The Gabor wave front set

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by  Luigi Rodino

THE GLOBAL WAVE FRONT SET AND THE
SHORT-TIME FOURIER TRANSFORM
LUIGI RODINO AND PATRIK WAHLBERG
Abstract. We give a detailed and partly original account of the
microlocalcharacterizationbyHo¨rmanderofthe globalwavefront
2
set, as the set of phase space directions without rapid decay of
1
0 the short-time Fourier transform. As an application we show the
2 inclusion for the global wave front set

l WF(aw(x,D)u) WF(u), u S′(Rd), a S0 ,
u ⊆ ∈ ∈ 0,0
J where S0 denotes the Ho¨rmander symbol class of order zero and
0,0
4 parameter values zero.
2

]
A
F 0. Introduction
.
h
t The paper concerns the global wave front set of tempered distribu-
a
tions on Rd, introduced by H¨ormander in [8].
m
The global wave front set WF(u) T (Rd) 0 of u S (Rd)
[ ∗ ′
consists of the complement of all dire⊆ctions in t\he{ p}hase sp∈ace, such
1
that there exists a symbol a in a Shubin class Gm where a is nonchar-
v
8 acteristic in the direction, and aw(x,D)u S(Rd). Noncharacteristic
2 ∈
in the direction z T (Rd) 0 means that
6 0 ∈ ∗ \{ }
5
a(z) ε z m, ε > 0,
.
7 | | ≥ | |
0 when z belongs to an open conic subset of T (Rd) 0 that contains
∗
2 \{ }
z , and z is large.
1 0
| |
: H¨ormander showed in [8] that (x ,ξ ) / WF(u) 0 if and only if
v 0 0 ∈ \{ }
the short-time Fourier transform with respect to a Gaussian window ϕ
i
X
r V u(x,ξ) = F(uϕ( x))(ξ), (x,ξ) R2d,
ϕ
a ·− ∈
where F is the Fourier transform, decays rapidly (polynomially) in a
conical neighborhood of (x ,ξ ).
0 0
H¨ormander’s proof of this microlocal characterization is brief and
omits several details. In this paper we give a detailed proof of the
characterization, and some generalizations and applications.
The microlocal characterization of the global wave front set can be
seen as an equality of two sets, and in the proof one shows that each set
isincludedintheother. Foroneoftheinclusions, wepresentanoriginal
proof using as a tool localization operators. The opposite inclusion is
much more complicated to show, and here we present a proof that is
based on H¨ormander’s ideas in [8, Proposition 6.8]. Moreover we show
1
2 L. RODINOANDP. WAHLBERG
that the microlocal characterization does not depend on the Gaussian
window function, and in fact any nonzero Schwartz function can be
used as a window function.
As an application of the microlocal characterization we show the
inclusion for the global wave front set
(0.1) WF(aw(x,D)u) WF(u), u S (Rd), a S0 ,
⊆ ∈ ′ ∈ 0,0
where S0 denotes the H¨ormander class of symbols on R2d, defined by
0,0
the property that a derivative of any order is uniformly bounded, and
whereaw(x,D)denotestheWeyl quantization. Thisinclusionisproved
using time-frequency techniques, Sj¨ostrand symbol classes, and results
by Gr¨ochenig concerning the almost diagonalization in phase space
for pseudodifferential operators with symbols in weighted Sj¨ostrand
classes.
The paper is organized as follows. After the preliminary Section
1, we treat conical cutoff functions and partitions of unity, and some
inclusions for the global wave front set in Section 2. Some of these
inclusions were originally proved by H¨ormander in [8], but as a service
to the reader we prefer to present a selfcontained account, rededucing
some results. Section 3 treats the equality of the global wave front set
and the set of phase space directions without rapid decay, and Section
4 treats the inclusion (0.1). Finally, we make a brief comparison in
Section5betweentheglobalwavefrontsetandCoriascoandManiccia’s
S-wave front set [2], which is another type of global wave front set.
1. Preliminaries
The transpose of a matrix A is denoted At, and the inverse transpose
of A GL(R,d) is written A t. An open ball in Rd of radius δ > 0
−
∈
centered at the origin is denoted Bδ = x Rd : x < δ , and the
d { ∈ | | }
unit sphere in Rd is denoted S = x Rd : x = 1 . The overline
d 1
− { ∈ | | }
A denotes either the closure of a measurable set A Rd, or as in f
⊆
the complex conjugate of a function f on Rd, the choice being clear
from the context. The Fourier transform of f S(Rd) (the Schwartz
∈
space) is defined by
Ff(ξ) = f(ξ) = f(x)e i x,ξ dx,
−h i
ZRd
where x,ξ denotestheinnebrproductonRd. Thestandardmulti-index
h i
notationforpartialdifferentialoperatorsisused, andD = (D ,...,D )
1 d
where D = i∂/∂x . We denote translation by T f(y) = f(y x),
j j x
− −
modulation by M f(y) = ei y,ξ f(y), x,y,ξ Rd, and phase space
ξ h i
∈
translation by Π(z) = M T , z = (x,ξ) R2d. Given a window func-
ξ x
∈
tion ϕ S(Rd), the short-time Fourier transform (STFT) is defined
∈
by
V f(z) = (f,Π(z)ϕ), f S (Rd), z R2d,
ϕ ′
∈ ∈
3
where ( , ) denotes the conjugate linear action of S on S, consistent
′
· ·
with the inner product ( , )L2. The cross-Wigner distribution of f,g
S(Rd) is defined by · · ∈
W(f,g)(x,ξ) = f(x+τ/2) g(x τ/2) e i τ,ξ dτ, (x,ξ) R2d.
−h i
− ∈
ZRd
We write W(f,f) = W(f), and denote x = (1+ x 2)1/2 for x Rd.
h i | | ∈
We write f(x) . g(x) provided there exists C > 0 such that f(x)
≤
Cg(x) for all x in the domain of f and g. For s R, the weighted
∈
LebesguespaceL1(Rd)isdefinedbythenorm f = f s .
s k kL1s(Rd) k h·i kL1(Rd)
A set A Rd is called conic if it is closed with respect to multipli-
⊆
cation by all positive reals.
Definition 1.1. For a D (Rd) the conic support conesupp(a) is the
′
∈
set of all x Rd 0 such that any conic open set Γ containing x
x
∈ \ { }
satisfies:
supp(a) Γ is not compact in Rd.
x
∩
Thesymplectic groupSp(d,R)consists ofallmatricesA GL(2d,R)
∈
that satisfy
σ(Az,Az ) = σ(z,z ), z,z R2d,
′ ′ ′
∈
where the symplectic form σ is defined by
σ((x,ξ),(x,ξ )) = x,ξ x,ξ .
′ ′ ′ ′
h i−h i
We work with symbols in Shubin classes, introduced in [12] and defined
as follows.
Definition 1.2. For m R, Gm is the subspace of all f C (R2d)
∞
∈ ∈
such that for every α,β Nd there exists a constant C > 0 so that
α,β
∈
∂α∂βa(x,ξ) C (x,ξ) m α β
| x ξ | ≤ α,βh i −| |−| |
is satisfied for every (x,ξ) R2d. Gm is a Fr´echet space with respect
∈
to seminorms defined by
sup (x,ξ) m+α+β ∂α∂βa(x,ξ) , (α,β) N2d.
h i− | | | | x ξ ∈
(x,ξ) R2d
∈ (cid:12) (cid:12)
(cid:12) (cid:12)
We have Gm = S(R2d)(cid:12). We denot(cid:12)e G = Gm. Let
m R ∞ m R
(a ) be a sequ∈ence of symbols such that a Gmj and m∈ as
j j 0 j j
j ≥+ , anTdset m = max m . Then there∈exists a sSymbo→l a−∞Gm,
j 0 j
un→ique∞modulo S(R2d), such≥ that ∈
n 1
a− − aj ∈ Gm′n, n ≥ 1, m′n = mj anx mj
j=0 ≥
X
4 L. RODINOANDP. WAHLBERG
(cf. [12, Proposition 23.1]). This is called an asymptotic expansion and
denoted a a . The symbol a may be constructed as
∼ j 0 j
≥
P ∞
(1.1) a(z) = χ(z/t )a (z)
j j
j=0
X
where χ C (R2d) satisfies 0 χ 1, χ(z) = 0 for z 1/2,
∞
∈ ≤ ≤ | | ≤
χ(z) = 1 for z 1, and (t ) R is a sufficiently rapidly increasing
j
| | ≥ ⊆
sequence of positive numbers.
The Weyl quantization is the map from symbols to operators defined
by
x+y
aw(x,D)f(x) = (2π) d ei x y,ξ a ,ξ f(y) dy dξ
− h − i
2
ZZR2d (cid:18) (cid:19)
for a S(R2d) and f S(Rd). The latter conditions can be relaxed
∈ ∈
in various ways. In particular, if a Gm for m R then aw(x,D) is
continuousonS(Rd), andextending∈bydualityit∈followsthataw(x,D)
is continuous on S (Rd) [12]. By the Schwartz kernel theorem, any
′
continuous linear operator S(Rd) S (Rd) can be written a Weyl
′
quantization for a unique a S (R7→2d). The Weyl quantization can be
′
∈
expressed in terms of the cross-Wigner distribution as
(1.2)
(aw(x,D)f,g) = (2π) d(a,W(g,f)), f,g S(Rd), a S (R2d).
− ′
∈ ∈
The Weyl product # is the product on symbol pairs corresponding to
operator composition:
aw(x,D)bw(x,D) = (a#b)w(x,D).
The Weyl product is a bilinear continuous map
(1.3) # : Gm Gn Gm+n.
× →
We have the following asymptotic expansion for the Weyl product (cf.
[12, Theorem 23.6 and Problem 23.2]):
( 1)β
(1.4) a#b(x,ξ) ∼ −α!β|! | 2−|α+β|Dxβ∂ξαa(x,ξ)Dxα∂ξβb(x,ξ).
α,β 0
X≥
In a few places we need to consider the Kohn–Nirenberg quantiza-
tion, defined by
a(x,D)f(x) = (2π) d ei x,ξ a(x,ξ)f(ξ)dξ, a Gm, f S(Rd).
− h i
∈ ∈
ZRd
Since any continuous linear operator bS(Rd) S′(Rd) has a unique
Weyl symbol in S (R2d) as well as a unique 7→Kohn–Nirenberg symbol
′
in S (R2d), a bijective mapping from the Kohn–Nirenberg symbol to
′
the Weyl symbol, denoted T, can be defined for such operators. This
means that
(1.5) a(x,D) = (Ta)w(x,D),
5
and if a Gm then also Ta Gm (cf. [12, Corollary 23.2]). Further-
∈ ∈
more we have the asymptotic expansions (cf. [12, Theorem 23.3])
α
1 1 | |
(1.6) Ta(x,ξ) Dα∂αa(x,ξ),
∼ α! −2 x ξ
α 0 (cid:18) (cid:19)
X≥
α
1 1 | |
(1.7) T 1a(x,ξ) Dα∂αa(x,ξ).
− ∼ α! 2 x ξ
α 0 (cid:18) (cid:19)
X≥
H¨ormander [8] introduced the following concepts in order to define
a global type of wave front set as a conic subset of the phase space.
Definition 1.3. Given a Gm, a point in the phase space z
0
∈ ∈
T (Rd) (0,0) is called non-characteristic for a provided there exists
∗
\{ }
A,ε > 0 and an open conic set Γ T (Rd) (0,0) such that z Γ
∗ 0
⊆ \{ } ∈
and
(1.8) a(z) ε z m, z Γ, z A.
| | ≥ h i ∈ | | ≥
Remark 1.4. More precisely, H¨ormander’s definition is the less restric-
tive condition
a(tz ) εtm, t A.
0
| | ≥ ≥
for some A,ε > 0.
For a Gm we denote the characteristic set of a by char(a) and
∈
define it as the set of all (x,ξ) T (Rd) (0,0) such that (x,ξ) is
∗
∈ \ { }
not non-characteristic according to Definition 1.3. Note that
conesupp(a) char(a) = T (Rd) 0 , a Gm.
∗
∪ \{ } ∈
Definition 1.5. Ifu S (Rd)thenWF(u)istheset ofallphasespace
′
points (x,ξ) T (Rd∈) (0,0) such that a Gm and aw(x,D)u S
∗
∈ \{ } ∈ ∈
implies that (x,ξ) char(a).
∈
Hence (0,0) = (x,ξ) / WF(u) if and only if there exists m R and
a Gm such th6at aw(x,∈D)u S and (x,ξ) / char(a). It follo∈ws from
∈ ∈ ∈
these definitions that WF(u) is closed and conic in T (Rd) (0,0) .
∗
\{ }
Remark 1.6. NotethattheKohn–Nirenbergquantizationgivesanequiv-
alent definition in Definition 1.5. In fact, if there exists a Gm such
that aw(x,D)u S and 0 = z / char(a), then there exis∈ts an open
0
∈ 6 ∈
conic set Γ T (Rd) (0,0) containing z such that (1.8) holds for
∗ 0
⊆ \{ }
some ε,A > 0. From (1.7) we obtain T 1a = a + b where b Gm 1,
− −
∈
and therefore for z Γ and z A, we have fore some C > 0
∈ | | ≥
(T 1a)(z) a(z) b(z) ε z m C z m 1
− −
≥ | |−| | ≥ h i − h i
C
(cid:12)(cid:12) (cid:12)(cid:12) = ε z m 1 z −1
h i − ε h i
(cid:18) (cid:19)
ε
z m
≥ 2h i
6 L. RODINOANDP. WAHLBERG
provided z A for A > 0 sufficiently large. Thus z / char(T 1a)
1 1 0 −
and (T 1a|)|(x≥,D)u = aw(x,D)u S. ∈
−
∈
To each linear symplectic map χ Sp(d,R) is associated a unitary
∈
operator U on L2(Rd), determined up to a complex factor of modulus
χ
one, such that
(1.9) U 1aw(x,D)U = (a χ)w(x,D)
χ− χ ◦
for a S (R2d) (cf. [3,7]). U is an homeomorphism on S and on S .
′ χ ′
∈
According to [8, Proposition 2.2] the global wave front set is symplec-
tically invariant as follows for u S (Rd).
′
∈
(1.10) (x,ξ) WF(u) χ(x,ξ) WF(U u) χ Sp(d,R),
χ
∈ ⇐⇒ ∈ ∀ ∈
or, in short, WF(U u) = χWF(u) for all χ Sp(d,R) and all u
χ
∈ ∈
S (Rd).
′
2. Partitions of unity and basic inclusions for the global
wave front set
First we prove a lemma that shows that there exists functions in G0
supported in a given open cone and equal to one on a smaller open
cone.
Lemma 2.1. LetΓ,Γ Rd be two conicopensets such thatΓ S
′ ′ d 1
Γ. There exists a functi⊆on χ C (Rd) such that 0 χ 1, sup∩p(χ−) ⊆
∞
∈ ≤ ≤ ⊆
Γ, conesupp(χ) Γ, χ(x) = 1 for x Γ and x 1, and
′
⊆ ∈ | | ≥
sup x α ∂αχ(x) < + α Nd.
| |
h i | | ∞ ∀ ∈
x Rd
∈
Proof. Let g C (Rd) satisfy 0 g 1, suppg Γ y : 1/2 <
∈ c∞ ≤ ≤ ⊆ ∩ {
y < 2 and g(x) = 1 for x Γ y : 1 y √2 . Set
′
| | } ∈ ∩{ ≤ | | ≤ }
√2x
χ(x) = g .
x
!
h i
Then we have 1 √2x/ x √2 for x 1, and therefore χ(x) = 1
≤ | h i| ≤ | | ≥
when x Γ and x 1.
′
∈ | | ≥
There exists ε > 0 such that supp(g)+Bε Γ. If x (Rd 0 ) Γ
d ⊆ ∈ \{ } \
and y Bδ then
∈ d
√2(x+y)
= z +w
x+y
h i
where z = √2x/ x+ y (Rd 0 ) Γ and w < ε provided δ > 0
h i ∈ \{ } \ | |
is sufficiently small. It follows that χ(x + y) = g(z + w) = 0 for all
y Bδ, which implies supp(χ) Γ.
∈ d ⊆
Let x (Rd 0 ) Γ, and define
∈ \{ } \
y
Γ = y Rd 0 : (Rd 0 ) Γ+Bε/√2 .
x ∈ \{ } y ∈ \{ } \ d
(cid:26) | | (cid:27)
7
Then Γ is open, conic, and x Γ . Suppose y Γ and χ(y) =
x x x
∈ ∈
g(√2y/ y ) = 0. Since y/ y = z + w where z (Rd 0 ) Γ and
h i 6 | | ∈ \ { } \
w < ε/√2, we have
| |
√2 y √2y √2 y
z | | = w | | supp(g)+Bε Γ,
y y − y ∈ d ⊆
h i h i h i
and since Γ is conic we obtain the contradiction z Γ. From this
∈
argument we may conclude that χ(y) = 0 for all y Γ , which implies
x
∈
conesupp(χ) Γ.
⊆
It remains to estimate the derivatives. We have
(2.1) ∂αχ(x) = (∂βg)(√2x x 1) C xγ x γ α
− α,β,γ −| |−| |
h i h i
β α γ
X≤ X
where the sum over γ is finite and C R. In fact, (2.1) can be
α,β,γ
∈
verified by induction, starting from
∂χ d ∂g √2x ∂ √2x
k
(x) =
∂x ∂x x ∂x x
j k ! j !
k=1 h i h i
X
d ∂g √2x ∂g √2x
= √2 x x x 3 + √2 x 1.
j k − −
∂x x − h i ∂x x h i
k ! j !
Xk=1 h i (cid:16) (cid:17) h i
The estimate (2.1) yields
∂αχ(x) C x α
α −| |
| | ≤ h i
for some C > 0, for all x Rd. (cid:3)
α
∈
As a corollary we obtain conical partitions of unity:
Corollary 2.2. Let Γ T (Rd) (0,0) be a closed conic set, let
∗
⊆ \ { }
Γ T (Rd), 1 j k, be open conic sets, and suppose that
j ∗
⊆ ≤ ≤
k
Γ Γ .
j
⊆
j=1
[
Thenthere existssymbols 0 a G0, 1 j k, such that supp(a )
j j
≤ ∈ ≤ ≤ ⊆
Γ , conesupp(a ) Γ for 1 j k, k a 1, and
j j ⊆ j ≤ ≤ j=1 j ≤
k P
(2.2) a (z) = 1, z Γ, z 1,
j ′
∈ | | ≥
j=1
X
where Γ Γ and Γ T (Rd) (0,0) is open and conic.
′ ′ ∗
⊆ ⊆ \{ }
Proof. Set
D = z R2d : 1/4 < z < 4 ,
{ ∈ | | }
D = z R2d : 1/2 z 2 .
′
{ ∈ ≤ | | ≤ }
8 L. RODINOANDP. WAHLBERG
Let 0 < ε 2 √2 be chosen such that
≤ −
k
Γ D +Bε Γ D.
∩ ′ 2d ⊆ j ∩
j=1
[
By [7, Theorem 1.4.5], there exist (g )k C (R2d) such that g 0,
j j=1 ⊆ c∞ j ≥
suppg Γ D for 1 j k, and k g (z) 1 with equality
j ⊆ j ∩ ≤ ≤ j=1 j ≤
when z Γ D +Bε . Set a (z) = g (√2z/ z ) for 1 j k. By
∈ ∩ ′ 2d j jP h i ≤ ≤
theproofofLemma 2.1, a G0, supp(a ) Γ andconesupp(a ) Γ
j j j j j
∈ ⊆ ⊆
for 1 j k. Set
≤ ≤
z
Γ = z R2d 0 : Γ+Bε/√2 .
′ ∈ \{ } z ∈ 2d
(cid:26) | | (cid:27)
Then Γ is conic, open and Γ Γ.
′ ′
⊆
Suppose z Γ and z 1. Then w := √2z/ z Γ D , thus
′ ′ ′
∈ | | ≥ h i ∈ ∩
ε/√2
w/ w = x+y with x Γ and y B . Hence
| | ∈ ∈ 2d
1 ε ε
< 1 < x < 1+ ,
2 − √2 | | √2
and since 1 w √2 we obtain
≤ | | ≤
1
< w x < √2+ε 2.
2 | || | ≤
We may conclude w x Γ D and w y Bε , and therefore w =
| | ∈ ∩ ′ | | ∈ 2d
w x+ w y Γ D +Bε . Thus k a (z) = k g (w) = 1, which
| | | | ∈ ∩ ′ 2d j=1 j j=1 j
proves (2.2). (cid:3)
P P
Next we shall prove inclusions for the action of pseudodifferential
operators on global wave front sets in terms of the conic support. We
will need the following lemma.
Lemma 2.3. If u S (Rd), a Gm, aw(x,D)u S and 0 = z /
′ 0
char(a), then there e∈xists b G2m∈such that bw(x,D)∈u S, b =6b +∈b
2m ′
∈ ∈
where b 0 and b G2m 2, and z / char(b).
2m ′ − 0
≥ ∈ ∈
Proof. The asymptotic expansion (1.4) gives a#a = a 2 + b′ where
b G2m 2. Set b = a 2 +b G2m. The assumption a|w|(x,D)u S
′ − ′
giv∈es bw(x,D)u = aw(|x|,D)aw(∈x,D)u S since aw(x,D) maps S∈into
∈
itself continuously. The assumption z / char(a) implies that there
0
∈
exists conic open set Γ T (Rd) (0,0) such that z Γ and such
∗ 0
⊆ \{ } ∈
that (1.8) is satisfied for some ε,A > 0. Together with b G2m 2, this
′ −
∈
gives for z Γ and z A
∈ | | ≥
b(z) a(z) 2 b(z) ε2 z 2m C z 2m 2
′ −
| | ≥ | | −| | ≥ h i − h i
= ε2 z 2m 1 ε 2C z 2
− −
h i − h i
for some C > 0, which shows(cid:0)that z / char(cid:1)(b). (cid:3)
0
∈
9
The following result treats inclusion relations for global wave front
sets with respect to the action of pseudodifferential operators with
symbols in the Shubin classes. The first inclusion in (2.3) is identical
to [8, Proposition 2.5]; we give a proof here for completeness’ sake.
Proposition 2.4. If u S (Rd) and a Gm then
′
∈ ∈
WF(aw(x,D)u) WF(u) conesupp(a)
⊆
(2.3) WF(u)\ WF(aw(x,D)u) char(a).
⊆ ⊆
Proof. Suppose first that 0 = z / WF(aw(x,D)u) and z[/ char(a).
0 0
6 ∈ ∈
Then there exists b Gn, n R, and an open conic set Γ containing
z such that bw(x,D∈) aw(x,D∈)u S and
0
∈
a(z) ε z m, z Γ, z A,
| | ≥ h i ∈ | | ≥
b(z) ε z n, z Γ, z A,
| | ≥ h i ∈ | | ≥
for some ε,A > 0. The asymptotic expansion (1.4) gives b#a = ba+c
where c Gm+n 2. Therefore we have, provided z Γ and z A,
−
∈ ∈ | | ≥
for some C > 0
b#a(z) b(z)a(z) c(z) ε2 z n+m 1+ε−2C z −2 .
| | ≥ | |−| | ≥ h i h i
By possibly augmenting A it follows that z /(cid:0)char(b#a). Th(cid:1)us z /
0 0
∈ ∈
WF(u) which means that we have proved the third inclusion in (2.3).
The second inclusion is trivial, and it remains to prove the first inclu-
sion.
If 0 = z / conesupp(a) then there exists an open cone Γ contain-
0
6 ∈
ing z such that K = supp(a) Γ is compact. By Lemma 2.1 we can
0
∩
choose b G0 such that supp(b) Γ and z / char(b). According to
0
∈ ⊆ ∈
(1.4) we have b#a a where supp(a ) K for all j 0. There-
∼ j 0 j j ⊆ ≥
fore b#a equals, modulo≥ S(R2d), a symbol supported in K. Hence
P
b#a S(R2d) which implies bw(x,D)aw(x,D)u S. This proves
∈ ∈
z / WF(aw(x,D)u), and WF(aw(x,D)u) conesupp(a) follows.
0
∈ ⊆
What remains is to prove the inclusion
(2.4) WF(aw(x,D)u) WF(u).
⊆
Suppose 0 = z / WF(u), that is, there exists b Gn where n R
0
such that z6 / c∈har(b) and bw(x,D)u S. By L∈emma 2.3 we m∈ay
0
∈ ∈
assume that b = b +b where b 0, b Gn 1 and
n ′ n ′ −
≥ ∈
b (z) ε z n, z Γ, z A
n
≥ h i ∈ | | ≥
for an open conic set Γ T (Rd) (0,0) such that z Γ, and for
∗ 0
⊆ \{ } ∈
someε,A > 0. LetΓ T (Rd) (0,0) beanopenconicset such that
′ ∗
⊆ \{ }
z Γ and Γ S Γ. Let χ G0 be chosen according to Lemma
0 ′ ′ 2d 1
∈ ∩ − ⊆ ∈
2.1 with respect to Γ and Γ, that is, suppχ Γ, 0 χ 1, and
′
⊆ ≤ ≤
χ(z) = 1 for z Γ and z 1. Set b (z) = χ(z)b(z)+(1 χ(z))ε z n.
′ 0
∈ | | ≥ − h i
10 L. RODINOANDP. WAHLBERG
For z / Γ we have b (z) = ε z n, whereas for z Γ we have, for some
0
∈ | | h i ∈
C > 0,
b (z) = χ(z)b (z)+(1 χ(z))ε z n +χ(z)b(z)
0 n ′
| | | − h i |
χ(z)b (z)+(1 χ(z))ε z n χ(z)b(z)
n ′
≥ − h i −| |
ε z n C z n 1
−
≥ h i − h i
ε
z n,
≥ 2h i
provided z A for A > 0 sufficiently large. This implies that
1 1
| | ≥
b HGn,n which denotes the set of hypoelliptic symbols in Gn (cf.
0
∈
[12, Definition 25.1]). According to [12, Theorem 25.1] there exists
c HG−n,−n such that c#b0 = 1+r1 where r1 S(R2d). Let g G0
∈ ∈ ∈
be chosen with aid of Lemma 2.1, satisfying 0 g 1, suppg Γ
′
≤ ≤ ⊆
and z0 / char(g). Set h = g#a#c Gm−n. Inserting c#b0 = 1+r1, we
∈ ∈
have
h#b = g#a#c#b +g#a#c#(b b )
0 0
−
= g#a+g#a#c#(b b )+r
0 2
−
where r S(R2d). Since supp(g) supp(b b ) is compact, (1.4)
2 0
implies th∈at g#a#c#(b b ) S(R∩2d). The−refore g#a = h#b + r
0 3
where r S(R2d). Fro−m bw(x∈,D)u S it now follows that
3
∈ ∈
gw(x,D)aw(x,D)u = hw(x,D)bw(x,D)u+rw(x,D)u S.
3 ∈
Wehaveshownz / WF(aw(x,D)u),concludingtheproofof (2.4). (cid:3)
0
∈
Corollary 2.5. If a Gm, u S (Rd) and char(a) = , then
′
∈ ∈ ∅
WF(aw(x,D)u) = WF(u).
The following result gives a detailed proof of [8, Proposition 2.4],
which is lacking in the reference.
Proposition 2.6. If u S (Rd), a Gm, and
′
∈ ∈
conesupp(a) WF(u) = ,
∩ ∅
then aw(x,D)u S.
∈
Proof. TheassumptionandProposition2.4implythatWF(aw(x,D)u) =
∅bw. (Txh,uDs)faorw(axn,yDz)0u∈ TS∗(,Radn)d\{(0,0)} there exists bz0 ∈ Gmz0 such that
z0 ∈
b (z) ε z mz0, z Γ , z A,
| z0 | ≥ h i ∈ z0 | | ≥
for some A,ε > 0 and for some conical open set Γ T (Rd) (0,0)
z0 ⊆ ∗ \{ }
containing z . From the compactness of S we may conclude that
0 2d 1
there exists an integer n > 0, symbols b − Gmj, conical open sets
j
∈
Γ T (Rd) (0,0) , for 1 j n, and A,ε > 0 such that
j ∗
⊆ \{ } ≤ ≤
b (z) ε z mj, z Γ , 1 j n, z A,
j j
| | ≥ h i ∈ ≤ ≤ | | ≥
(2.5) bw(x,D)aw(x,D)u S for 1 j n,
j ∈ ≤ ≤