z,u) E SP-N2/2+(n+nz)/4 are supported in small compactly based cones in (IRn x R.n) x (IRN1 \0) and (R.n x JRnz) x (IRN2 \0), respectively. Here nz = dimZ. We must show that if :F(x,y) = [ JRN1
ei 0 such that as x --+ 0, I:F/,.,.(x)l > clxlp.-m-n. Thus, we have
:Ff,.,.
fl. Lfoc
if J.L- m- n ~ -nfp.
(6.2.33)
On the other hand, using stationary phase (cf. Lemma 2.3.3) one sees that for y near sn- 1 1/,.,.(y)l ~ (dist(y,sn- 1 ))"'-(n+l)/2 ,
J.L
< (n+ 1)/2,
meaning that
!,.,. E I?
{:=:?
J.L > (n + 1)/2- 1/p.
Substituting this into (6.2.33) shows that :F cannot be bounded on £P for p 2::: 2 if (m + n + 1/p- (n + 1)/2) > nfp, that is,
-(n -1)(1/2- 1/p) < m. This shows tpat the estimates in Theorem 6.2.1 are sharp for p 2::: 2. By duality, the same holds for p < 2 and one can adapt the above arguments to see that the Lipschitz estimates are also sharp.
Remark. If X = Y is a compact C 00 manifold of dimension n and if P E '111~ 1 (X) is elliptic then, for a given t, eitP E f>(X,X;Ct) where Ct is the canonical rel~ion described in the remark after Theorem 6.2.2. Since Ct is a canonical graph we conclude that Otp =
(n- 1)11/p- 1/21, (6.2.34)
lleitP !IILip(o-(n-1)/2)
~ c IIIII Lip (o)•
where C remains bounded for t in any compact time interval. We claim that, for all but a discrete set of times t, these estimates cannot be improved. In view of the above discussion, this amounts to showing that the differential of Ct --+ X x Y must have full rank somewhere unless t belongs to a discrete exceptional set. To see that this is the case, we note that the condition T = p(x, e) in the full canonical relation of eitP means that if (x, e) - cP(y, t, e) is a (local) phase function for the operator, then ¢~ = p(¢e(y, t, e), e). Let us suppose that dllxxY does not have full rank for a given time to. We
6. 2. Regularity Properties
185
shall then see how these facts imply that for all t near to the rank of Ct -+ X x Y has to be maximal somewhere. To show this, we first note that, given xo, we can always choose ~o E Ex0 = {~ E T;0 X\O: p(xo,~) = 1} so that this cosphere has all n - 1 principal curvatures positive at ~0· If coordinates are chosen so that ~o = (0, ... , 0, 1), this is equivalent to the statement that (82 pf8~ja~k) 1 e'f.' (yo, to, ~o) is invertible if~ = (e' ~"), = (6' ... '~r ), and cl>'/.;f.• (yo, to, ~o) = 0 if either j or k is > r. (In the case where r = 0 one would modify this in the obvious way taking ~" = 0 To make use of this, we notice that at (YO, t, ~o) we must have
e
, _ (cl>'/.'f.' + O(t- to) tPf.f. O(t- to)
O(t- to) ) (t- to)p'{"f." + O((t- t 0 ) 2 )
·
( 6·2·35 )
But this matrix must have the largest possible rank, n -1, fort close to
t0 , because the (n- r) x (n- r) Hessian of the homogeneous function ~11 -+
p(xo, 0, ~") must have largest rank n- r - 1, due to the fact that {~": (0, ~") E Ex0 } has non-vanishing Gaussian curvature. One reaches this conclusion about the rank of cl>'/.f. after noting that det (
~: ~:) ~ tm-r,
t small,
if At= A+O(t), with A being a non-singular rxr matrix, Bt, Ct = O(t) and if Dt = tD + O(t2 ) with D being an (m- r) x (m- r) non-singular matrix. Since Proposition 6.1.5 says that the rank of the differential of the projection of Ct at 'YO = (ci>~(YO, t, ~o), ~o, YQ, cl>~(yo, t, ~o)) E Ct is n plus the rank of cl>'/.f.(YO, t, ~o), we conclude that this differential must have maximal rank 2n- 1 at -ro, which finishes the proof. Similar remarks apply to the regularity properties of solutions to strictly hyperbolic differential equations. Specifically, let L(x, t, Dx,t) = D~ + Ej!: 1 Pj(x, t, Dx)D;n-j be a strictly hyperbolic differential operator of order m. Then if {/j} j!:(/ are the data for the Cauchy problem
186
6. Fourier Integral Operators we have, for instance, that the solution satisfies m-1
lluiiLP(X) :5 C
L
11/;IIL:p_/X).
j=O
ll.i=1 (
Moreover, if T - Aj (x, t, e)) is a factorization of the principal symbol of L and if, for every t, at least one of the roots >.; is a nonzero function of T* X \ Q-that is, elliptic-this result cannot be improved.
6.3. Spherical Maximal Theorems: Take 1 In this section we shall present some maximal theorems which are related to Stein's spherical maximal theorem. The latter is the higherdimensional version of the circular maximal theorem proved at the end of Section 2.4. It says that, if n ~ 3 and p > nf(n- 1), f(x- ty)dcT(y)'P dx)lfp :5 Cpii/IILP(Rn)• ( { supl { }Rn t>O lsn-1
f E S. (6.3.1)
Notice that the spherical means operators which are involved are Fourier integral operators of order -(n-1)/2. This is because their distribution kernels are t-"6o(1-lx- Ylft), where 6o is the Dirac delta distribution. Consequently, we can write the spherical mean operator corresponding to the dilation t as
(211")-1 {
}Rn
1
00
t-neiO(l-lx-ylft] f(y) d(}dy.
-oo
Since there is just one theta variable, the order must be 1/2- 2n/4 = -(n-1)/2 as claimed. Notice also that the phase function satisfies (6.2.3) and hence the spherical means operators are a smooth family of Fourier integral operators of order -(n-1)/2, each of which is locally a canonical graph. In this section we shall always deal with Fourier integral operators Ft E I~mp(X,Y;Ct),t E I, where X andY are C 00 manifolds of a common dimension n and I C R. is an interval. We say that {Ft} is a smooth family of operators in I~mp if there are finitely many nondegenerate phase functions tPtJ(x, y, 9) and symbols a;,t(x, y, 9) so that, fortE I, we can write Ftf(x)
='~}y}RNi "" { { ei¢•.;(x,y,O)at,;(x,y,9)f(y)d9dy. 1
We require that both the ¢t,j and the atJ be smooth functions of t
6.3. Spherical Maximal Theorems: Take 1
187
with values in S 1 (r;) and sm-N;/2+n/2(r;), respectively, where r; is an open conic subset of (X x Y) x (JR.N; \ 0). In addition, we require that the symbols be supported in r; and vanish for (x, y) not belonging to a fixed compact subset of X x Y. Usually, we shall deal with smooth bounded families in 1:mp> by which we mean that, in addition to the above, we have the following uniform bounds for (t,x,y,9) E I X r;:
ID~,y,tD~t,j(X, y, 9)1 ~ Ca-y(1 + 191) 1-lal, ID~,y,tD~at,j(X, y, 9)1 ~ Ca-y(1 + l91)m-(N;-n)/2-lal.
(6.3.2)
We can now state the main result of this section. Later on we shall see that it can be used to recover Stein's theorem (6.3.1). Theorem 6.3.1: Let :Ft E /:mp(X, Y; Ct), t E [1, 2], be a smooth family of Fourier integral operators which belongs to a bounded subset of 1:mp· Assume also that, for every t E [1, 2], Ct is locally a canonical graph. Then if ap = (n- 1)11/p- 1/21, we have for p > 1
(J
sup I:Ftf(x)IP dx) 1/P
~ CllfiiLP,
m
< -ap- 1/p.
(6.3.3)
tE(1,2J
In particular, if n ~ 3 and p > nf(n- 1), the maximal inequality holds if m = -(n- 1)/2.
The proof involves a straightforward application of Lemma 2.4.2. We fix {3 E Clf((1/2,2)) satisfying E/3(2-ks) = 1, s > 0, and define for k = 1,2, ... :Fk,tf(x)
=~If eitP•.;(x,y,fJ) nt,;(x, y, 9) /3(19l/2k) f(y) d9dy. 3
Then, if we set :Fo,t = :Ft- E~ 1 :Fk,t• it follows that :Fo,t has a bounded compactly supported kernel and hence the maximal operator associated to it is trivially bounded on all £P spaces. On account of this, (6.3.3) would follow from showing that fork= 1, 2, ... ( / sup I:Fk,tf(x)IPdx) 11P ~ C2k(m+ap+l/p) llfiiLP·
(6.3.31)
tE(1,2J
But if we apply Lemma 2.4.2, we can dominate the pth power of the left side by
lxf I:Fk 'd(x)IP dx
188
6. Fourier Integral Operators Theorem 6.2.1 implies that the first term satisfies the desired bounds and also that
( [ IFk,tf(x)IPdx f/p' :S C{2k(m+op) = C(2k(m+op)
11/llpt/p' 11/llp)p-1.
Since (6.3.2) implies that 2-kdt_.rk,t is a bounded family in I:mp> we can also estimate the last factor:
If we integrate in t and combine these two estimates, we get (6.3.31). The last part of the theorem just follows from the fact that, for n ~ 3, O:p + 1/p is smaller than (n -1)/2 precisely when p > nf(n -1). Notice that when n = 2 there are no exponents for which this is true. In fact, for p < 2, a:p + 1/p is > 1/2, while for p ~ 2, a:p + 1/p 1/2. This explains why maximal theorems like the circular maximal theorem are harder to prove than their higher-dimensional counterpart. Momentarily, we shall see that the last statement in Theorem 6.3.1 cannot hold in this level of generality when n = 2. An additional condition is needed which takes into account the t dependence of the canonical relations Ct. To make this more precise, let us state a corollary of Theorem 6.3.1. This deals with averaging over hypersurfaces given by
=
Sx,t
= {y E IR.n: 4't(x,y) = 0}.
Here, the defining function is assumed to be a smooth function of (t, x, y) E [1, 2] X IR.n X IR.n. We also assume that both V' x4't and V' y4't never vanish, and, moreover, that the Monge-Ampere matrix associated to 4't is non-singular: det (
~ afxi~) 1 o.
(6.3.4)
If this is the case, we say that the family of surfaces satisfies the rotational curvature condition of Phong and Stein, and we have the following:
Corollary 6.3.2: Let Sx,t be as above and let dux,t denote Lebesgue measure on the hypersurface. Fix71(x, y) E CSO(JR.n xlR.n) and (for f E S) set
Atf(x)
=
1
S.,,,
71(x, y)f(y) dux,t(y).
189
6.3. Spherical Maximal Theorems: Take 1 Then, if n
~
3, it follows that
II tE(1,2) sup IAtf(x)lllv•(R") ~ Cp 11/IILP(R")•
if P > nf(n- 1).
(6.3.5)
To see that this follows from the last part of Theorem 6.3.1, we write Atf(x) = (211")- 1
Ji: ei9~•(x,y)170(x,
y) f(y) d9dy,
where 710 is a C 00 function times TJ· This is a bounded family in ~~~p- 1 >1 2 . By (6.2.3), each operator is locally a canonical graph since the Monge-Ampere condition is satisfied, and hence (6.3.5) is just a special case of Theorem 6.3.1. There are two extreme cases which should be pointed out. One is when Sx,t = { x} + t · S, where S is a fixed hypersurface. In this case the operators are (essentially) translation-invariant and (6.3.4) is satisfied if and only if the hypersurface S has nonvanishing Gaussian curvature. The other extreme case is when the rotational curvature hypothesis is fulfilled because of the way the surfaces change with x and y and not because of the presence of Gaussian curvature. For instance, if ~t(x,
y) = (x, y} - t,
(6.3.6)
the operator At involves averaging over the hyperplane Sx,t = {y : (x, y} = t}. IfTJ(x, y) vanishes near the origin, this family still satisfies the curvature hypothesis (6.3.4) despite the fact that the Gaussian curvature and all the principal curvatures vanish identically on Sx,t·
Remark. The last example shows how the above results cannot extend in this level of generality to two dimensions. For if ~t is given by (6.3.6) and 0 $ TJ E Cif is nontrivial then (6.3.5) can never hold for a finite p if n = 2. This is because, given E > 0, there is a measurable subset n~ = [-1, 1] X (-1, 1] satisfying IO~I < E but having the property that a unit line segment can be continously moved in n~ until its orientation is reversed. Thus, n~ contains a unit line segment in every direction. If one takes f to be the characteristic function of an appropriate translate and dilate of this Kakeya set n~, the left side is ~ 1 while the right side is < e 1/P, providing a counterexample to the possibility of the twodimensional theorem. Let us conclude this section by showing how Theorem 6.3.1 can be used to prove maximal theorems involving smooth hypersurfaces which shrink to a point as t --+ 0. Specifically, we assume that Bx.t c R.n is a
I
190
6. Fourier Integral Operators family of C 00 hypersurfaces which depend smoothly on the parameters (x,t) E R.n x [0, 1]. We then put Sx,t
=X+ tSx,t·
Our assumptions are that: (i) Given any subinterval [a, 1] with a > 0 the surfaces Sx,t satisfy the rotational curvature hypothesis in Corollary 6.3.2, and (ii) the "initial hypersurface" Bx,o always has everywhere nonvanishing Gaussian curvature. Corollary 6.3.3: Let TJ E cr(IRn x R.n) and suppose that Sx,t satisfies the two conditions described above. Then if n ~ 3 and if dnx,t now denotes Lebesgue measure on Bx,t we have
r_
sup I II O 0 and b < T since the canonical relation associated to At is Ct = {(x,e,y,TJ): (x,e) = Xt(Y,TJ)}, where Xt: T*X\0--+ T*X\0 is the canonical transformation obtained by flowing for time t along the I
Hamilton vector field associated to V'Egik(x)f.;f.k, if Egik(x)f.;f.k is the cometric. To prove Corollary 6.3.3 we notice that Theorem 1.2.1 implies that, for small t, we can write the operators in (6.3. 7) as the sum of two terms, each of which is of the form :Ftf(x)
={
}Rn
eicp(x,t,e) {1 + 1tf.12) -(n-1)/4 a(x, t, tf.) ](f.) df..
6.9. Spherical Maximal Theorems: Take 1 The symbol satisfies ID~ItD{a(x, t, e)l ::::; Ca-y(1 phase function must be of the form
+ len-lal,
191 while the
where 1/J(x, e) is one of the two phase functions occurring in the Fourier transform of surface measure on Bx 0· Since we can estimate the maximal operator corresponding to dilations t E [2-io, 1], it suffices to show that if :Ft is as above then 1
II
I:Ftf(x)IIILP(R") $
sup . 0 1, Ft(x) is bounded on LP with a fixed constant Cp,Re(z) if Re(z) < -1. Since Ft(x) = ~x) an application of the analytic interpolation lemma yields (6.3.711 ). This completes the proof of Corollary 6.3.3.
Notes
Notes For historical comments about the development of the theory of Fourier integral operators we refer the reader to Hormander [5]. We have consciously presented here only the bare minimum of material for use in the next chapter and Hormander's paper is also an excellent source for the interested noninitiated reader who wishes to go further. See also Treves [1] and Hormander [7] whose expositions we have also followed in the first section. The L 2 regularity theorem for Fourier integral operators goes back to Eskin [1] and Hormander [5]. Eskin proved a local version, while Hormander proved the global version we have stated. The LP and pointwise regularity theorem is more recent and is due to Seeger, Sogge, and Stein [1], although many partial results were known, including those of Beals [1], Littman [1], Miyachi [1], and Peral [1]. See also Sugimoto [1]. The proof of the regularity theorem used ideas from the study of Riesz means in Fefferman [3], Cordoba [1], and Christ and Sogge [1]. In some ways the basic idea behind the decomposition of the operators in the proof is also similar to that used in the analysis of the solution to the wave equation using plane waves (see John [1]). The spherical maximal,.theorem is due to Stein [3] and the variable coefficient versions of the spherical maximal theorem are due to Sogge and Stein [3]. See also Oberlin and Stein [1], where the mapping properties of the operator corresponding to cf>t(x, y) = t - (x, y} were studied. For background about the Kakeya set and discussions about applications of such maximal theorems we refer the reader to Falconer [1]. The role of rotational curvature in Fourier ahalysis was introduced by Phong and Stein [1] in their study of singular Radon transforms.
193
Chapter 7 Local Smoothing of Fourier Integral Operators
In this chapter we shall prove estimates for certain Fourier integral operators which send functions of n variables to functions of n + 1 variables. We shall deal with a special class which contains the solution operators for the Cauchy problem associated to variable coefficient wave equations. The estimates we obtain are better than those which follow trivially from the sharp regularity estimates for Fourier integral operators in Theorem 6.2.1. We call these estimates local smoothing estimates. In Section 3 we shall see that these local smoothing estimates can be used to improve many of the estimates for maximal operators in Section 6.3. In particular, we shall be able to prove the natural variable coefficient version of the circular maximal theorem proved in Section 2.4, which includes estimates for averages over geodesic circles. The argument will involve an adaptation of the arguments in Section 2.4, and, among other things, we shall require a variable coefficient Kakeya maximal theorem which contains estimates for a maximal operator involving averages over thin "geodesic rectangles." The reason for the terminology is because the phrase "local smoothing" was first used for certain types of estimates for dispersive equations which go back to Kato, Sjolin, and Vega. In the case of the solution to the Schrooinger equation, w(x,t) = (eit~ J)(x), the local smoothing estimate says that if f E L2(1Rn), then w(x, t) E L~; 2 ,loc(!Rn X IR). Note that this is a big improvement over the fixed-time estimate llw(' 1 t)11£2(Rn) = II/II£2(Rn)·
7.1. Local Smoothing in Two Dimensions On the other hand, if one considers the seemingly related hyperbolic operator I -+ u(x,t) = eit~ I, there can be no local smoothing in L 2 • This is because-in sharp contrast to the solution operator for Schrooinger's equation-the hyperbolic operator is of a local nature, by which we mean that the kernel K(x, t; y) and all of its derivatives are O(lx- Yi-N) for any N if lx- Yi > 2t. Because of this, and the fact that here we also have llu( · ,t)IIL2(Rn) IIIIIP(Rn)• one concludes that u(x, t) can, in general, only be in L?oc(IR.n x JR.) if I is in L 2 (JR.n). It is not also hard to see that if O:p = (n -1)11/p -1/21, then, for 1 < p < 2, one can in general only say that u(x, t) E £P-ap, 1oc (IR.n x JR.) if I E LP(IR.n), which is just a trivial consequence of Theorem 6.2.1. For 2 < p < oo, the situation is much different. For every p there is an e = e(p, n) such that u(x, t) E L~-ap,loc(IR.n x JR.) if I E LP(IR.n). Since we saw at the end of Section 6.2 that, in general, u( · , t) only belongs to L~ap,loc(JR.n) if I E LP(IR.n), we conclude that, when measured in
=
LP, p > 2, eit~ I has much better properties as a distribution in (x, t), rather than in x alone. In two dimensions this local smoothing estimate just follows from inequality (2.4.27); however, we shall see that this estimate holds for variable coefficient Laplacians in all dimensions. The class of Fourier integral operators we shall deal with satisfy the so-called cinematic curvature condition. This is just the natural homogeneous version of the Carleson-Sjolin condition for non-homogeneous oscillatory integral operators which was stated in Section 2.2.
7.1. Local Smoothing in Two Dimensions and Variable Coefficient Kakeya Maximal Theorems Before we can state the local smoothing estimates we need to go over the hypotheses. From now on Y and Z are to be C 00 paracompact manifolds of dimension nand n + 1, respectively. As usual, we assume that n ~ 2. We shall consider a class of Fourier integral operators 1 JP.-l/ 4 (Z, Y;C), which is determined by the properties of the canonical relation C. Notice that our assumptions imply that C c T* Z \0 xT*Y \0 is a conic submanifold of dimension 2n + 1. 1 To be consistent with the convention regarding the orders of Fourier integral operators given in Chapter 6, we prefer to state things in terms of order p. - 1/4 so that when the operators are written in terms of oscillatory integrals with n theta variables, such as when one uses a generating function, the symbols will have order p.. The possible confusion arises from the fact that Z and Y have different dimensions.
195
196
7. Local Smoothing of Fourier Integral Operators To guarantee nontrivial local regularity properties of operators
r
E
p~- 1 14 (Z, Y;C), we shall impose conditions on C which are based on the
properties of the following three projections:
c /
l
(7.1.1)
z
T*Y\0
The condition has two parts: first, a natural non-degeneracy condition which involves the first two projections, and, second, a condition involving the principal curvatures of the images of the projection of C onto the fibers of T* Z \ 0. To describe the first condition, let nT·Y and llz denote the first two projections in (7.1.1). We require that they both be submersions, that is,
rankdllT·Y rankdllz
=2n,
(7.1.2)
= n + 1.
(7.1.3)
Together, these make up the non-degenerjl-Cy requirement. As a side remark, let us point out that if Y and Jl were of the same dimension, then, as we saw in the last chapter, (7.1.2) would imply that C is locally a canonical graph. Also, in this case the differential in (7.1.3) would automatically be surjective; however, since we are assuming that the dimensions are different, (7.1.2) does not imply that llz is a submersion. In order to describe the curvature condition, l~t zo E llzC and let llT· z be the projection of C onto the fiber T.zo* Z \ 0. Then, clearly, •o
r .zo
(7.1.4)
= llT·•o z(C)
is always a conic subset of T;0 Z \ 0. In fact, r Zo is a smooth immersed hypersurface in T;0 Z \0. In order to see this, note that the first assumption, (7.1.2), implies that the differential of the projection of C onto the whole space T* Z \0 must have constant rank 2n+ 1. (See formulas (7.1.7) and (7.1.21) below.) Furthermore, since the differential of C -+ T* Z \0 splits into the differential in the Z direction and in the fiber direction, we see that in view of (7.1.3)
rank dllT·•o z
=n
and therefore, by the constant rank theorem, dimensional hypersurface.
(7.1.5)
r zo
is a smooth conic n-
197
7.1. Local Smoothing in Two Dimensions Now in addition to the non-degeneracy assumption we shall impose the following Cone condition: For every ( E do not vanish.
r zo, n -
1 principal curvatures (7.1.6)
Since r zo is conic this is the maximum number of curvatures which can be nonzero. Clearly (7.1.6) does not depend on the choice of local coordinates in Z since changes of variables in Z induce changes of variables in the cotangent bundle which are linear in the fibers. If (7.1.2), (7.1.3), and (7.1.6) are all met then we say that C satisfies the cinematic curvature condition. This condition is of course related to the Carleson-Sjolin condition since the non-degeneracy condition is just the homogeneous analogue of (2.2.2) and the cone condition is just the homogeneous replacement for the curvature condition (2.2.4). Let us now see how our assumptions can be reformulated in two useful ways, if we use local coordinates. First, we note that we can use the proof of Proposition 6.2.4 to see that (7.1.2) and (7.1.3) imply that, near a given (zo, (o, yo, 170) E C, local coordinates can be chosen so that C is given by a generating function. Specifically, there is a phase function rt~(z, 71) such that Cis parameterized by rt~(z, 71)- (y, 71}, that is, C can be written (locally) as { (z, rt~~(z, 71), rt~~(z, 17), 71) : 71 E IRn \0 in a conic neighborhood of 170}. (7.1.7) In this case, condition (7.1.2) becomes rank rt~z" 71 -= n,
(7.1.2')
which of course means that if we fix zo, then, as before,
r Zo
= { ft'~ ( zo, 71) : 71 E IRn \ 0 in a conic neighborhood of 170} C T;0 Z \ 0
must be a smooth conic submanifold of dimension n. So if
and (} E sn is normal to directions for which
r Zo
at
(I
it follows that
±(}
are the unique
(7.1.8)
198
7. Local Smoothing of Fourier Integml Opemtors The condition that n - 1 principal curvatures be nonzero at ( then is just that
a2
'
rank ( 817;a1Jk) (rpz(zo, 17), 8} = n- 1,
if 7J, 8 are as in (7.1.8). (7.1.6')
It is also convenient to give a formulation which is in the spirit of the wave equation. This involves a splitting of the z variables into "timelike" and "spacelike" directions. If, as above, we work locally, then (7.1.2') guarantees that we can choose coordinates z = (x, t) E IRn x lR vanishing at a given point zo such that, first of all, " = r ank 'Px'1
(7.1.9)
n,
and second rp~ ~ 0,
if 1J :f: 0.
In other words, ro must be of the form ro = {(rp~(0,7J),q(rp~(0,7J))}, for some q satisfying q(e) ~ 0 if e :f: 0. This is because, if (e, r) are the variables dual to (x, t), then ro does not intersect the T axis. The cone condition, (7.1.6), just translates here to the condition that rankqee = n- 1. Since r x,t must have the same form for small (x, t), we see that local coordinates can always be chosen so that C is of the form C = {(x, t,
e,
T,
y, 1]~.: (x, ~)
= Xt(Y,_1J), T = q(x, t, e) :f: 0},
(7.1.10)
where, by (7.1.9),
Xt
is a canonical transformation
(7.1.2")
and " = - n- 1. rank qee
(7.1.6")
We can now state the main result of this chapter. Theorem 7.1.1: Suppose that :F E [P.- 114 (Z, Y;C) where, as before, C satisfies the non-degeneracy condition (7.1.2), (7.1.3), and the cone condition (7.1.6). Then :F: L~omp(Y) -+ Lf0 c(Z) if JL $ -(n- 1}(1/21/p) + e and e < e(p), with
e(p) = {
,Jp.
4 $p
< oo, (7.1.11)
H! - ~),
2 < p $ 4.
Remark. This result is not sharp-at least in higher dimensions. However, since the result has the most interesting consequences in two
199
7.1. Local Smoothing in Two Dimensions dimensions we just state it this way for the sake of simplicity. A natural conjecture would be that for p ~ 2n/(n- 1) one should be able to take e(p) = 1/p in the theorem. If this result were true, one could use it to prove sharp estimates for Riesz means in R.n by estimating the operators in (2.4.5) using Minkowski's integral inequality and a scaling argument. Also, one can use the counterexample which was used to prove the sharpness of Theorem 6.2.1 to see that for 2 < p < 2n/(n-1) there cannot be local smoothing of all orders < 1/p, and in fact the best possible result would just follow from interpolating with the trivial £ 2 estimate if the conjecture for 2n/(n- 1) were true. Different arguments are needed to handle two dimensions versus higher dimensions. So we shall prove the two-dimensional case in this section and then turn to the case of higher dimensions in Section 2. Before turning to the proofs, though, let us state a corollary. If M is either a C 00 compact manifold of dimension n or R.n we consider the Cauchy problem
((8/&t) 2 - ag)u(x, t) = 0, {
(7.1.12}
ult=o= /, (aj&t)ult=O =g. Here a9 is a Laplace-Beltrami operator which is assumed to be the usual Laplacian in the Euclidean case. By the results at the end of Section 6.2, it follows that iff E L~(M) and g E £~_ 1 (M) then u( ·, t) E ~-orp (M) if O:p = (n-1)11/p-1/21. Furthermore, this result is sharp for all nonzero times in the Euclidean and all but a discrete set of times in the compact case. Since the canonical relation for the solution to the wave equation has the form (7.1.10) with q = J'"£gik(x)e;ek, the solution operator satisfies the cinematic curvature condition. Therefore, Theorem 7.1.1 gives local smoothing for the wave equation. Corollary 7.1.2: Let u be the solution to the Cauchy problem (7.1.12). Then, if I CR. is a compact interval and if e < e(p),
Similarly, eitP f belongs to L~-orp+c,loc(M x R.) iff E L~(M), provided that P E \II ~1 (M) is self-adjoint and elliptic and the cospher:es Ex = {e : p(x, e) = ±1} c T; M \ 0 all have non-vanishing Gaussian curvature.
200
7. Local Smoothing of Fourier Integral Operators Orthogonality arguments in two dimensions We now turn to the first step in the proof of the local smoothing estimates for n = 2. Since local coordinates can be chosen so that C is of the form (7.1.7), we can use the proof of Proposition 6.1.4 to conclude that we may assume that F is of the form
F l(x, t) = [
ei.(x, t, 17) f(f1) df1.
The square of the left side of (7.1.14) is dominated by
Lll lv1-v21:::::2k L :f'rl 10.
2tiiL2(R3) ·
k
Since we are assuming that a has small conic support the summation only involves indices k < log ..X 1/ 2 . We shall need to make a further decomposition based on k. To this end, we fix p E C8"((-1, 1}) satisfying p(u) = 1 for lui < 1/4 and LjEzp(u- j) 1. We then set
=
:F'{;{f(x, t) =
I
eirp(x,t,TJ)
a~·,{(x, t, 17) j(17) d17,
with (7.1.15) Note that this decomposition involves localizing T = rp~ to intervals of height ~ ..\2-k. These are larger than the intervals of height ..X 112 which were used in Section 2.4 to prove the constant coefficient local smoothing estimates. Since rp~ is not assumed to be constant in x and t, this different localization is needed for the integration by parts arguments which are to follow. Let ""'" denote the partial Fourier transform in the t variable. Then if I{ k is the interval of length 2-k ,xi+e and center j2-k ..X, the proof of Lem~a 2.4.4 shows that
:F'{•{ (f)(x, t) = 2_ f '
.
21T }TEl'.>.,k
where R~1 has an £ 4
-+
eiTt
(:F~·{ (!))"' (x, r) dr + R~·i(f)(x, t), '
'
L 4 norm which is O(..x-N). Thus, since :Ff,kf =
E;:::::2 k :F~1 (/), one can use Plancherel's theorem in the t variable, as in (2.4.28), to reach the desired conclusion II:F>..flli4(R3)
1 ~ c..xe'\:" L....J 2k 211("' L....J I ""' L....J k
it J2 lvt -v21:::::2k
1 2 :f":t.i1U):f":2,;2U)I2) ' 11 >.,k >.,k L2(R3)
202
7. Local Smoothing of Fourier Integml Opemtors On account of this, the desired inequality, (7.1.14), would follow if we could show that there is a fixed constant C, independent of ..X and k, for which
(7.1.16) This is the main inequality. Notice that, in going from (7.1.14) to (7.1.16), we have lost a factor of 2k/ 4 in the L 4 bounds. To prove (7.1.16), we shall need to exploit the cone condition. Recall that this condition says that, if 0 is a small conic neighborhood of supp71 a, then rx,t
= {(rp~(x,t,71),rp~(x,t,71)): 71 E 0}
is a C 00 conic hypersurface in R.3 \ 0 which has the property that at every point (e, r) E r x,t one principal curvature is nonzero. To use this, let us define subsets (7.1.17) which depend on the scale ..X and also on k. By the non-degeneracy hypothesis (7.1.9), r~:{ is contained in a sector around ( (rp~(x, t, 11r), rp~(x, t, TJr)) which has angle~ ..X - 1/ 2 . Here 11r are the unit vectors occurring in the defiThe sets r~:{ have height ~ ..\Tk. So r~:{ is basically a ..X1 / 2 X nition of
xr.
..XTk rectangle in the tangent plane tor~~ at (rp~(x, t, 11r), rp~(x, t, 11r)).
On account of this, it is a geometrical fact that, if we consider algebraic sums of the sets in (7.1.17), then if lv- v'l = dist ((vt, V2), (z1, v~)) is larger than a fixed constant, dist (rVt,jl x,t
+ rV2,j2 rV~,jl + rV~,h) x,t • x,t x,t
> c..\j(nVl .,>.
-
+ .,v2) _ "I).
(TJV~ ),.
+ TJV~), ),. • (7.1.18)
where if, as we are assuming, a has small conic support, the constant c > 0 depends only on rp. To see this, let us fix "heights" Tt, T2 ~ ..X so that the cross-sections f~~t1 = {( T) E r~i : T = Tf} are nonempty. Here we are assuming, as we may, that 0 < rp~. It is not hard to check that the sets r~·{ have been chosen to have the largest height so that { ( T) E ' r~i 1 + r~~ih : T = Tl + 1'2} is contained in a tubular neighborhood of
e,
e,
7.1. Local Smoothing in Two Dimensions
203
width 0(2k) around f~i 1 + f~~th. Using the curvature of r z 1t one can prove that if lv- v'l is larger than a fixed constant dist (fvtJt + fv21i2 fv~Jt + fv~J2) ~ ~~(TJv1 + TJv2) _ (TJv~ + TJv~)~
X1t
X1t ' x 1t
x 1t
~
>.
>.
>.
which, by the previous observations, leads to (7.1.18), since
~I(TJ~1
>.
'
+TJ~2 )
(TJ~~ + TJ~~)i ~ 2klv- v'l. Next, we claim that (7.1.18) leads to the bounds
1/
:F':l .it (!):F':21j2 (!)F.'; ljl (!).r!'2J2 (!) dxdtl >.lk
>.lk
>.lk
>.lk
(7.1.19)
One sees this by first noticing that the left side equals the absolute value of
J
eici> a~ttt (x, t, TJ)a~2t2 (x, t,{) I
I
with ~ = rt~(x, t, TJ)
+ rt~(x, t, {) -
rt~(x, t, TJ') - rt~(x, t, {').
To estimate this term we need the following lower bounds for the gradient of the phase function on the support of the integrand:
1Vx 1 t~l ~ c~I(TJ~1 + T/~2 )- (TJ~~ + TJ~~)I ~ ~(~ -l/21v- v'l· max ITJ~; - TJ~~ 1),
lv- v'l ~CO (7.1.20)
IVx~l ~ cd(TJ + {)- (TJ1 +{')I- c2~(max ITJ~; - TJ~~ 1) 2. (7.1.21)
Here c; are fixed positive constants. The first inequality in the first lower bound is equivalent to (7.1.18), while the second inequality there follows from the fact that the unit vectors {TJn are separated by angle ~ ~-l/ 2 . To prove (7.1.21) one notices that, by homogeneity, rt~~(x, t, TJ) = rt~~11 (x, t, TJ) · TJ· Hence we can write Vx~ = rt~~71 (x,t,TJ~1 )((TJ+{)- (TJ1 +e'))
+ (rt~~11 (x, t, TJ)- rt~~11 (x, t, TJ~1 ))TJ + ·· ·- (rt~~(x,t,{')- rt~~11 (x,t,TJ~1 }){'.
204
7. Local Smoothing of Fourier Integral Operators The first term has absolute value ~ ell (17 + e) - (77' + e') I because det.-lf2r- m)g(e, r).
Using the definition (7.1.15) and the proof of Lemma 2.4.4 one sees that, if j is fixed, there must be an integer m(j) such that pm ~1 has I
LP(JR2 )-+ LP(JR3 ) norm 0(>.-N) if lm- m(j)l ~ >. 112 Tk>.c. To prove this claim, one merely uses the fact that the symbol in (7.1.15) vanishes if cp~ does not belong to an interval of length 2-k >. which depends only on j. Using this claim, one repeats the arguments which lead to (7.1.16) to verify that
(L IF~:iU>I 2 ) 112 IIL'(R3) ~ c>.c(Tk>.l/2) 114 11( 2: IPm~:iU)I 2 ) 112 IIL'(R3) + c IIIIIL (R II
v,j
4
2 )·
v,j,m
Combining this with the last inequality means that we would be done if we could prove that
II(
2: IPm~:iU)I 2 ) 112 IIL'(R3) :5 c>.c IIIIIL (R 4
(7.1.24)
2 )·
v,j,m
1·
To prove this, we recall that suppp C ( -1, 1) and p(u) = 1 for lui < So, if we define a~1m = a~·,{;(x, t, TJ)p((>.- 1 12 cp~(x, t, 17)-m) /8) and then set
then
_ pm:F':,j,mll ll pm:P:.i >.,k >.,k LP-+LP = 0(>.-N). Consequently, (7.1.24) must follow from
II(
2: 1Pm~:i·m(J)I 2 ) 112 IIL'(R3) ~ c>.c IIIIIL (R 4
(7.1.24')
2 )·
v,j,m
To prove this, a couple of observations are in order. We first notice that if m is fixed then there are 0(1) indices j for which ~·i·m =F 0. I
Also, the symbols, a~·~m(x, t, TJ), of these operators vanish if cp~ [>.1/2m- 8>.1/2, >.112m'+ 8>. 112]. In addition, if we let Q(x, t, v, m) = {TJ: x~(TJ)p((>. -l/ 2 cp~(x, t, 17)- m)/8) =F
0},
fl.
206
7. Local Smoothing of Fourier Integral Operators then this set is comparable to a cube of side-length >.. 1/ 2 • In particular, each one intersects at most Co cubes in a given >.. 1/ 2 lattice of cubes in R2 . Since It'~ is smooth, Co can be chosen to be independent of the relatively compact set K = {(x, t) : a(x, t, 71) # 0}. With this in mind, we set
Thus I = E I w In addition, if, for a given (x, t, v, m), we let I(x,t,v,m) c Z 2 denote those JL for which .r~:t·ml~£(x,t) # 0, it follows from the properties of the Q(x, t, v, m) that Card(I(x,t,v,m)) ~C1,
(7.1.25)
where C1 is an absolute constant. Similarly, for fixed v and (x, t) E K there must be a uniform constant c2 such that (7.1.26) Finally, we let .1(v) d..enote those JL for which .B(I711/>..)x~(71) x p(>.-l/211l_JLI)p(>.- 1 1~-JL2) does not vanish identically and note that JL belongs to only finitely many of the sets .1(v) since supp,B(I711/>..)x~(71) is contained in the set {71: 1711 ~ >.., and 171~ -71/17111 ~ c>..- 112 }. We now turn to (7.1.241). Let K~·fm(x, t; y) be the kernel of I
pm.rrfm. We shall see that we .have the uniform bounds (7.1.27) Thus, since, for fixed m, .rr:t·m vanishes for all but finitely many j, the Schwarz inequality, (7.1.25), and (7.1.26) yield LIPm.rr:fml(x,t)1 2 mj
I
~ c:E I m,j
I~£(Y)I 2 1K~:fm(x,t;y)ldy
2:
~£EI(x,t,v,m)
~£E.7(v)
~ c'f2:
2:
II~£(Y)I 2 1K~:fm(x, t; y)l dy
m,j ~£EI(x,t,v,m) ~£E.7(v)
~ C"
I
L ~£E.7(v)
II~£(Y)I 2 su~ IK~;{•m(x, t; y)l dy. m,J
7.1. Local Smoothing in Two Dimensions
207
From this we see that for a given g(x, t)
1/ ~
1Pm-0i'mf(x,t)l 2 g(x,t)dxdtl
v,J,m
5, C
JL
lf14 (y)l 2 sup{ v
14
f
sup 1Kr1·m(x, t; y)llg(x, t)l dxdt} dy.
JR3 j,m
'
(7.1.28)
We have already seen in Lemma 2.4.6 that IICE 14 lf14 12)112114 5, Cllfll4· Therefore, since the left side of (7.1.24') is dominated by the supremum over all llgll2 = 1 of the left side of (7.1.28), we would be done if we could prove the maximal inequality
(f
supl
JR2 v
f
supiKr·fm(x,t;y)lg(x,t)dxdtl 2 dy) 112
}Ra j,m
'
5, CllogAI 312 IIgiiL2(R3)·
(7.1.29)
To prove the missing inequalities (7.1.27) and (7.1.29) we need to introduce some notation. If N C JR.3 x IR.2 \ 0 is a small conic neighborhood of suppa~, we define the smooth curves 'Yy,17
= {(x, t) : cp~(x, t, 17) = y, (x, t, 17) EN}.
(7.1.30)
Then one estimate we need is the following. Lemma 7.1.3: If 11r are the unit vectors occurring in the definition of
:Ff, then, given any N, (7.1.31)
Proof: Since the kernel of pm is 6o(x-y)Al/2p(Alf2(s-t))eim~- 112 (t-s), it suffices to show that the kernel .Kr·fm(x, t; y) of .rr·fm satisfies the estimates in (7.1.31). But ' '
.Kr·fm(x, t; y) = f '
~2
ei(.p(x,t,TJ)-(y,TJ)) a~·tm(x, t, 17) d11. '
After using a finite partition of unity, we may assume that the (x, t)support of the symbol is small. Since we are assuming that cp~ f; 0 and hence cp~17 f; 0, it follows from the implicit function theorem that 'Yy,17A is of the form (x(t), t) near K = SUPPx,ta if this set is small enough. Also, we are assuming that det cp~17 f; 0, so if K is small enough there must be a c > 0 such that
208
7. Local Smoothing of Fourier Integral Operators But V 11 rp is homogeneous of degree zero and therefore
IV11 rp(x,t,f1)- V11 rp(x,t,11nl ~ c>.- 112 ,
11 E suppx~·
Combining these two bounds shows that there is a d
IV11 [rp(x, t, 17) -
> 0 such that
I
(y, 17}] ~ c' dist ((x, t)!Yy, 11;:),
provided that dist ((x,t),'Yy, 11;:) is larger than a fixed multiple of >.- 1/ 2. Since O::a~1m(x, t, 17) = 0(>.-lol/2) and since, for fixed (x, t), this symbol vanishes for 11 outside a set of measure 0(>.), the desired estimate I for K~·fm follows from integration by parts. ' It is clear that (7.1.31) implies the uniform L 1 estimates (7.1.27) for the kernels. Using (7.1.31) again, we shall see that the maximal estimates (7.1.29) follow from the Kakeya maximal estimates in the next subsection. Variable coefficient Kakeya maximal functions For later use in proving the higher-dimensional local smoothing estimates, and since the result is of independent interest, we shall state a maximal theorem which is more general than the one needed to prove (7.1.29). We now assume that Z and Y are as in Theorem 7.1.1, with the dimension of Y being n ~ 2 and dim Z = n + 1. To state the hypotheses in an invariant way, let C satisfy the non-degeneracy conditions (7.1.2) and (7.1.3). It then follows that
'Yy, 11
= {z E Z: (z,(,y,17) E C,
some(},
is a C 00 immersed curve in Z which depends smoothly on the parameters (y, 17) E llT·Y(C). Let us fix a smooth metric on Z and define the "6-tube"
Rt,
11 =
{z: dist (z,-yy, 11 ) < 6}.
The variable coefficient maximal theorem then is the following. Theorem 7.1.4: Let C satisfy the non-degeneracy conditions (7.1.2) and (7.1.3) as well as the cone condition (7.1.6). Then, i/0 < 6 < ~ and a E C~(Y x Z) ( {
sup
}y TJEIIT;Y(C)
Iy, 1(~6 0
y,T]
) {
6
lR,,., 5:
a(y, z) g(z) dzl2 dy) 1/2 n-2
3
cr-2 llog6l2llgiiL2(Z)·
(7.1.32)
7.1. Local Smoothing in Two Dimensions
209
Since the canonical relation associated to the operator in the proof of the two-dimensional version of Theorem 7.1.1 is given by (7.1.7) (with z = (x,t)), it is clear from (7.1.31) that (7.1.32) implies (7.1.29) by taking 6 = >.- 112 ,2>.- 112 , .... Before turning to the proof, let us state one more consequence. If (Y, g) is a compact Riemannian manifold, then for a given y E Y and 8 E TyY, let 'Yy,e(t) be the geodesic starting at yin the direction 8 which is parameterized by arclength. Then, if we fix 0 < T < oo and let
Rt,
11 =
{(x,t): dist (x,'Yy,B(t))
< 6, 0 ~ t ~ T},
one can use the Legendre transform, mapping TY -+ T*Y, (see, e.g., Sternberg [1]) to see that a special case of Theorem 7.1.4 is: sup 8 Vol
(~
y,B
)1
lg(z)idxdt
£2(Y)
R!. 9
~C6_n; 2 llog6l~llgiiL2(YxR)·
Proof of Theorem 7 .1.4: For the sake of clarity, let us first present the arguments in the case where n = 2 and then explain the modifications that are need to handle the higher-dimensional case. We write z = (x, t) in what fOllows. We may work locally and assume that 'Yy, 11 is of the form (7.1.30) where of course we are now assuming that N C JR3 x JR.2 \ 0. We may assume that (0,0,0; 1,0) E N. Also, by (7.1.2') and (7.1.61 ), we may assume for the sake of convenience that coordinates have been chosen so that 2
and
a8 2 1Pz3, =F 0 '12
at z = 0, 1J = (1, 0).
(7.1.33)
To proceed, we fix a E C~(JR.2 ) satisfying 0. ~ 0. Then for o:(z, 8) E C~ supported in a small neighborhood of (0, 0) we put
o:0 (z, 8; TJ) = o:(z, 8) a(617) and define Aog(y, 8) =
11 ei((1 A~·k. Since e is bounded, there are O(log A) terms and hence (7.1.36') would follow from the uniform estimates
lls~p IA~·kg(y, 9)ljjL2(R2) ~ C II911£2(R3)•
k
= 0, 1, 2,... .
(7.1.3611 )
We can now invoke (7.1.37). By applying it and Schwarz's inequality we see that (7.1.3611 ) would be a consequence of the estimates
(//1 (!)j A~·kg(y,9)12
dOdy) 1/2
~ C(A-1/42-k/2)1-2j II9II£2(R3)• j
= 0, 1. (7.1.42)
However, (7.1.40) implies that, on the supports of the symbols (8/89)(cp~(z,cos9,sin9),17) = 0(A 1122k) and hence (8/89)A~,k behaves like A1/22k A~·k, so we shall only prove the estimate for j = 0. It turns out to be easier to prove the estimate for the adjoint operator because this allows us to use the Fourier transform. Specifically, if " - " denotes the partial Fourier transform with respect toy, then the desired estimate is equivalent to
II/Jei(cp~(z,cos9,sin9),1J) /3(
2-k A1/2e)
x f3(1171/A)ao(z,9;17)i(17,9)d17d911
< CA- 1/ 4 2-k/2 11/11£2·
L 2 (dz) -
However, if we make the change of variables 17-+ A17, 8 would be a consequence of the following.
-+
e, this in turn
I
Lemma 7.1.5: Let bk(z; 17, e) satisfy l~bkl ~ C0 Va, as well as bk = 0 ifeitherlzl > 1,1171 fl. [1,2], ore fl. [2k- 1A- 1/2,.2kA- 1/2] when k = 1,2, ... ,[21l'logA 112] and e fl. [O,A- 112] when k = 0. Then, if we
212
7. Local Smoothing of Fourier Integral Operators also assume that bk vanishes in a neighborhood of the 112 axis (so that
(>(z;71,8) is C 00 on suppbk}, it follows that T>.,k f(z) =
f
JR3
ei>.4>(z;7J,e) bk(z; 71, 8) f(71, 8) d71d8
satisfies
(7.1.43) On the other hand, if r( z; 71, 8) = 0 for 1711 fl. [1, 2] and 71 outside a small neighborhood of the 112 axis, or for 8 outside a small neighborhood of the origin, then
n>< f(z)
= {
JR3
ei>.4>(z;7J,B) r(z; 71. 8) f(11. 8) d77d8,
satisfies IIR>. fiiL2(R3) $ CA -a/ 2 11fiiL2(R3)·
Proof: Let us first prove the estimates for the operators T>.,k. We use a modification of the argument used in the proof of the Carleson-Sjolin theorem (Theorem 2.2.1, (2)). We first note that, after perhaps using a smooth partition of unity, we may assume that bk has small support. Then the square of tl& left side of (7.1.43) equals #
f f
JR3 JR3
H> is a C 1 function of 8 2 . But (7.1.39) implies that, in the coordinates where 8 ~ 0 is replaced by 8 2 , the Hessian of 4> is non-singular. This and the fact that (> is C 1 in (71. 8 2 ) implies that" there must be a c > 0 such that
IVz [4>(z; 71. 8)- 4>(z; 77'. 8')] I ~ cl (77- 77 ,82 1
(8') 2 ) I
on the support of the symbol if bk has small enough support. Consequently, since both 4> and the bk are uniformly smooth in the z variables, a partial integration gives the bounds IH>.,kl $ CN(1 + Al71- 77'1)-N (1
+ Al82 -
(8') 2 1)-N,
for any N. But if k =F 0, 182 - (8') 2 1~ A- 112 2kl8- 8'1 on the support of the kernel and hence the L 1 norm with respect to either of the pairs of variables is O(A- 2 2-kA- 112 ). One reaches the same conclusion for
213
7.1. Local Smoothing in Two Dimensions k = 0 if one recalls that in this case the kernel vanishes for 181 > >. - 1/ 2. Using these estimates, one concludes that (7.1.44) must be :5 c>.- 2 2-k>.- 1 1 2 11/11~. which finishes the proof of {7.1.43). If werecall that our assumptions imply that deta2 ~(z;TJ,8)/8z8(TJ,8) "I 0 on supp r, modifications of this argument give the estimate for R>.. I The proof of the higher-dimensional maximal estimates follows the same lines. First, if w(8) are local coordinates near w(O) = (1, 0, ... , 0) E sn- 1, then one sets
A6 g(y, 8) =
f ei((rp~(z,w(B)),11)-(y,17)] o:0 (z, 8; TJ) g(z) dTJdz,
f
}Rn+l }Rn
where now o:0 (z, 8; TJ) = o:(z, 8)a(6TJ) with o: E ego supported near the origin and a E Cif(!Rn) satisfying a~ 0. The maximal inequality would then follow from IAog(y, 8)111 :5 c6-(n-2)/2llog 6I3/2119IIL2(Rn+l)· II sup 8 L2(Rn) This time we shall want to use the following higher-dimensional version of (7.1.37}: sup IF(8)1 2 BERn-1
:5 Cn-1
L lal+l/3l$n-1
(I
188 Fl 2
d(J) 112 (I~~ Fl 2 d(J) 112 , (7.1.371 )
if F(O) = 0 when 8; = 0 for some 1 :5 j :5 n - 1. To apply it, as before, we must break up the operators. First, if A~ are the dyadic operators, it suffices to show that the associated maximal operators send L2(JRn+l) -+ L2(1Rn) with norm O(>.(n- 2)/ 2 log >.). To see this, we note as before that (7.1.2') and (7.1.6') imply that ~(z;TJ,8) = (.)xr(TJ) o:0 (z, 8; TJ) g(z) dTJdz,
8) 12 d8dy) 1/2 :5 c>.lal-1/2119112·
This follows from the argument used to estimate the remainder terms in Lemma 7.1.5. By (7.1.37') the maximal operator associated to R~ has
214
7. Local Smoothing of Fourier Integml Opemtors £ 2 bound O(.~(n- 2 )1 2 ). So, if we let and it suffices to show that
Rg,
.Ag
be the difference between
lls~p IAgg(y, 9)1112 $ C>. (n-2)/2
log >.llgll2·
Ag
(7.1.45)
If we let f!(9, TJ) =min± dist (w(9), ±TJ/ITJI), then, by the above discussion, we may assume that f!(9, TJ) is small on the support of the symbol, ii0 , of .Ag, and, hence, 18(Jf!l $ Calfll 1-lal there. With this in mind, we then set for k = 1, 2, ... -.>.k
A 6 • g(y,9)
=II
ei'P(z;1J,9)
{3(2-k >.1/2n(9, TJ)) fJ(ITJI/ >..) iio(z, 9; TJ) g(z) d1Jdz,
-.>.o -.>. -.>. k and A6 • = A6 - Lk> 1 A6 • . Then arguing as before shows that (7.1.45) follows from the estimates
(lllfYo A~·kg(y,9)i2
d9dy) 1/2:::;; C(>..-1/4 2-k/2)1-2laiiiYII2·
-.>. •k ~ (>. 1 /2 2k ) 1a 1A -.>. •k , the arguments for o = 0 give the Since 8(j A 6 6 bounds for general o. To prove this, just like before, one ~timates the adjoint operator in order to use the Fourier transform. Since we are assuming that cp satisfies (7.1.2') and (7.1.61), the arguments that were used to handle the two-dimensional case can easily be modified to show that this operator satisfies the desired estimates. I
7.2. Local Smoothing in Higher Dimensions In this section we shall prove the local smoothing estimates in Theorem 7.1.1 corresponding to n ~ 3. The orthogonality arguments are much simpler here because we can use a variable coefficient version of Strichartz's £ 2 restriction theorem for the light cone in JRn+I. Applying this £ 2 -+ Lq local smoothing theorem, we use a variation of the arguments in Section 5.2 that showed how, in the favorable range of exponents for the lJl -+ £ 2 spectral projection theorem, one could deduce sharp lJl -+ lJl estimates for Riesz means from lJl -+ £ 2 estimates for the spectral projection operators. To prove the higher-dimensional lJl -+ lJl local smoothing theorem, in addition to the the Kakeya maximal estimates just proved, we shall need the following sharp £ 2 -+ Lq local smoothing estimates whose straightforward proof will be given at the end of this section.
7. 2. Local Smoothing in Higher Dimensions
21
Theorem 7.2.1: Suppose that :FE JP.-l/ 4(Z, Y;C) and that C satisfies the non-degeneracy assumptions (7.1.2)- (7.1.3) and the cone condition (7.1.6). Then :F: L~omp(Y)-+ Li!,c(Z) if2(n+ 1)/(n-1) $ q < oo and JL ~ -n(1/2- 1/q) + 1/q. Remark. Using the Sobolev embedding theorem and the £ 2 boundedness of Fourier integral operators it is not hard to see that if one just assumes (7.1.2) and (7.1.3) then :F : L~omp(Y) -+ Li!,c(Z) if JL $ -n(1/2 - 1/q) and 2 ~ q < oo. Thus, Theorem 7.2.1 says that, under cinematic curvature, there is local smoothing of order 1/q for q as in the theorem. Using the counterexample that was used to prove the sharpness of Theorem 6.2.1, one can see that the above £ 2 -+ Lq local smoothing theorem is sharp. In the model case, where C is the canonical relation for the solution to the wave equation in IR.n, it is equivalent to Strichartz's restriction theorem
The reason for this is that, by duality and Plancherel's theorem, the last inequality is equivalent to
f ei((x,{)+tlell j(e)__!!{_ I II }Rn leil/2 L•(Rn+i) $ CII/IIL2(Rn)•
q = P1 = 2 ,/(z) =
f
}Rn
ei,(~. 71) j(71) d71,
a>,(z, 71) =
/3(1'71/ A)a(z, 71).
If we assume that cp satisfies (7.1.21) and (7.1.6'), then, by summing a geometric series, it suffices to show that for e > 0
(7.2.1)
216
7. Local Smoothing of Fourier Integral Operators To prove this we need to make an angular decomposition of these operators. So we let {xn. v = 1, ... , N(A) ~ A(n- 1)/ 2 , be the homogeneous partition of unity that was used in the proof of Theorem 6.2.1. We then put
:Frf(z)
=I
eicp(z,TJ) x~(17)a>.(z,17)/(17)d17.
We now make one further decomposition so that the resulting operators will have symbols that have rrsupports that are comparable to A112 cubes. To this end, if p E C~(( -1, 1}) are the functions which were used in the two-dimensional orthogonality arguments, we set
F;.•j /(z)
=I eicp(z,TJ)a~,j(z,
17) /(17) d17,
with
a~,j (z, 17) =X~ (17)P(A - 1/ 2 1171 - j) a>.(z, 17).
A couple of important observations are in order. First, if nv,j ~>.
= suppTJ a>.v,j( z, 17) ,
(7.. 2 2)
then the Q~,j are all comparable to cubes of side-length A1/ 2 which are contained in the annulus {17 : 1171 ~ A}. This, along with the fact that the symbols satisfy the natural estimates associated to this support property,
!a]~a~•j(z, 11 )!::; Ca,"f(1 + 117 1)-lal/2 ~ A-lal/2,
(7.2.3)
makes the decomposed operators much easier to handle. The square function that will be used in the proof of (7 .2.1) will involve operators which are related to the F;..,j operators. In fact, the main step in the proof of the orthogonality argument is to establish the following result which, for reasons of exposition, is stated in more generality than what is needed here. Proposition 7.2.2: Fix c > 0. Then, given any N, there are finite M(N) and CN such that whenever Q C JRn+l is a cube of side-length A-1/2-e and 2(n + 1)/(n- 1) $ q < oo II:F>.IIIL•(Q) $ CNIQI-! A-~+n(~-!>-!
L lli~M(N)
II(L IF;..;~/1 2 ) 112 IIL•(Q) v,j
+ CNA-NII/IIL•(R") · (7.2.4)
21
7.2. Local Smoothing in Higher Dimensions Here
-0;~f(z)
=I
ei.,i z,17 =
;I
'J
d Qv,j 17 'F .>. ,
l~a~',{(z, 17)1 :5 CaA -lal/ 2 'Vo..
(7.2.5)
If the phase function cp is fixed, the constants in (7.2.4) and (7.2.5) depend on only finitely many of those in (7.2.3). The first operator -0;~ will just be an oscillatory factor times -0';, while, for i ~ 1, the operators in (7.2.4) will involve derivatives of the symbol and the phase function. '1\uning to the proof, it is clear that (7.2.4) would be a consequence of the following uniform upperbounds valid for z E Q:
IIF.>.fiiL•(Q) :5
eN( A-~ An(!-~)-~ L (L 1-0;~f(z)l 2 ) 112 +A-NIIfiiL•(R")). lii~M(N) v,j
1
(7.2.4') We may assume that 0 E Q and we shall prove the estimate for z = 0. To proceed, given v, j for which Q~·j is nonempty, we choose 17~,j E Q~·j and set
c~·j (z)
=I
ei[.f(z) = Ev,j eicp(z,f1~';)c~'j (z). Therefore, if we set
~;~f(z) =
A-lll/2 ~I ei[cp(z,f1)-cp(z,f1~';)] a~,j (z, 17) /(77) d7J,
then since lziiAill/ 2 :5 1, the left side of (7.2.4') is dominated by CA-NIIfllq plus
L
IlL eicp(z,f1~';) ~~f(O)IILq
lli~M v,j
Since 117 - 17~,; I :5 C A112 for 17
· (Q)
E
supp17 a~,j it is clear that the symbol
of~;~ satisfies (7.2.5). Therefore, by the last majorization, we would be done if we had the following discrete version of the £ 2 -+ Lq local smoothing theorem.
Lemma 7 .2.3: Let Q and 17~,; be as above. Then if Q is contained in a small relatively compact neighborhood of supp17 a>. (so that cp is well defined}
ll~eicp(z,f1~·;>cv,;IILq(Q) :5 CA-~ An-~(~ 1~,;12f/2. V 1J
V 1J
1
Proof: As before we""assume 0 E Q. If we then normalize the phase function by setting
t/>(z, 17) = cp(z, 17) - cp(O, 17), then the desired estimate is equivalent to the following: s 1 1 1 " ei~(z,f1~';)~,jll < CA-~ An(2"-q>-q lcv,j12)1/2. (7.2.7) II ~ ~~.
(L
V 1J
V 1J
To see that this is a corollary of the £ 2 -+ Lq local smoothing theorem, we let Q~,j be the cube of side-length A1/ 2 in IR.n that is centered at 17~,;. If A is sufficiently large and if we let
bv,j(z,7J) = { { . ei[~(z,f1)-~(Z,f1~';)] d7J}-1. X v,;(7J), >. JQ~·' Q,. it follows that, for z E Q,
lb~,j (z, 17)1 :5 CIQ~,j 1- 1 = CA -n/2 • To see this, note that t/>(z, 17) - t/>(z, 17~,;) = 0 when 17 = 17~,; and V 17 ¢(z,7J) = O(A- 112-e) for z in Q. Hence, lt/>(z,7J)- t/>(z,7J~,j)l < ! for large A, giving the estimate. Similar considerations show that
~~b~'i(z,7J)I :5 CaA-n/ 2 Alal/ 2,
z E Q.
(7.2.8)
21
7.2. Local Smoothing in Higher Dimensions These estimates are relevant since the quantity inside the Lq norm in (7.2.7) equals
J
eicf>(z,TJ)
2; b~,j (z, 17) ~,j d17. V,]
So, if we write
.
b~'1 (z, 17)
a b~J.(u1, 0, ... , 0, 17) du1 = b~·1.(0, 17) + loz' -a 0
+ .. ·+
U1
rl ... rn+l _!!_.,,_a_b~·j(u,17)du,
lo
lo
aul
aun+l
then it follows that the left side of (7.2.7) is dominated by
+
I ... I
lukl~.>.-l-•
II/ eicf>(z,TJ) ""_!!__ ~ au1
... _a_bv,j (u, 17)cv,j d7711 8un+l .>.
Lq(Q)
du.
v,J
Since b~,j vanishes unless 1171 ~A, Theorem 7.2.1 and Plancherel's theorem imply that this is majorized by
+An(i-~)-~
/···/
lukl9-l-•
ll"'_!!_ ... aun+l ~ au1
_a_bv,j(u,17)cvJII .>.
£2(dTJ)
du.
v,J
However, using (7.2.8) and the fact that the sets Q~·j have finite overlap shows that this in turn is
~cAn.¢ [C01, Co].
(7.2.11)
To apply this, we notice that, if :F is the operator in (7.2.10), then, by Littlewood-Paley theory and the fact that q > 2, we have 00
II:F/llq $ Cqll (~= j=O
00
IL;:F/1 2) 112 llq $ Cq (~= IIL;:F/II~) 112 . j=O
Note that :F = L::F2k· So, if/; is defined by }j(17) = /3(117lf2i)f(17), we
222
7. Local Smoothing of Fourier Integral Operators can use (7.2.11) to see that, if (7.2.101) were true, then the last term would be majorized by
(~= ll/i11~) 112
00
+L
j
2-Nj
11/112 :::; Cll/112·
j=O
Since we have verified the claim we are left with proving (7.2.10'). We shall actually prove the dual version (7.2.10")
However, since
I IF~g(y)
12
dy =
I F>.F~g(x,
t) g(x, t) dxdt
:::; IIF>.J19IILP (R"+l) IIYIILP(Rn+l), 1
this in turn would follow from
IIF>..1"~giiLP 1 (Rn+l) :5 CllgiiLP(R"+l)
l
1
:5 P :5 2 ~:l)
•
(7.2.12)
Recall that .1">. is of order JLq-1/4 with JLq = -nl1/q-1/21+1/q. Thus, since the canonical relation of .r is given by (7.1.10), the composition theorem implies that F>.J-1 is a Fourier integral operator of order 2JLq1/2 with canonical relation
C 0 C* = {(x, t, ~~ T, y, S, 1J, a) : (x, ~) = Xt o x_;- 1 (y, 1J), T = q(x, t, ~),a = q(y, s, 17) }.
(7.2.13)
So, if we assume, as we may, that a(x, t, 17) is supported in a small conic set, it follows that the kernel .1">..1"). is of the form {
ei[¢(x,t,s,7J)-{y,7J)]
b>,(X, t, S, 1J) d7J,
'R"
modulo C 00 , where b>. E S2~-'Q vanishes unless 1171 ~ >. and t - s is small. Consequently, if we let O:>.(x,t,s,7J) = >.- 21J.qb>.(x,t,s,7J) E SO, then we see that (7.2.12) would be a consequence of the following estimates. I Lemma 7.2.4: Let l/J be as above and suppose that o: E SO vanishes unless and t - s is small. Then, if we define the dyadic operators g>.g(x, t)
=
III
ei[¢(x,t,s,7J)-{y,7J)]p(I7JI/ >.) o:(x, t,
s, 17) g(y, s) d7Jdyds,
7.2. Local Smoothing in Higher Dimensions
22
it follows that IIQ>.giiLP'(R"+l):::;
C>.
1 1 2 2n( .. - - ) - ~ p' p' llgiiLP(R"+l)•
1:::; p:::;
2(n+ll) --fttr.
(7.2.14)
Furthermore, the constants remain bounded if o as above belongs to a bounded subset of SO. Proof: Since the symbol of 9>. vanishes unless 1171 be O(.~n). Hence
~
>., the kernel must
ll9>.giiL""(R"+l) $ C).n llgiiLl(R"+l)·
So, if we apply theM. Riesz interpolation theorem, we find that (7.2.14) would follow from the other endpoint estimate:
ll9>.giiLP (Rn+l) $ C>.llgiiLP(R"+l)• 1
P = 2 ~:31 ) •
(7.2.141 )
To prove this we need to use (7.2.13) to read off the properties of the phase function tf>. First, since Xt o x;- 1 = Identity when t = s, it follows that (x, y, 17) -+ tf>(x, t, t, 17)- (y, 17} must parameterize the trivial Lagrangian. In other words,
t/>(x, t, s, 17) = (x, 17}
if t = s.
Also, since T = q(x, t, e) in (7.2.13), it follows that, when t q(x, t, 17). So we conclude further that
t/>(x, t, s, 17)
= s,
= (x, 17} + (t- s)q(x, s, 17) + (t- s) 2 r(x, t, y, s, 17).
4>~
=
(7.2.15)
Using this we can estimate 9>.. More precisely, we claim that if we set
Tt,sf(x)
=II
ei[4>(x,t,s,7J)-(y,7J)].B(I771/>.)o(x, t, s, 17)f(y) d17dy,
and if o is as above, then !ttl
n-1
I!Tt,sfiiL""(R") :5 C>. ---r It- si--r llfiiLl(R")·
(7.2.16)
This estimate is relevant because we also have 11Tt,sfiiL2(R")
:5 C llfiiP(R")•
since the zero order Fourier integral operators Tt,s have canonical relations which are canonical graphs. By interpolating between these two estimates we get
224
7. Local Smoothing of Fourier Integral Operators And since 1 - (1/p- 1/p') = (n - 1)/(n + 1) for this value of p, we get (7.2.141 ) by applying the Hardy-Littlewood-Sobolev inequality-or, more precisely, Proposition 0.3.6. The proof of (7.2.16) just uses stationary phase. Assuming that t-s > 0, we can make a change of variables and write the kernel of Tt,s as (t _ s)-n
·~
! e'[{ t-s ,,.,)+q(x,s,,.,)+(t-s)r(x,t,y,s,,.,)) x
.8(1771/A(t- s)) a(x, t, s, 11/(t- s)) d17.
(7.2.17)
=
To estimate the integral we first recall that, by assumption, rank q:;,., n - 1. So, if N is a small conic neighborhood of supp71 o and if t is close to s, the sets Sx,t,y,s = { 17 EN: q(x, s, 17)
+ (t- s )r(x, t, y, s, 17) = ±1}
have non-vanishing Gaussian curvature. In addition, since q =F 0 for 17 =F 0, it follows that q~ =F 0 there as well. Consequently, for t close to s x-y
V,.,[( t _ S ,7J) + q(x, S,1J) + (t- s)r(x,t,y,s,7J)] =j; 0, unless
lx- Yl ~ It- sl.
If we put these two facts together and use the polar coordinates associated to Sx,t,y,s, we conclude that Theorem 1.2.1 implies that (7.2.17) must be
Since this of course implies (7.2.16), we are done.
I
7.3. Spherical Maximal Theorems Revisited Using the local smoothing estimates in Theorem 7.1.1 we can improve many of the maximal theorems in Section 6.3. We shall deal here with smooth families of Fourier integrals :Ft E rn(X, X; Ct), t E I, having the property that if we set :Ff(x, t) = :Ftf(x), then :F E pn-l I4 (X x I, X; C), where the full canonical relation C satisfies the non-degeneracy conditions and the cone condition described in Section 7.1. Our main result then is the following. Theorem 7.3.1: Let X be a smooth n-dimensional manifold and suppose that :Ft E Im(X, X;Ct), t E [1, 2J, is a smooth family of Fourier integral operators which belongs to a bounded subset of I~mp· Suppose
7.3. Spherical Maximal Theorems Revisited
22
further that the full canonical relation C C T*(X x [1, 2]) \0 x T* X\ 0 associated to this family satisfies the non-degeneracy conditions (7.1.2)(7.1.3) and the cone condition (7.1.6). Then, if e(p) is as in Theorem 7.1.1 and 2 < p < oo, II sup IFtf(x)lllv•(X) S: Cm,p llfiiLP(X)• tE[l,2]
+ e(p).
if m < -(n -1)(!- ~)- ~
(7.3.1)
If we define .rk,t as in the proof of Theorem 6.3.1, then Theorem 7.1.1 yields
(1 /)(ft)i.rk,tf(x)lp dxdtr'p 2
S: Ce 2 kj 2k[m+(n-l)(l/2-l/p)-e]llfllp,
E
< e(p).
By substituting this into the proof of Theorem 6.3.1, we get (7.3.1). Remark. A reasonable conjecture would be that for p ~ 2n/(n-1) the mapping properties of the maximal operators associated to the family of operators should be essentially the same as the mapping properties of the individual operators. By this we mean that for p ~ 2n/(n- 1) (7.3.1) should actually hold for all m < -(n- 1)(1/2 - 1/p). This of course would follow from showing that there is local smoothing of all orders< 1/p for this range of exponents. Using Theorem 7.3.1 we can give an important extension of Corollary 6.3.2 which allows us to handle the case of n = 2 under the assumption of cinematic curvature. Specifically, if we consider averaging operators
Atf(x) =
1s.,,.
f(y) TJ(x, y) dax,t(Y),
TJ E C(f,
associated to C 00 curves Sx,t in the plane which vary smoothly with the parameters, then we have the following result. Corollary 7.3.2: Let C C T*(R.2 x [1, 2]) \0 x T*R. 2 \0 be the conormal bundle of the C 00 hypersurface
S
= {(x,t,y): y E Sx,t} C
(R.2
X
[1,2])
X
R.2 .
Then if C satisfies the non-degeneracy condition (7.1.2) and the cone condition (7.1.6)
II tE[1,2] sup IAtf(x)IIILP(R2) S: Cp llfiiLP(R2)•
P
> 2.
(7.3.2)
226
7. Local Smoothing of Fourier Integml Opemtors Note that, as we saw in Section 6.3, At is a Fourier integral operator of order-!. Also, when n = 2, -(n-1)(!-1/p)-1/p =-!.So (7.3.2) follows from Theorem 7.3.1 since, if we set Ff(x, t) = Atf(x), then F is a conormal operator whose canonical relation is the conormal bundle of S. It is not hard to adapt the counterexample that was used to show that the circular maximal operator can never be bounded on .LP(R.2 ) when p $ 2 and see that the same applies to (7.3.2). Also, as we pointed out in Section 6.3, the Kakeya set precludes the possibility of Corollary 7.3.2 holding without the assumption of cinematic curvature. More precisely, we saw that (7.3.2) cannot hold for any finite exponents if Sx,t = {y : (x, y} = t}. ButS= {(x, t, y): (x, y} = t} is a subspace of R.5 and hence the cone condition cannot hold for the rotating lines operators, since, if C is the conormal bundle of S, then the images of the projection of C onto the fibers of T*(R.2 x R.) are just subspaces, meaning that the cones in (7.1.6) are just linear subspaces which of course cannot have any non-vanishing principal curvatures. We can also estimate maximal theorems corresponding to curves in R.2 which shrink to a point. Specifically, we now let Sx,t
=X
+ t Sx,t,
where Sx,t are C 00 curves depending smoothly on (x, t) E R.2 x [0, 1J. If we define new averaging operators by setting Atf(x) =
~-
f(x- ty) dux,t(y),
S,.,,
where dux,t denotes Lebesgue measure on Sx,t, then we have the following result. Corollary 7.3.3: Let C C T*(R.2 x (0, 1J) \0 x T*R.2 \0 be the conormal bundle of S = {(x,t,y): Y E Sx,t,t
> 0}
C (R. 2 X
(0, 1J) X R.2 .
Assume thatC satisfies the non-degeneracy condition (7.1.2) and the cone condition (7.1.6). Then, if the initial curves Sx,O have non-vanishing curvature,
II sup 1Atf(x)IIILP(R2) O
