18.155 LECTURE 8, 2015
6 OCTOBER
RICHARD MELROSE
Abstract. Notes before and after lecture – if you have questions, ask!
Read: Friedlander and Joshi – the chapter on wavefront set.
Before lecture
I decided to slip a discussion of wavefront set in at this point, it will take two
lectures but most of the material is here. I will off-load at least one estimate to
Problem set 4 dealing with the decay in cones of convolutions. It is not hard and I
may actually do it in class but it is important to understand why things work.
(1)
• One item on the homework – I mentioned it I think but did not give a proof
– is that if u ∈ S 0 (Rn ) has compact support then û ∈ C ∞ (Rn ) and is of
slow growth (converse false).
• We deal with ‘conic’ subsets of Rn \ 0. A closed cone is a subset of Rn \ 0
which is invariant under scaling ξ 7−→ tξ, t > 0. It is determined by its
intersection with Sn−1 = {|ξ| = 1} ⊂ Rn \0. Then a conic set is open/closed
if this is true of its intersection with the sphere. Each point η 6= 0 in Rn has
open (closed) neighbourhoods corresponding to (small) balls in the sphere
so of the form
ξ
η
{ξ ∈ Rn \ 0; |
−
| < } resp ≤ > 0.
|ξ| |η|
• A (measureable, usually smooth) function v : Rn \ 0 −→ C is said to decay
rapidly in an open cone Γ ⊂ Rn \ 0 if for each closed cone γ ⊂ Γ and each
N there exists CN,γ such that
(2)
|v(ξ)| ≤ CN,γ (1 + |ξ|)−N ∀ ξ ∈ γ.
• If u ∈ C −∞ (Ω) then
(3)
WF(u) ⊂ Ω × (Rn \ 0), (p, η) ∈
/ WF(u) ⇐⇒
c decays rapidly in an open cone around η.
∃ ψ ∈ Cc∞ (Ω), ψ(p) 6= 0, ψu
• This is the same sort of ‘backwards’ definition we had for the singular
support. In particular
(4)
WF(u) ⊂ Ω × (Rn \ 0) is (relatively) closed and conic
since the defining condition is open and conic.
• If p ∈
/ singsupp(u) then
(5)
({p} × (Ω \ 0)) ∩ WF(u) = ∅.
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2
RICHARD MELROSE
Indeed, there exists ψ ∈ Cc∞ (Ω) with ψ(p) 6= 0 and ψu ∈ Cc∞ (Rn ) so
c ∈ S(Rn ).
ψu
• Let’s go a little in the opposite direction. Suppose that u ∈ C −∞ (Ω) and
we set
(6)
Cη (u) = π1 (WF(u) ∩ (Ω × {η})
for some fixed η ∈ Sn−1 . This is necessarily closed – as the image of a closed
set – and we can use a partition of unity argument to show that
(7)
ψ ∈ Cc∞ (Ω), supp(ψ) ∩ Cη = ∅ =⇒
c rapidly decreasing in an open cone around η
ψu
Indeed, for each q ∈ supp(ψ) we know from the discussion above that there
c is rapidly
is an open set Ωq containing q such that if φ ∈ Cc∞ (Ωq ) then φu
decreasing in a cone around η. So this is true for
Pfinite collection of qj
such that the Ωqj cover supp(ψ). But then ψ =
ψj is a finite sum of
j
c is rapidly decreasing in an open cone
ψj ∈ Cc∞ (Ωqj ) and it follows that ψu
around η – the intersection of the cone for each qj .
• What we want to get at is the inverse of (5) which can be expressed in
terms of the projection onto the first facto
(8)
π1 : Ω × (Rn \ 0) 3 (p, ξ) 7−→ p ∈ Ω.
Namely, (5) says that π1 (WF(u)) ⊂ singsupp(u) since if a point is not in
singsupp(u) then its preimage does not meet WF(u). The more interesting
part is that in fact
(9)
π1 (WF(u)) = singsupp(u) ∀ u ∈ C −∞ (Ω).
‘The wavefront set is a refinement of the singular support’ containing information about the ‘direction’ (really co-direction) of singularities.
• We are aiming for (9) and the main step is:
(10)
WF(φu) ⊂ WF(u), φ ∈ C ∞ (Ω), u ∈ C −∞ (Ω).
Clearly this is true with WF replaced by singsupp . Going back to the
definition it is enough to show that
Lemma 1. If ψ ∈ Cc∞ (Ω), φ ∈ C ∞ (Ω) and u ∈ C −∞ (Ω) then
c rapidly decreasing in an open cone Γ ⊂ Rn \ 0 =⇒
(11) ψu
d is rapidly decreasing in Γ.
ψφu
Proof. Since ψ has compact support we can choose χ ∈ Cc∞ (Ω) such that
χψ = ψ and then ψφu = ψ(χφ)u. So, it is enough to suppose that φ ∈
Cc∞ (Ω). Setting w = ψu ∈ Cc−∞ (Rn ) we are given the rapid decrease, in the
c
sense of (2) for v = ŵ and we want to conclude the rapid decrease of φw.
The Fourier transform is given in terms of convolution
Z
c
φw(ξ)
= (2π)−n φ̂(ξ − η)ŵ(η)dη
(take the inverse FT of the right side to see this). Here φ̂ ∈ S(Rn ), so it is
rapidly decaying. In the homework you show that, indeed, rapid decay of
L1
3
c in that cone. So we, or rather
ŵ in an open cone implies rapid decay of φw
you, have proved the Lemma.
• So, back to (9). We already know the inclusion of the left in the right.
Thus, we need to show that if (5) holds then p ∈
/ singsupp(u). What we
know is that for each η ∈ Rn \ 0 there is a ψ ∈ Cc∞ (Ω) with ψ(p) 6= 0 and
ˆ rapidly decreasing in an open cone around η. Clearly it is enough to
ψu
consider η ∈ Sn−1 , which is compact, and the covering by the intersections
of the cones with the sphere. So it follows that there is a finite collection
d
of ψj ∈ Cc∞ (Ω),ψj (p) 6= 0 with ψ
j u rapidly decreasing in an open cone
n
Γj ⊂ (R \ 0) such that ∪j Γj = Rn \ 0 cover the whole space. Now, we
can find one function ψ ∈ Cc∞ (Ω) with ψ(p) 6= 0 and supp(φ) ⊂ {ψj 6= 0}
for all j. So each ψj = φj ψ where φj ∈ Cc∞ (Ω) – defined to be ψ/ψj where
this makes sense and 0 otherwise. Applying (11) of the Lemma, it follows
c is rapidly decreasing in Γj for all j. So it is rapidly decreasing, but
that ψu
that implies that ψu is in all the Sobolev spaces, so ψu ∈ Cc∞ (Ω) and hence
p∈
/ singsupp(u) which is what we were aiming for!
• Now, check that
(12)
u, v ∈ Cc−∞ (Rn ) =⇒ WF(u∗v) ⊂ {(z, ξ) ∈ Rn times(Rn \0); ∃ x ∈ Rn with (x, ξ) ∈ WF(u) and (z−x, ξ) ∈ WF(v)
In fact it is enough for one of the terms to have compact support, as usual.
Why is this true? When we look at a particular point (p, η) we only need
to consider ψ(u∗v) where ψ ∈ Cc∞ (Rn ) has support near p, Even if only one
of u or v has compact support K we can multiply the other by a compactly
supported smooth cutoff which is equal to one near p − K without changing
ψ(u ∗ v) and the WFs only decrease, so the condition in (12) remains true
if it was true before. So it is okay to assume that both terms have compact
support. For a fixed η ∈ Sn−1 , consider
(13) Ku = π1 (WF(u) ∩ (Rn × {η}) b Rn , Kv = π1 (WF(v) ∩ (Rn × {η}) b Rn .
By assumption Ku and p − Kv are disjoint, so we can choose a cutoff
ψ ∈ Cc∞ (Rn ) which is one near Ku and with support not meeting p − Kv .
Similarly we can choose ψ 0 ∈ Cc∞ (Rn ) which is equal to 1 near p − Kv but
with supp(ψ) ∩ (p − supp(ψ 0 ) = ∅. This divides the convolution up into four
terms
(14) u∗v = (ψu)∗(ψ 0 v)+((1−ψ)u)∗(ψ 0 v)+(ψu)∗((1−ψ 0 )v)+((1−ψ)u)∗((1−ψ 0 )v).
The first term has support disjoint from p. Each of the others has one or
other factor with Fourier transform rapidly decreasing in a cone around η,
by (1) and (7) and since the Fourier transform of the convolution is the
product of the Fourier transform and the other term is of slow growth it
\
follows that χ(u
∗ v) is rapidly decreasing in a cone around η if χ ∈ Cc∞ (Rn )
has support close enough to p.
• Characteristic variety Charm (P ) and elliptic set Ellm (P ) of a polynomial
of degree m.
• Microlocal elliptic regularity
(15)
WF(P (D)u) ⊂ WF(u) ⊂ WF(P (D)u) ∪ Rn × Charm (P ).
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RICHARD MELROSE
Department of Mathematics, Massachusetts Institute of Technology
E-mail address: rbm@math.mit.edu