Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 461 стр.

UptoLike

ВУЗ: 

МГУ | Москва

Составители: 

Рубрика: 

s ≥ 0 H
s
(R
d
)
S(R
d
)
b
f(ξ) f
L
2
(R
d
, dx) ⊂ H
s
(R
d
) , s ≥ 0.
s < 0 H
s
(R
d
)
S(R
d
)
∀(f ∈ S(R
d
)) : kf |
e
H
n
(R
d
)k
2
=
Z
X
0≤|m|≤n
|D
m
x
f(x)|
2
dx.
kf |
e
H
n
(R
d
)k
2
= (2π)
−d
Z
X
0≤|m|≤n
|ξ
m
|
2
b
f(ξ)|
2
dξ.
∃(C
1
, C
2
) : C
1
(1 + |ξ|
2
)
n
≤
X
0≤|m|≤n
|ξ
m
|
2
≤ C
2
(1 + |ξ|
2
)
n
,
k |
e
H
n
(R
d
)k k | H
n
(R
d
)k
k |
e
H
n
(R
d
)k ∼ k | H
n
(R
d
)k.
s = n + α , 0 < α < 1
∀(φ ∈ S(R
d
)) : kφ |
e
H
n , α
(R
d
)k
2
=
X
0≤|m|≤n
Z
|D
m
x
φ(x)|
2
dx+
X
|m|=n
ZZ
|x − y|
−(d+2α)
|D
m
x
φ(x) − D
m
y
φ(y)|
2
dxdy. , 0 < α < 1 , n = 0 , 1 . . .
S(R
d
) k | H
n+α
(R
d
)k
k |
e
H
n , α
(R
d
)k
k | H
n+α
(R
d
)k ∼ k |
e
H
n , α
(R
d
)k.
   Òåîðåìà äîêàçàíà.
   Èç òåîðåìû (6.4.1) ñëåäóåò, ÷òî ïðè s ≥ 0 ïðîñòðàíñòâî H s (Rd ) ìîæ-
íî îòîæäåñòâèòü ñ ïîïîëíåíèåì ïðîñòðàíñòâà S(Rd ) ïî ìåòðèêå (6.109),
ãäå fb(ξ) -ïðåîáðàçîâàíèå Ôóðüå-Ïëàíøåðåëÿ ôóíêöèè f , è ñïðàâåäëèâî
âêëþ÷åíèå
                      L2 (Rd , dx) ⊂ H s (Rd ) , s ≥ 0.
Ïðè s < 0 ïðîñòðàíñòâî H s (Rd ) åñòü ïðîñòðàíñòâî ðàñïðåäåëåíèé, êîòî-
ðûå äåéñòâóþò ïî ïðàâèëó (6.111).
   Íà ïðîñòðàíñòâå S(Rd ) îïðåäåëèì íîðìó
                                                             
                                         Z   X
       ∀(f ∈ S(Rd )) : kf | H
                            e n (Rd )k2 =       |Dxm f (x)|2  dx.
                                                        0≤|m|≤n


Èç ðàâåíñòâà Ïàðñåâàëÿ ñëåäóåò, ÷òî
                                             
                                Z   X
            e n (Rd )k2 = (2π)−d 
       kf | H                         |ξ m |2  fb(ξ)|2 dξ.                            (6.113)
                                            0≤|m|≤n


Òàê êàê
                                              X
          ∃(C1 , C2 ) : C1 (1 + |ξ|2 )n ≤              |ξ m |2 ≤ C2 (1 + |ξ|2 )n ,
                                            0≤|m|≤n


òî íîðìû k      e n (Rd )k è k
               |H                    | H n (Rd )k ýêâèâàëåíòíû:

                         k      e n (Rd )k ∼ k
                               |H                     | H n (Rd )k.

   ñëó÷àå s = n + α , 0 < α < 1 ïîëîæèì

                                def
                                    X Z
         d            n , α d 2
∀(φ ∈ S(R )) : kφ | H
                    e (R )k =            |Dxm φ(x)|2 dx+
                                           0≤|m|≤n
 X ZZ
          |x − y|−(d+2α) |Dxm φ(x) − Dym φ(y)|2 dxdy. , 0 < α < 1 , n = 0 , 1 . . .
 |m|=n
                                                                                       (6.114)

Òåîðåìà 6.4.2. Íà ïðîñòðàíñòâå S(R ) íîðìà k      d
                                                                      | H n+α (Rd )k   ýêâèâà-
ëåíòíà íîðìå    k     e n , α (Rd )k
                     |H                :

                     k       | H n+α (Rd )k ∼ k        e n , α (Rd )k.
                                                      |H                               (6.115)

                                           449

Страницы