18.155 LECTURE 9, 2015
RICHARD MELROSE
Abstract. Notes before and after lecture – if you have questions, ask!
Read:
Before lecture
Mostly in notes for Lecture 8.
After lecture
I don’t think I was crystal clear (to say the least) about some of this, so let me
try to fix it, keeping the symmetry between factors in the convolution and changing
the order from the lecture. The main technical result for today is:
Proposition 1. If u, v P Cc´8 pRn q then
(1) WFpu ˚ vq Ă tpy ` z, ηq P Rn ˆ pRn zt0uq; py, ηq P WFpuq AND pz, ηq P WFpvq.
There is a small technical result which is useful which I think I mentioned last
time:Lemma 1. If u P Cc´8 pRn q and Rn ˆ tηu X WFpuq “ H then ûpξq is rapidly
decreaseing in a cone around η.
Proof. For each p P supppuq the assumption is that pp, ηq R WFpuq, so there exists
p is rapidly decreaseing in a cone around η. This
φ P Cc8 pRn q, φppq ­“ 0 such that φi
covers the compact set supppuq by the open sets where φ ­“ 0 so this has a finite
subcover, i.e. there are finitely many φi P Cc8 pRn q such that the tφi ­“ 0u cover
y
supppuq and each φ
i u is rapidly decreaseing in a cone around η, and so all in open
cone which is their intersection. Now
ř we can find a partition of unity χi subordinate
to this cover of supppuq so u “ pχi uq but each χi u “ ψi φi u where ψi P Cc8 pRn q
i
(since φi ­“ 0 on supppχi qq. From the result about convolutions in the homework
y
for this week, each χ
i u is also rapidly decreasing in this same cone, and hence so if
û, their sum.
Proof of Proposition. The content of (1) is that if px, ηq P Rn ˆ pRn zt0uq and the
two sets
(2)
K1 “ ty P Rn ; py, ηq P WFpuq, K2 “ tz P Rn ; pz, ηq P WFpvq satisfy x R K1 ` K2 ,
then px, ηq R WFpu ˚ vq. So, we hold x and η fixed with this property. Given this
disjointness we can find two cutoff functions χ1 , χ2 P Cc8 pRn q such that χ1 “ 1
1
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RICHARD MELROSE
near K1 , χ2 “ 1 near K2 but x R supppχ2 q ` supppχ2 q. Now, divide the convolution
up accordingly:
(3)
u˚v “ pχ1 uq˚pχ2 vq`pp1´χ1 quq˚pχ2 vq`pχ1 uq˚pp1´χ2 qvq`pp1´χ1 quq˚pp1´χ2 qvq.
For the last three terms we use the fact that py, ηq R WFpp1 ´ χ1 quq for any y and
similarly pz, ηq R WFpp1 ´ χ2 qvq for any z. To these distributions the Lemma above
applies, and shows that they have Fourier transforms which decrease rapidly in a
cone around η. Taking the Fourier transform of the each of the last three terms gives
the product of the Fourier transforms where one (or both) is rapidly decreasing in
a cone around η and the other is at most of slow growth. So it follows that these
three terms do not have pp, ηq in their wavefront set for any p, in particular for
p “ x. [Of course the decomposition depends on this xs. So, we only need to look
at the first term. Since we know that
supppu ˚ V q Ă supppuq ` supppvq
we see that x R supppu ˚ vq and so in particular px, ηq is not in the wavefront set of
this term either, so px, ηq R WFpu ˚ vq as desired.
Now, we can use this to prove ‘microlocal elliptic regularity’. Recall that for
polynomial of degree m, P “ Pm ` P 1 where Pm is homogeneous of degree m and
P 1 is of order at most m ´ 1 and then we set
CharpP q “ tξ P Rn zt0u; Pm pξq “ 0u,
(4)
EllpP q “ pRn zt0uq z CharpP q.
So EllpP q is an open cone. If Γ Ă EllpP q is a closed cone then it intersects the
sphere in a compact set on which Pm does not vanish, so
|Pm pξq| ě c|ξ|m on Γ.
Since |Pm´1 | ď C|ξ|m´1 in |ξ| ą 1 it follows that for a large enough constant
(5)
|P pξq| ě c{2|ξ|m in Γ X t|ξ| ě Cu.
Thus, near infinity on a closed cone in EllpP q, P is ‘as big as a polynomial of degree
m can be’. This is why it is the elliptic set for P not just Pm .
Theorem 1. If u P C ´8 pΩq, Ω Ă Rn open then
(6)
WFpP pDquq Ă WFpuq Ă WFpP pDqqu Y pΩ ˆ CharpP qq .
So in the elliptic set P pDq behaves as though it is elliptic! The left hand inclusion
is straight forward, it is the right side which is not so obvious.
To prove this we will use the properties of convolution above and also a constructive result.
Proposition 2. If η P EllpP q then there exists Q P Cc´8 pRn q with WFpQq Ă
t0u ˆ pRn zt0uq such that
(7)
P pDqQ “ δ ´ e, p0, ηq R WFpeq.
From the left half of (6) we know that WFpeq Ă t0u ˆ pRn zt0uq.
Proof of Theorem. First we can get rid of the open set Ω. What we need to show is
that if pp, ηq P Ω ˆ pRn zt0uq, η P EllpP q (i.e. η R CharpP qq and pp, ηq R WFpP pDquq
then pp, ηq R WFpuq. The conclusion follows if we replace u by v “ φu where
LECTURE 9
3
φ P Cc8 pΩq and φppq “ 1 in a neighbourhood of p. Then g “ P pDqv “ φpP pDquq`h
where h comes from terms in which φ is differentiated at least once – so p R suppphq
and pp, ηq P WFpP pDquq is equivalent to pp, ηq P WFpP pDqvq. Thus we can assume
that u P Cc´8 pRn q in the first place.
So we look at a particular η P EllpP q and we want to show that pp, ηq R
WFpP pDquq implies pp, ηq R WFpuq. Now, we use Q from Proposition 2 constructed
for this particular η. Observe that both Q and u have compact supports, so convolution makes sense and
P pDqpQ ˚ uq “ Q ˚ pP pDquq “ pP pDqQq ˚ u “ δ ˚ u ´ e ˚ u.
So we can write this as the identity
(8)
u “ Q ˚ pP pDquq ` e ˚ u.
The construction of Q gives us p0, ηq R WFpeq. Now we apply Proposition 1. Since
WFpQq Ă t0u ˆ pRn zt0uq
WFpQ ˚ pP pDquq Ă WFpP pDquq S pp, ηq.
On the other hand WFpeq Ă t0u ˆ pRn zt0uq and also p0, ηq R WFpeq so
WFpe ˚ uq X pRn ˆ tηuq “ H
for any u. Combining these we conclude that pp, ηq R WFpuq and the theorem is
proved.
Proof of Proposition 2. Our hypothesis is that η is an ‘elliptic point’ for P (which
is the opposite of being a characteristic point, or vector), so (5) holds for a conic
neighbourhood of η. Suppose ψ P Cc8 pRn q is equal to 1 in a neigbourhood of 0.
Consider the function
˙
ˆ
η
´1 ξ
q P pξq´1 P C 8 pRn q, ą 0 small.
(9)
q “ p1 ´ χpξqqχ p ´
|ξ
|η|
The first term cuts out a big ball around the origin and the second term, which is
smooth outside the origin, localizes to a small cone around η. So for small ą 0 we
know that |P pξq| ě c{2|ξ|m on the support of the product of the first two terms.
The whole function is therefore smooth and in fact
(10)
q P S ´m pRn q, i.e. |Dα q| ď Cα p1 ` |ξ|q´m´|α|
is a symbol of order ´m. The leading estimate for this is
|qpξq| ď Cp1 ` |ξ|q´m
which follows from the bound from below on P. Taking derivatives, any deriavative
acting on the first term in (9) makes the support compact. The second factor is
homogeneous of degree 0 so applying Dξβ makes it homogeneous of degree ´|β|. So
in fact the product of the first two terms in (9) is in S 0 pRn q. We know that in the
region where it is smooth, Dα P ´1 “ pα P ´1´|α| where pα is a polynomial of degree
at most pm ´ 1q|α| and combining all these things gives (10).
We then define
(11)
Q “ χGq,
the inverse Fourier transform of q cut off near 0. We showed before that Gq is
smooth outside the origin for any symbol, so the same is true of Q. Moreover, if we
look at P pDqQ then the terms that come from differentiating χ have support away
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RICHARD MELROSE
from 0 and are smooth, so it is equal to χpP pDqGqq plus a term in Cc8 pRn q. Now,
the Fourier transform of P pDqGqq is
ˆ
ˆ
˙˙
η
´1 ξ
P pξqq “ 1 ´ p1 ´ χpξqq 1 ´ χ p ´
q
` χpξq
|ξ
|η|
from (9). The last term has compact support and the second one is rapidly decaying
in a cone around η. So indeed (7) holds, with e coming from these terms, plus the
previously ignored smooth terms.
Department of Mathematics, Massachusetts Institute of Technology
E-mail address: rbm@math.mit.edu