class="bi x0 y0 w1 h1"
class="bi x0 y0 w1 h2"
L
2
(R
n
)
L
r
(R
2n
) 1 ≤ r ≤ 2
S

(R
2n
)
L
r
(R
2n
) 2 < r ≤ ∞
τ
τ
W
a
a(x, ξ)
R
n
x
× R

a
a ∈ L
q
(R
2n
) L
p
(R
n
) (p, q)
q ≤ min{p, p

}
1
p
+
1
p

= 1.
S(R
2n
) L
q
(R
2n
) (q < ∞)
L
q
(R

+
= {0, 1, 2, }
R C
√
−1 = i.
n ∈ N \ {0} Z
n
+
= {α = (α
1
, α
2
, , α
n
), α
j
∈ Z
+
, j =
1, 2, , n}
R
n
= {x = (x
1
, x
2
, , x
n
), x
j


j=1
x
2
j

1
2
.
∂
∂x
1
,
∂
∂x
2
, ,
∂
∂x
n
R
n
∂
1
, ∂
2
, , ∂
n
−i∂
1

n
)
| α |=
n

j=1
α
j
α D
α
= D
α
1
1
D
α
2
2
D
α
n
n
a
α
R
n
| α |≤ m P (x, D)
R
2n
P (x, ξ) =

α
2
2
∂
α
n
n
,
∂
α
x
∂
α
ξ
∂
α
D
α
x
D
α
ξ
f g R
n
D
α
(fg) =

β≤α


α
β

=

α
1
β
1

α
2
β
2



α
n
β
n


α
j
β
j

=
α

n
, k ∈ Z
+
C
k
(Ω) = {u : Ω −→ C, u k},
C(Ω) = {u : Ω −→ C
}
C
k
0
(Ω) = {u : Ω −→ C | u ∈ C
k
(Ω),
u },
C
∞
(Ω) =
∞

k=1
C
k
(Ω), C
∞
0
(Ω) =
∞

k=1

n
) −→ L
2
(R
n
) }
B(L
2
(R
n
)) || · ||
∗
.
D(Ω) ϕ ∈ C
∞
0
(Ω)
{ϕ
j
}
∞
j=1
C
∞
0
(Ω)
ϕ ∈ C
∞
0
(Ω)

)
S(R
n
) = {ϕ ∈ C
∞
(Ω) |
x∈R
n
| x
α
D
β
ϕ(x) |< +∞, ∀α, β ∈ Z
n
+
}
{ϕ
k
}
∞
k=1
S(R
n
)
ϕ ∈ S(R
n
) S(R
n
)
k→∞




(1+ | x |
2
)
m

|α|≤m
| D
α
ϕ(x) |



, ∀ϕ ∈ D(R
n
).
S

(R
n
)
C
∞
0
(R
n
) S(R
n

)
≤|| f ||
L
1
(R
n
)
|| g ||
L
r
(R
n
)
.
f ∗ g f
g
f, g ∈ S(R
n
) f ∗ g ∈ S(R
n
).
ϕ ∈ L
1
(R
n
)

R
n
ϕ(x)dx = a.

ˆ
f(ξ) = (2π)
−
n
2

R
n
e
−ix·ξ
f(x)dx, ξ ∈ R
n
.
f ∈ L
1
(R
n
)
ˆ
f(ξ) =

R
n
e
−2πix·ξ
f(x)dx, ξ ∈ R
n
.
f
ϕ(x) = e

)
(f ∗ g)
ˆ
(ξ) = (2π)
n
2
ˆ
f(ξ)ˆg(ξ), ξ ∈ R
n
.
ϕ ∈ S(R
n
) α
(D
α
ϕ)
ˆ
(ξ) = ξ
α
ˆϕ(ξ), ξ ∈ R
n
,
(D
α
ˆϕ)(ξ) = ((−x)
α
ϕ)
ˆ
(ξ), ξ ∈ R
n

−n
(D
1
a
ˆ
f)(ξ), ξ ∈ R
n
,
(T
y
f)(x) = f(x + y), x ∈ R
n
,
(M
y
f)(x) = e
ix·y
f(x), x ∈ R
n
,
(D
a
f)(x) = f(ax), x ∈ R
n
,
y ∈ R
n
a ∈ R \ {0}.
T
y

n
),
∨
g(x) = (2π)
−
n
2

R
n
e
ix·ξ
g(ξ)dξ, x ∈ R
n
, g ∈ S(R
n
)
∨
g
g F
−1
g.
A X X
F : S(R
n
) −→ S(R
n
)
L
2

(ϕ) =

R
n
f(x)ϕ(x)dx, ϕ ∈ S(R
n
),
T
f
f
S(R
n
) S

(R
n
).
m S
m
R
2n
α β
C
α,β
α β
| D
α
x
(D
β

n
).

|α|≤m
a
α
(x)D
α
R
n
a
α
R
n
x∈R
n
| (D
β
a
α
)(x) |< ∞ , | α |≤ m,
β σ ∈ S
m
σ(x, ξ) =

|α|≤m
a
α
(x)ξ
α

R
n
f
k
dx −→

R
n
fdx.

A×B
| f(x, y) | d(x, y) < ∞
A × B A B

A


B
f(x, y)dy

dx =

B


A
f(x, y)dx

dy =


.
(ρ(q, p))
−1
= ρ(−q, −p)
q, p ∈ R
n
f g ∈ S(R
n
) V (f, g) R
2n
V (f, g)(q, p) = (2π)
−
n
2
< ρ(q, p)f, g > , q, p ∈ R
n
,
< , > L
2
(R
n
)
f g
< , > L
2
(R
2n
)
f g ∈ S(R
n

)
V (αf + βg, h) = αV (f, h) + βV ( g, h)
V (h, αf + βg) = αV (h, f) + βV (h, g)
h ∈ S(R
n
)
f g ∈ S(R
n
)
V (f, g)
ˆ
(x, ξ) = (2π)
−
n
2

R
n
e
iξ·p
f(x +
p
2
)
g(x −
p
2
)dp, x, ξ ∈ R
n
.

2
, x ∈ R
n
,
(1.5)
I
ε
(x, ξ)
= (2π)
−
n
2

R
n

R
n
e
−
ε
2
|q|
2
2
e
−ix·q−iξ·p


R

e
−i(x−y)·q
e
−
|εq|
2
2
dq)f(y +
p
2
)
g(y −
p
2
)dy

dp
=

R
n
e
−iξ·p


R
n
ε
−n
e

p
2
), y ∈ R
n
.
(1.7) (1.8)
I
ε
(x, ξ) =

R
n
e
−iξ·p
(F
p
∗ ϕ
ε
)(x)dp, x, ξ ∈ R
n
.
ϕ
ε
(x) = ε
−n
ϕ(
x
ε
), x ∈ R
n

∗ ϕ
ε
)(x) |≤|| F
p
||
L
∞
(R
n
)
|| ϕ
ε
||
L
1
(R
n
)
=|| F
p
||
L
∞
(R
n
)
|| ϕ ||
L
1
(R

)
= (2π)
n
2
(2π)
−
n
2

R
n
e
−ix·0
e
−
|x|
2
2
dx = (2π)
n
2
ϕ(0) = (2π)
n
2
.
(1.9) (1.11) (1.12)
ε→0
I
ε
(x, ξ) = (2π)

V (f, g)(q, p)dqdp
= (2π)
n
V (f, g)
ˆ
(x, ξ), x, ξ ∈ R
n
.
(1.13) (1 .14)
(2π)
n
V (f, g)
ˆ
(x, ξ) = (2π)
n
2

R
n
e
−iξ·p
f(y +
p
2
)
g(y −
p
2
)dp, x, ξ ∈ R
n


R
n
e
−iξ·p
f(y +
p
2
)
g(y −
p
2
)dp, x, ξ ∈ R
n
.
f g
W ig(f, g)(x, ξ) =

R
n
e
−2πiξ·p
f(y +
p
2
)
g(y −
p
2
)dp, x, ξ ∈ R

∼
W : S(R
2n
) −→ S(R
2n
)
∼
W F (x, ξ) = ( 2π)
−
n
2

R
n
e
−iξ·p
F (x +
p
2
, x −
p
2
)dp, x, ξ ∈ R
n
,
F ∈ S(R
2n
). (1.17)
<
∼

R
n
(
∼
W F
1
)(x, ξ)
(
∼
WF
2
)(x, ξ)dξ

dx
=

R
n


R
n
F
1
(x +
p
2
, x −
p
2

(x +
p
2
, x −
p
2
)dpdx,
F
1
, F
2
∈ S(R
2n
)
u = x +
p
2
v = x −
p
2
(1.18)
<
∼
W F
1
,
∼
W F
2
>=

∈ S(R
2n
) F
1
F
2
R
2n
F
1
(u, v) = f
1
(u)g
1
(v), u, v ∈ R
n
,
F
2
(u, v) = f
2
(u)g
2
(v), u, v ∈ R
n
.
(1.15) (1.17) (1.19) −(1.21)
< W (f
1
, g

(u, v)dudv
=

R
n

R
n
f
1
(u)
g
1
(v) f
2
(u)g
2
(v)dudv
=


R
n
f
1
(u)
f
2
(u)du


) × L
2
(R
n
) −→ L
2
(R
2n
)
|| W (f, g) ||
L
2
(R
2n
)
=|| f ||
L
2
(R
n
)
|| g ||
L
2
(R
n
)
, ∀f, g ∈ L
2
(R

n
f, g ∈ S(R
n
)
W (ρ(a, b)f, ρ(c, d)g)(x, ξ)
= e
i{(a−c)·x+(b−d)·ξ}
e
1
2
i(a·d−b·c)
W (f, g)(x +
b + d
2
, ξ +
a − c
2
)
x, ξ ∈ R
n
.
W (g, f) = W (f, g), f, g ∈ S(R
n
)
W (f) = W (f, f), f ∈ L
2
(R
n
)
W (ρ(a, b)f)(x, ξ) = W (f)(x + b, ξ −a), a, b, x, ξ ∈ R

.
σ ∈ S
m
, m ∈ R
W
σ
: S(R
n
) −→ S(R
n
)
ϕ ∈ S(R
n
) W
σ
ϕ R
n
(W
σ
ϕ)(x) = (2 π)
−n

R
n

R
n
e
i(x−y)·ξ
σ(

σ
: S(R
n
) −→ S(R
n
)
θ C
∞
0
(R
n
) θ(0) = 1
ε→0
(2π)
−n

R
n

R
n
θ(εξ)e
i(x−y)·ξ
σ(
x + y
2
, ξ)ϕ(y)dydξ
θ
x R
n

σ(x, ξ)W (f, g)(x, ξ)dxdξ
=
ε→0
+

R
n

R
n
θ(εξ)σ(x, ξ)W (f, g)(x, ξ)dxdξ
=
ε→0
+
(2π)
−
n
2

R
n

R
n
θ(εξ)σ(x, ξ)


R
n
e

σ(x, ξ)e
−iξ·p
f(x +
p
2
)
g(x −
p
2
)dpdx

dξ.
u = x +
p
2
v = x −
p
2
(1.24)
(1.24)

R
n

R
n
σ(x, ξ)W (f, g)(x, ξ)dxdξ
=
ε→0
+

2

R
n
g(v)


R
n

R
n
θ(εξ)σ(
u + v
2
, ξ)e
−i(v−u)·ξ
f(u)dudξ

dv
= (2π)
−
n
2

R
n
g(v)(W
σ
f)(v)dv = (2π)

2
|| σ ||
L
2
(R
2n
)
f g ∈ L
2
(R
n
) σ ∈ L
2
(R
2n
)