paper5.dvi


CUBO A Mathematical Journal

Vol.12, No¯ 03, (241–253). October 2010

On the Weyl Transform with Symbol in the

Gel’fand-Shilov Space and its Dual Space

YASUYUKI OKA

Department of Mathematics, Sophia University

7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan

email: yasuyu-o@hoffman.cc.sophia.ac.jp

ABSTRACT

In this paper, we claim two subjects. One is that the Weyl transform with symbol in

the Gel’fand-Shilov space S rr , r ≥
1
2

, is a trace class operator. The other one is that the

Weyl transform with symbol in the generalized function (S rr )
′, r ≥ 1

2
, is a continuous lin-

ear transformation from the Gel’fand-Shilov space S rr to (S
r

r )
′. As r > 1, Z. Lozanov-

Crvenković and D. Perišić have proved in [6] this result. Our second claim includes their

result.

RESUMEN

En este artículo afirmamos dos asuntos. El primero es que la transformada de Weyl con

símbolo en el espacio de Gel’fand-Shilov S rr , r ≥
1
2

, es un operador de clase trazo. El

segundo asunto es que la transformación de Weyl con símbolo en las funciones general-

izadas (S rr )
′, r ≥ 1

2
, es una transformación lineal continua del espacio Gel’fand-Shilov S rr

to (S rr )
′. Como r > 1, Z. Lozanov-Crvenković y D. Perišić probaron en [6] este resultado.

Nuestro resultado incluye su resultado.

Key words and phrases: Weyl transform, Gel’fand-Shilov space, Fourier-Wigner transform,

trace class operator, Schwartz’s kernel theorem.

Math. Subj. Class.: 46F05; 46F15; 81R15; 81S40.



242 Yasuyuki Oka
CUBO

12, 3 (2010)

1 Introduction

The subject of this article is to show the properties, as operators, of the Weyl transform with

the symbol in the Gel’fand-Shilov space S rr , r ≥ 1/2, and its dual space (S
r

r )
′, r ≥ 1/2.

The Weyl transform was first considered by Hermann Weyl arising in quantum mechan-

ics in [14] and the properties of the Weyl transform as operators have been studied by many

mathematicians. See for instance, [1], [6], [9], [11], [12], [13], [15] and others. These investi-

gations are mainly to consider the correspondence between the functional space, in which the

symbol belongs, and the operator class, in which the Weyl transform belongs.

There exist two remarkable results about these considerations: first, that the Weyl trans-

form with the symbol in Schwartz class is a trace class operator in [13]; and secondly, that

the Weyl transform with the symbol in (S rr )
′, r > 1, is a continuous and linear maps from

S
r

r (R
d ) to (S rr )

′(Rd ) in [6]. Our discussion is principally aimed at slightly developing these

two results. They depend on two areas of study: first, the correspondence between the Weyl

transform with the symbol in S rr , r ≥ 1/2, and the sequence space with some exponential
decrease, and secondly, the study of the Schwartz’s kernel theorem for (S rr )

′, r ≥ 1/2. We
consider these subjects in detail.

The plan of the paper is as follows. In the next section we introduce some properties

of the Gel’fand-Shilov space. In section 3 we treat the Weyl transform with symbol in S rr .

In section 4 we show the Schwartz’s kernel theorem for (S rr )
′ and the property of the Weyl

transform with symbol in generalized functions. Through this article we always treat the

index r ≥ 1/2.

2 The Gel’fand-Shilov Space S rr and its Dual (S
r

r )
′

First of all, we give some notations. We use a multi-index α ∈ Zd+, namely, α = (α1 ···αd ),
where αi ∈ Z and αi ≥ 0. So, for x ∈ Rd , xα = x

α1
1

··· xαd
d

and ∂αx = ∂
α1
x1

···∂αdxd , where ∂
α j
x j

= ( ∂
∂x j

)α j .

Definition 1 ([4]). Let A, B ∈ (0,∞)d . For r = (r1,··· , r d ) and r i ≥ 0, 1 ≤ i ≤ d,
S

r,B

r,A
(Rd ) = {ϕ ∈ C∞(Rd ) | ∀δ ∈ (0,∞)d , ∀ρ ∈ (0,∞)d , ∃Cδρ ≥ 0 s.t.

|xk∂qx ϕ(x)| ≤ Cδρ(A +δ)k(B +ρ)q kkr qqr , ∀k, q ∈ Zd+},
where

(A +δ)k = (A1 +δ1)k1 ···(Ad +δd )kd ,

(B +ρ)q = (B1 +ρ1)q1 ···(Bd +ρd )
qd .



CUBO
12, 3 (2010)

On the Weyl Transform ... 243

The space S
r,B

r,A
(Rd ) is a Fréchet space with the semi-norms

‖ϕ‖δρ = sup
x,k,q

|xk∂qx ϕ(x)|
(A +δ)k(B +ρ)q kkr qqr

, δi ,ρi = 1,
1

2
,
1

3
··· .

The space S rr (R
d ) is given by the inductive limit

S
r

r (R
d
) = lim

−→
A,B→∞

S
r,B

r,A
(R

d
).

The Gel’fand-Shilov space is the subspace of the Schwartz class S (Rd ).

Let a ∈ (0,∞)d be a = r
e A

1
r

. For any a, B ∈ (0,∞)d , we define the space S r,Br,a (Rd ) by

S
r,B

r,a (R
d ) = {ϕ ∈ C∞(Rd) | ∀δ,ρ ∈ (0,∞)d , ∃Cδρ > 0 s.t. |∂

q
x ϕ(x)| ≤ Cδρ(B +ρ)q qqr e−aδ|x|

1
r
,

∀k, q ∈ Zd+},
where aδ = r

e(A+δ)
1
r

and

‖ϕ‖δρ = sup
x,β

|∂βx ϕ(x)|

(B +ρ)βββr e−aδ|x|
1
r

.

The Gel’fand-Shilov spaces S rr (R
d) enjoy the following properties [4]:

Proposition 1. Let {ϕ j } be a sequence in S
r

r (R
d ). Then we obtain

ϕ j −→ 0 as j −→ +∞ in S rr

if and only if there are positive constants B and a such that

sup
x,β

|∂βx ϕ j (x)|

Bβββr e−a|x|
1
r

−→ 0 as j −→ +∞.

Proposition 2. (i) S rr ≡ {0} , 0 < r <
1
2

.

(ii) For r1 < r2, S
r1

r1
is included in S

r2
r2

and S
r1

r1
is dense in S

r2
r2

.

(iii) Let Ŝ rr be the image of the Fourier transform of S
r

r . Then Ŝ
r

r = S rr .

Remark 1. As r = 1, the Gel’fand-Shilov space S 1
1

(Rd ) is known to be isomorphism to the

space of test functions of the Fourier-hyperfunctions [7].

We define the Hermite functions {hn (x)}n=0,1,2··· on R
1 by

hn (x) = (2n n!)−
1
2 π

− 1
4 (−1)n e

x2

2

(

d

dx

)n

e
−x2

.



244 Yasuyuki Oka
CUBO

12, 3 (2010)

It is known that the set {hn (x)}n=0,1,2,··· is a complete orthonormal system in L
2(R1). That

is, for any f in L2(R1),

f (x) =
∞
∑

n=0
an hn (x) in L

2
(R

1
),

where an = ( f , hn) =
ˆ

R1

f (x)hn(x)dx. This expansion is called the Hermite expansions and

{an}n=0,1,2··· is called the Hermite coefficients. For d-dimensions, the Hermite functions on R
d

is defined by

hα(x) = hα1 (x1) ⊗···⊗ hαd (xd ), α ∈ Z
d
+ , x ∈ R

d
.

The set {hα(x)}α∈Zd+
is also a complete orthonormal system in L2(Rd ).

Proposition 3 ([17]). Let φ ∈ S rr (R
d ), r ≥ 1

2
. Then there exist some constants C > 0 and L ∈

(0,∞)d such that

φ =
∞
∑

|α|=0

(

φ, hα
)

hα and |
(

φ, hα
)

| ≤ C exp(−Lα
1
2r ).

Conversely, if |aα| ≤ C exp(−Lα
1
2r ) for some constants C > 0 and L ∈ (0,∞)d , then the series

∞
∑

|α|=0
aαhα(x) converges to a function in S

r
r (R

d ), where hα(x) is the Hermite function.

Definition 2. We denote by (S rr )
′(Rd) the dual space of the Gel’fand-Shilov space S rr (R

d).

3 The Weyl Transform with Symbol in S rr

As quantization from classical mechanics to quantum mechanics, H. Weyl introduced the

operator W (F) as follows: for any F ∈ S (R2d ),

W (F)ϕ(ξ) =
Ï

R2d
F(x, y)[π(x, y)ϕ](ξ)dxd y, ϕ ∈ L2(Rd ), (3.1)

where [π(x, y)ϕ](ξ) = ei(x·ξ+
1
2

x·y)ϕ(ξ+ y). We call this transform W (F) the Weyl transform with
symbol F. The Weyl transform W (F) is also expressed by the following matrix element: for

any ϕ, ψ ∈ L2(Rd ),

(W (F)ϕ,ψ) =
Ï

R
2d

F(x, y)(π(x, y)ϕ,ψ)dxd y

=
Ï

R2d
F(x, y)V (ϕ,ψ)(x, y)dxd y,

where V (ϕ,ψ)(x, y) is the Fourier-Wigner transform of ϕ and ψ defined by

V (ϕ,ψ)(x, y) = (2π)−
d
2

ˆ

Rd

e
ix·p

ϕ( p +
y

2
)ψ( p −

y

2
)d p.



CUBO
12, 3 (2010)

On the Weyl Transform ... 245

The Fourier-Wigner transform has the following property, see for example [5]. To be

definite, we shall repeat here the proof.

Proposition 4. Let ϕ,ψ ∈ S rr (R
d ), r ≥ 1

2
. Then V (ϕ,ψ) ∈ S rr (R

2d ) .

P roo f . It follows from Proposition 2 (iii) that a partial Fourier transform of the first

variables is a continuous map from S rr to S
r

r , so it suffices to show that if ϕ,ψ are in S
r

r (R
d ),

then ϕ( p +
y

2
)ψ̄( p −

y

2
) is in S rr (R

2d). Suppose ϕ,ψ ∈ S rr (R
d ). Since

p
α =

α
∑

|k|=0

(

α

k

)

(

p +
y

2

)k (

p −
y

2

)α−k
and y

β =
β
∑

|l|=0

(

β

l

)

(

p +
y

2

)l

(−1)|β−l|
(

p −
y

2

)β−l
,

we have that

p
α

y
β
∂
γ
p∂

δ
yϕ( p +

y

2
)ψ̄( p −

y

2
) =

α,β,γ,δ
∑

k,l,m,n

(

α

k

) (

β

l

) (

γ

m

)(

δ

n

)

(−1)|β−l|
(

p +
y

2

)k+l (
p −

y

2

)α+β−k−l

×∂mp ∂
n
yϕ( p +

y

2
)∂

γ−m
p ∂

δ−n
y ψ̄( p −

y

2
). (3.2)

Set u = p +
y

2
and v = p −

y

2
, then

(3.2) =
α,β,γ,δ

∑

k,l,m,n

(

α

k

) (

β

l

) (

γ

m

) (

δ

n

)

(−1)|β+δ−l−n|
(

1

2

)|δ|
u

k+l
v
α+β−k−l

∂
m+n
u ∂

γ−m+δ−n
v ϕ(u)ψ̄(v).

So we obtain that for any α, β γ, δ ∈ Zd+,

|pα yβ∂γp∂
δ
yϕ( p +

y

2
)ψ̄( p −

y

2
)| ≤

α,β,γ,δ
∑

k,l,m,n

(

α

k

) (

β

l

) (

γ

m

) (

δ

n

)

|uk+l ∂m+nu ϕ(u)||v
α+β−k−l

∂
γ−m+δ−n
v ψ̄(v)|

≤ C1C2
α,β,γ,δ

∑

k,l,m,n

(

α

k

)(

β

l

) (

γ

m

)(

δ

n

)

A
k+l
1 A

α+β−k−l
2

B
m+n
1 B

γ−m+δ−n
2

× (k + l)(k+l)r (α+β− k − l)(α+β−k−l)r(m + n)(m+n)r

(γ− m +δ− n)(γ−m+δ−n)r

(3.3)

for suitable constants A1, A2, B1, B2 ∈ (0,∞)d and C1, C2 > 0. Thus we have that

(3.3) ≤ C3 A
α+β
3

B
γ+δ
3

(α+β)(α+β)r(γ+δ)(γ+δ)r (3.4)

for some constants A3, B3 ∈ (0,∞)d and C3 > 0. Since

(α+β)(α+β)r ≤ eαr eβrααrββr and (γ+δ)(γ+δ)r ≤ eγr eδrγγrδδr ,



246 Yasuyuki Oka
CUBO

12, 3 (2010)

if we put A4 = A3 er and B4 = B3 er , then we have

(3.4) ≤ C3 A
α+β
4

B
γ+δ
4

α
αr
β
βr
γ
γr
δ
δr

.

Hence we obtain that there exist constants A4, B4 ∈ (0,∞)d and C3 > 0 such that

|pα yβ∂γp∂
δ
yϕ( p +

y

2
)ψ̄( p −

y

2
)| ≤ C3 A

α+β
4

B
γ+δ
4

α
αr
β
βr
γ
γr
δ
δr

for any α, β, γ and δ ∈ Zd+. This completes the proof of Proposition 4. ä

A straightforward computation with (3.1) shows that if F(x, y) ∈ S (R2d ), then we have

W (F)ϕ( p) =
ˆ

R
d

K ( p, p
′
)ϕ( p

′
)d p

′
, ϕ ∈ L2(Rd), (3.5)

where the kernel K ( p, p′) = F −1
1

F(
p+p′

2
, p′ − p). Here F −1

1
F denotes the inverse Fourier trans-

form of F in the first variables.

The Weyl transform has the following fundamental properties, see for example [15].

Proposition 5. (i) If the symbol F is in L1(R2d ), then the Weyl transform W (F) is a bounded

operator on L2(Rd),

(ii) Let the symbol F be in L2(R2d). Then the Weyl transform W (F) is the Hilbert Schmidt

operator on L2(Rd ). Conversely let Φ be the Hilbert-Schmidt operator. Then there exists

F ∈ L2(Rd ) such that
Φ = W (F).

We obtain the following result concerning on the property of the Weyl transform W (F)

with the symbol F in S rr (R
d ), r ≥ 1/2:

Theorem 1. Let W (S rr (R
2d)) be the set of all the Weyl transforms with the symbol in the

Gel’fand-Shilov space S rr (R
2d). Then

W (S rr (R
2d)) = {R ∈ B(L2(Rd )) |∃a, a′ ∈ (0,∞)d , ∃C > 0 such that

|(Rhα, hβ)| ≤ C e−a|α|
1
2r

e−a
′|β|

1
2r

, ∀α,β ∈ Zd+},

where B(L2(Rd )) is the set of all bounded operators on L2(Rd ) and hα, hβ are the Hermite

functions.

P roo f . Let G = {R ∈ B(L2(Rd )) |∃a, a′ ∈ (0,∞)d , ∃C > 0 such that |(Rhα, hβ)| ≤ C e−a|α|
1
2r

e−a
′|β|

1
2r

, ∀α,β ∈ Zd+}. By (3.5) and Proposition 2 (iii), it is apparent that the symbol F ∈



CUBO
12, 3 (2010)

On the Weyl Transform ... 247

S
r

r (R
2d ) if and only if the kernel K ∈ S rr (R

2d ). By proposition 3 and Fubini’s theorem, we

have that

|(W (F)hα, hβ)| ≤
∣

∣

∣

∣

(
ˆ

R
d

K ( p, p
′
)hα( p

′
)d p

′
, hβ( p)

)
∣

∣

∣

∣

=
∣

∣

∣

∣

Ï

R2 d
K ( p, p

′
)hα( p

′
)hβ( p)d p

′
d p

∣

∣

∣

∣

=
∣

∣(K , hα ⊗ hβ)
∣

∣

≤ C e−a|α|
1
2r

e
−a′|β|

1
2r

for some constants a, a′ ∈ (0,∞)d and C > 0. Therefore W (F) ∈ G . Conversely, let R1 ∈ G . Then

∞
∑

|α|=0
‖R1 hα‖2L2 (Rd ) =

∞
∑

|α|=0
(R1 hα, R1 hα)

≤
∞
∑

|α|=0

∞
∑

|β|=0
|(R1 hα, hβ)||(hβ, R1 hα)|

=
∞
∑

|α|=0

∞
∑

|β|=0
e
−2a|α|

1
2r

e
−2a′|β|

1
2r < +∞.

Hence R1 is the Hilbert-Schmidt operator. Therefore it follows from Proposition 5 that

there exists G ∈ L2(R2d ) such that W (G) = R1. Then from (3.5) there exists C > 0 and a, a′ ∈
(0,∞)d such that

|(R1 hα, hβ)| = |(W (G)hα, hβ)| = |(K , hα ⊗ hβ)| ≤ C e−a|α|
1
2r

e
−a′|β|

1
2r

.

By proposition 3, we obtain that K ∈ S rr (R
2d) and so is G. ä

Corollary 1. If F, G ∈ S rr (R
2d ), then there exists H ∈ S rr (R

2d ) such that W (H) = W (F)W (G)

P roo f . Let F, G ∈ S rr (R
2d ). Then we have that

|(W (F)W (G)hα, hβ)| = |(W (G)hα, W (F)∗hβ)|

≤
∑

γ

|(W (G)hα, hγ)||(hγ, W (F)∗hβ)|

=
∑

γ

|(W (G)hα, hγ)||(W (F)hγ, hβ)|

≤ C e−a|α|
1
2r

e
−b|β|

1
2r

∑

γ

e
−(a′+b′)|γ|

1
2r

= C′ e−a|α|
1
2r

e
−b|β|

1
2r



248 Yasuyuki Oka
CUBO

12, 3 (2010)

for suitable constants a, a′, b, b′ ∈ (0,∞)d and C, C′ > 0. Hence we obtain that W (F)W (G) ∈
G . Therefore it follows from Theorem 1 that H is in S rr (R

2d ) such that W (H) = W (F)W (G).
ä

Remark 2. It is known that W (F)W (G) = W (F ∗ 1
4

G), where

(F ∗ 1
4

G)(x, y) =
Ï

R2d
F(x −ξ, y −η)G(ξ,η)e

i
4
×2( y·ξ−x·η)

dξdη.

So if F, G ∈ S rr (R
2d ), then (F ∗ 1

4
G) ∈ S rr (R

2d) from Corollary 1.

Definition 3. We denote by S1 the family of all trace class operators defined as follows: for

A ∈ B(L2(Rd)), there exists an orthonormal basis {vk} of L2(Rd ) such that
∑

k

‖Avk‖L2 (Rd ) < ∞.

Proposition 6. Let A ∈ B(L2(Rd)). If {v j } is an orthonormal basis of L2(Rd ) then
∑

j

‖Av j‖L2 (Rd ) ≤
∑

j,k

|
(

Av j , vk
)

|.

P roo f . It is obvious as Av j = 0, so it suffices to show as Av j 6= 0. Let w j be an unit vector
for any index j. Then we have

∑

j

|
(

A
∗

w j , v j
)

| =
∑

j

|
(

w j , Av j
)

|

=
∑

j

∣

∣

∣

∣

∣

∑

k

(

w j, vk
)(

vk, Av j
)

∣

∣

∣

∣

∣

=
∑

j

∣

∣

∣

∣

∣

∑

k

(

A
∗

vk, v j
) (

w j, vk
)

∣

∣

∣

∣

∣

≤
∑

j,k

|
(

A
∗

vk, v j
)

|

=
∑

j,k

|
(

Av j , vk
)

|. (3.6)

Choose w j =
Av j

‖Av j‖
. The inequality now follows. Indeed from (3.6) we have

∑

j

|
(

w j , Av j
)

| =
∑

j

∣

∣

∣

∣

(

Av j

‖Av j‖
, Av j

)
∣

∣

∣

∣

=
∑

j

1

‖Av j‖
|
(

Av j , Av j
)

|2 =
∑

j

‖Av j‖ ≤
∑

j,k

|
(

Av j , vk
)

|. ä

We obtain the following result from Theorem 1 and Proposition 6:

Corollary 2. The Weyl transform W (F) with symbol in S rr (R
2d) is of the trace class S1.

Remark 3. Since the Gel’fand-Shilov classes are included in the Schwartz class, the preced-

ing Corollary 2 can be seen also as a consequence of the results of A. Voros [13], proving that

the Weyl transforms with symbol in the Schwartz class are trace operators.



CUBO
12, 3 (2010)

On the Weyl Transform ... 249

4 On the Weyl Transform with Symbol in (S rr )
′

We first show the Schwartz’s kernel theorem for (S rr )
′, r ≥ 1

2
, and give the property of the Weyl

transform with the symbol in (S rr )
′, r ≥ 1

2
, as a corollary of the Schwartz’s kernel theorem. S.

-Y. Chung, D. Kim and E. G. Lee proved the Schwartz kernel Theorem for (S 1
1

)′ in [2] and Z.

Lozanov-Crvenković and D. Perišić gave the Schwartz kernel theorem for (S rr )
′ as r > 1 in [6].

Our result includes their results.

We prove the following Schwartz’s kernel theorem for (S rr )
′, r ≥ 1

2
, along the idea in [2]:

Theorem 2. Let k be a continuous and linear operator from S rr (R
d2
x2

) to
(

S
r

r

)′
(R

d1
x1

), r ≥ 1
2

.

Then there exists K in
(

S
r

r

)′
(R

d1
x1

×Rd2x2 ), r ≥
1
2

, such that

〈kψ,ϕ〉 = 〈K ,ϕ⊗ψ〉,

where ϕ is in S rr (R
d1
x1

), r ≥ 1
2

, and ψ is in S rr (R
d2
x2

), r ≥ 1
2
.

To prove the Theorem 2, we begin from some preparations. We define the heat kernel

E(x, t) by

E(x, t) =
(

1
p

4πt

)d

e
− |x|

2

4t , (x, t) ∈ Rd × (0,∞).

The heat kernel enjoys the following properties:

· E(x, t) ∈ S (Rdx ),

·
ˆ

Rd

E(x, t)dx = 1,

and

·
(

∂

∂t
−∆

)

E(x, t) = 0 , in Rd × (0,∞).

Moreover we obtain the following estimate on the heat kernel E(x, t):

Proposition 7 ([16]). For any α ∈ Zd+, we have

|∂αx E(x, t)| ≤ E(x, t)(α!)
1
2 (2t)

−|α|
(1 +|x|)α, x ∈ Rd , 0 < t ≤

1

2
.

From this estimate, we immediately obtain the following properties:

Proposition 8. E(x, t) ∈ S rr (R
d
x ), r ≥

1
2
.

Proposition 9. Let E(x, t) is in S
r,B

r,a for any a, B > 0. Then for every T > 0 and ε > 0, there is
a constant C > 0 such that

‖E(x −·, t)‖δρ ≤ C exp[ε(|x|
1
r + (1/t)1/(2r−1))], x ∈ Rd , 0 < t < T.

(

r >
1

2

)



250 Yasuyuki Oka
CUBO

12, 3 (2010)

In the case where r = 1/2, we have the following inequality:

‖E(x −·, t)‖δρ ≤ Cε,t eε|x|
2

, x ∈ Rd , t > 0,
(

r =
1

2

)

.

Moreover we need the several propositions, which are the result of C. Dong and T. Mat-

suzawa [3], to prove Theorem 2 as follows:

Proposition 10 ([3]). Let ϕ(x) ∈ S r,Br,a (Rd ), r ≥ 1/2. Then

U(x, t) ≡
ˆ

Rd

E(x − y, t)ϕ( y)d y ∈ S r,Br,a (R
d
) , t > 0

and

U(x, t) → ϕ(x) in ∈ S r,Br,a (R
d
) as t → 0.

Proposition 11 ([3]). If Every C∞-function U(x, t) defined in Rd+1+ = {(x, t) | x ∈ R
d , t > 0}

satisfies the conditions:
(

∂

∂t
−∆

)

U(x, t) = 0 , in Rd+1+ ,

and for every T > 0 and ε > 0, there is a constant C > 0 such that

|U(x, t)| ≤ C exp[ε(|x|
1
r + (1/t)1/(2r−1))], x ∈ Rd , 0 < t < T.

(

r >
1

2

)

In the case where r = 1/2, U(x, t) has the following inequality:

|U(x, t)| ≤ Cε,t eε|x|
2

, x ∈ Rd , t > 0.
(

r =
1

2

)

Then U(x, t) can be expressed in the form U(x, t) = 〈u y , E(x − y, t)〉 with unique element u ∈
(

S
r

r

)′
(R

d
).

P roo f o f T heorem 2. We show the proof of theorem 2 as r > 1
2
. Since k is continuous,

the bilinear form B on S
r,B

r,a (R
d1 ) × S r,B

′

r,a′
(Rd2 ), for any a, B ∈ (0,∞)d1 and a′, B′ ∈ (0,∞)d2 ,

B(ϕ,ψ) = 〈kψ,ϕ〉 , ϕ ∈ S r,Br,a (Rd1 ) , ψ ∈ S
r,B′

r,a′
(Rd2 )

is separately continuous. Since S
r,B

r,a (R
d1 ) and S

r,B′

r,a′
(Rd2 ) is Fréchet space, B is continuous.

Hence we obtain that there exists a constant Ca,a′,B,B′ > 0 such that

|〈kψ,ϕ〉| ≤ Ca,a′,B,B′ ‖ϕ‖δρ‖ψ‖δ′ρ′ (♯).

Set for (x1, x2) ∈ Rd1 ×Rd2 and t > 0,

K t(x1, x2) = 〈kE(x2 −·, t), E(x1 −·, t)〉 .



CUBO
12, 3 (2010)

On the Weyl Transform ... 251

Now we show K t converges in
(

S
r

r

)′
(R

d1 ×Rd2 ) as t → 0. By (♯) and Proposition 9, for any
ε, ε′ > 0, there exists a constant Cε,ε′ > 0 such that

|K t(x1, x2)| ≤ Cε,ε′ exp[ε(|x1|
1
r + (1/t)1/(2r−1))] exp[ε′(|x2|

1
r + (1/t)1/(2r−1))].

Moreover we obtain
(

∂

∂t
−∆

)

K t(x1, x2) = 0 .

Therefore, by Proposition 11, there exists K0 ∈
(

S
r

r

)′
(R

d1 ×Rd2 ) such that K0 = lim
t→0

K t in
(

S
r

r

)′
(R

d1 ×Rd2 ).

For ϕ ∈ S rr (R
d
x1

), ψ ∈ S rr (R
d
x2

),

〈K t , ϕ⊗ψ〉 =
Ï

R
d1 +d2

K t(x1, x2)ϕ(x1)ψ(x2)dx1 dx2

=
Ï

R
d1 +d2

〈kE(x2 − y2, t)ψ(x2), E(x1 − y1, t)ϕ(x1)〉 dx1 dx2.

Since the Riemann sum of an integral converges in S rr , we obtain

〈K t , ϕ⊗ψ〉 = 〈k
ˆ

R
d2

E(x2 − y2, t)ψ(x2) ,
ˆ

R
d1

E(x1 − y1, t)ϕ(x1)〉.

Therefore, by Proposition 10, we obtain

〈K0,ϕ⊗ψ〉 = 〈kψ,ϕ〉 ,

as t → 0. ä

Similarly, we can also show the proof of Theorem 2 as r = 1
2
.

Remark 4. Z. Lozanov-Crvenković and D. Perišić also proved the Schwartz kernel theorem

for the spaces of tempered ultradistributions in [6] by means of the Hermite expansions.

We define the Weyl transform with symbol T ∈ (S rr )
′ by

〈W (T)ϕ,ψ〉 = 〈T, V (ϕ,ψ)〉, ϕ,ψ ∈ S rr (R
d
),

where V (ϕ,ψ̄) is the Fourier-Wigner transform of ϕ and ψ̄. It follows from Proposition 4 that

this definition is well defined. M. Cappiello, T. Gramchev and L. Rodino also showed this

subject in [1]. We obtain the following result from Theorem 2.

Corollary 3. The map W from S (R2d) to the space of bounded operators on L2(Rd ), defined

by

W (F)ϕ(ξ) =
Ï

R2d
F(x, y)[π(x, y)ϕ](ξ)dxd y , ϕ ∈ L2(Rd),

extends uniquely to a bijection from (S rr )
′(R2d), r ≥ 1/2, to the space of continuous linear maps

from S rr (R
d ), r ≥ 1/2, to (S rr )

′(Rd), r ≥ 1/2.



252 Yasuyuki Oka
CUBO

12, 3 (2010)

P roo f . Let k be a continuous linear map from S rr (R
d ) to (S rr )

′(Rd). By Theorem 2, for

any k, there exists K ∈ (S rr )
′(R2d) such that

〈kϕ,ψ〉 = 〈K ,ϕ⊗ψ〉 , ϕ,ψ ∈ S rr (R
d
).

So we have

〈kϕ,ψ〉 = 〈K ,ϕ⊗ψ〉

= 〈F1SK , V (ϕ,ψ̄)〉, (4.1)

where F1 is the Fourier transform of the first variable and S is defined by Sh(a, b) = h(a + b2 ,
a − b

2
). Set T = F1SK ,

(4.1) = 〈T, V (ϕ,ψ)〉

= 〈W (T)ϕ,ψ〉.

Since F1SK ∈ (S rr )
′(R2d ), for any k, there exists T ∈ (S rr )

′(R2d) such that k = W (T).
ä

Remark 5. Z. Lozanov-Crvenković and D. Perišić gave the similar result for (S rr )
′ as r > 1 in

[6].

References

[1] CAPPIELLO, M., GRAMCHEV, M.T. AND RODINO, L., Gelfand-Shilov spaces, pseudo-

differential operators and localization operators, in Modern Trends in Pseudo-

Differential Operators, Editors: Toft, J., Wong, M.W. and Zhu, H., Birkhäuser, 297–312.

[2] CHUNG, S.-Y., KIM, D. AND LEE, E.G., Schwartz kernel theorem for the Fourier hyper-

functions, Tsukuba J. Math., Vol. 19, N.2 (1995), 377–385.

[3] DONG, C. AND MATSUZAWA, T., S -space of Gel’fand-Shilov and differential equations,

Japan. J. Math. Vol. 19, N.2, (1994), 227–239.

[4] GEL’FAND, I.M. AND SHILOV, G.E., Generalized Functions Vol. 2, Academy of Sciences

Moscow, U.S.S.R, 1958.

[5] GRöCHENIG, K. AND ZIMMERMANN, G., Spaces of test functions via the STFT, J. Func-

tion Spaces Appl., 2 (2004), 25–53.

[6] LOZANOV-CRVENKOVIć, Z. AND PERIšIć, D., Kernel theorem for the space of tempered

ultradistributions, Integral Transforms and Special Functions, Vol. 18, N.10, October

(2007), 699–713.



CUBO
12, 3 (2010)

On the Weyl Transform ... 253

[7] NAGAMACHI, S. AND MUGIBAYASHI, N., Hyperfunction quantum field theory, Commun.

math. Phys., 46 (1976), 119–134.

[8] OKA, Y., N-representation for S and S ′, Sophia Univ. Master’s Thesis, 2002.

[9] POOL, J.C.T., Mathematical aspects of the Weyl correspondence, J. Math. Phys. vol. 7,

N.1, January (1966), 66–76.

[10] SIMON, B., Distributions and their Hermite expansions, J. Math. Phys. vol. 12, N.1

(1971), 140–148.

[11] SIMON, B., The Weyl transform and L p functions on phase space, Proc. Amer. Math.

Soc., 116 (1992), 1045–1047.

[12] TOFT, J., Continuity properties for modulation spaces with applications to pseudodiffer-

ential calculus, I, J. Funct. Anal., 207 (2), (2004), 399–429.

[13] VOROS, A., An algebra of pseudodifferential operators and the asymptotics of quantum

mechanics, J. Funct. Anal., 29 (1978), 104–132.

[14] WEYL, H., The Theory of Groups and Quantum Mechanics, Dover, New York, 1950.

[15] WONG, M.W., Weyl Transforms, Springer-Verlag, New York, Inc., 1998.

[16] YOSHINO, K. AND OKA, Y., Asymptotic expansions of the solutions to the heat equations

with hyperfunctions initial value, Commun. Korean Math. Soc., 23 (2008), N.4, 555–565.

[17] ZHANG, G.-Z., Theory of distributions of S type and pansions, Chinese Math. Acta., 4

(1963), 211–221.