Лекции по функциональному анализу для начинающих специалистов по математической физике. Арсеньев А.А. - 465 стр.

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МГУ | Москва

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∀(x ∈ R
d
, y ∈ R
d
, |m| ≤ n , f ∈ S(R
d
)) :
|D
m
x
f(x + y) − D
m
x
f(x)||y|
−α
≤ C(d , α)kf | H
d/2+n+α
(R
d
)k.
∀(m , |m| ≤ n , f ∈ S(R
d
)) :
kf | C
n , α
0
(R
d
)k ≤ C(d , α)kf | H
d/2+n+α
(R
d
)k,
∀(f ∈ S(R
d
)) : kf | C
n , α
0
(R
d
)k ≤ C(d , α)kf | H
d/2+n+α
(R
d
)k.
f ∈ H
d/2+n+α
(R
d
),
f
n
(x) ∈ S(R
d
) f
n
kf − f
n
| H
d/2+n+α
(R
d
)k → 0 , n → ∞.
f
n
H
d/2+n+α
(R
d
)
kf
n
− f
m
| C
n , α
0
(R
d
)k ≤ C(d , α)kf
n
− f
m
| H
d/2+n+α
(R
d
)k
f
n
(x)
C
n , α
0
(R
d
)
f
0
(x) ∈ C
n , α
0
(R
d
) f
0
(x)
f
0 ≤ |m| ≤ s m
f ∈ H
s
(R
d
)
L
2
(R
d
)
D
m
f(x) = (2π)
−d
lim
R→∞
Z
|ξ|<R
exp(i(x , ξ))ξ
m
b
f(ξ)dξ
m = (m1 , m2 , . . . md) , ξ = (ξ1 , ξ2 , . . .)
D
m
= D
m1
x1
. . . D
md
xd
,
ξ
m
= ξ1
m1
. . . ξd
md
.
Ïîýòîìó

       ∀(x ∈ Rd , y ∈ Rd , |m| ≤ n , f ∈ S(Rd )) :
       |Dxm f (x + y) − Dxm f (x)||y|−α ≤ C(d , α)kf | H d/2+n+α (Rd )k.

Àíàëîãè÷íî äîêàçûâàåòñÿ íåðàâåíñòâî

               ∀(m , |m| ≤ n , f ∈ S(Rd )) :
               kf | C0n , α (Rd )k ≤ C(d , α)kf | H d/2+n+α (Rd )k,        (6.123)

Îêîí÷àòåëüíî ïîëó÷àåì:

   ∀(f ∈ S(Rd )) : kf | C0n , α (Rd )k ≤ C(d , α)kf | H d/2+n+α (Rd )k.    (6.124)

Ïóñòü ðàñïðåäåëåíèå
                                f ∈ H d/2+n+α (Rd ),
è ïîñëåäîâàòåëüíîñòü ôóíêöèé fn (x) ∈ S(Rd ) çàäàåò ðàñïðåäåëåíèÿ fn ,
êîòîðûå óäîâëåòâîðÿþò óñëîâèþ:

                    kf − fn | H d/2+n+α (Rd )k → 0 , n → ∞.

Òàê êàê ïîñëåäîâàòåëüíîñòü fn ñõîäèòñÿ â ìåòðèêå ïðîñòðàíñòâà H d/2+n+α (Rd ),
îíà ôóíäàìåíòàëüíà â ìåòðèêå ýòîãî ïðîñòðàíñòâà. Èç íåðàâåíñòâà

        kfn − fm | C0n , α (Rd )k ≤ C(d , α)kfn − fm | H d/2+n+α (Rd )k

ñëåäóåò, ÷òî ïîñëåäîâàòåëüíîñòü ôóíêöèé fn (x) ôóíäàìåíòàëüíà â ìåò-
ðèêå ïðîñòðàíñòâà C0n , α (Rd ) è ïîýòîìó â ìåòðèêå ýòîãî ïðîñòðàíñòâà
ñõîäèòñÿ ê ôóíêöèè f0 (x) ∈ C0n , α (Rd ). ßñíî, ÷òî ôóíêöèÿ f0 (x) çàäàåò
ðàñïðåäåëåíèå f .
    Òåîðåìà äîêàçàíà.
    Åñëè 0 ≤ |m| ≤ s, òî îáîáùåííîé ïðîèçâîäíîé ïîðÿäêà m ôóíê-
öèè f ∈ H s (Rd ) íàçûâàåòñÿ ðàññìàòðèâàåìàÿ êàê ýëåìåíò ïðîñòðàíñòâà
L2 (Rd ) ôóíêöèÿ
                                  Z
                   def   −d
            m
          D f (x) = (2π) lim          exp(i(x , ξ))ξ m fb(ξ)dξ      (6.125)
                               R→∞
                                  |ξ|

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