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Rn
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R
Rn
L2(Rn)
C()
DK
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R
f : R R
sop(f) [M, M]
L
L/2> M
f
L
f(x) =nN
cn(L)e2i( nL)x
cn(L) = 1
L
L/2L/2
f(t)e2(nL)tdt
x [M, M]
f(x) =nN
1
L
L/2L/2
f(t)e2i(nL)tdte2i(
nL)x
f(x) =nN
1
L
f(t)e2i(nL)tdte2i(
nL)x
f 0
R [M, M]
fnL
:=
f(t)e2inL
tdt
f
L
nN1
Lf(n/L) e2(nL)x
f(t)e2itd
f(x) =
f(t)e2itd
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f() =
f(x)e2itdx
f L1(R)
f() =
f(x)e2ixdx
F :=f
f L1(R)
f || 0
f Ly f
f1
f
R
|f()| = |
Rnf(x)e2ixdx
Rn
|f(x)|dx= f1
g=
kj=1
cjRj Rj =
ni=1
[ajl , bjl ]
() = Rn
Rje2ixdx
=
R
[aj1,bj1](x1)[aj
2,bj2](x2) [ajn,bjn](xn)e
2ix11 e2ixnndx
=n
l=1
R
[ajl,bjl](xl)e
2ixlldxl
[a,b]
[a,b] =R
[a,b](x)e2ixdx=
ba
e2ixdx
e2ix
2i
ba
=e2ib
2i
e2ia
2i
0
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R
[ajl,bjl](xl)e
2ixlldxl
0
nl=1
R
[ajl,bjl](xl)e
2ixlldxl
0
Rj 0 g() 0
f L1(Rn) >0
g
f g1 < f() = f() g() +g()
(f g)() + |g()| f g1+ |g()|
Rn
f(+ h) f() = Rn
f(x)
e2ix(+h) e2ix
dx=Rn
e2ix
e2ixh 1
f(x)dx
f(+ h) f() Rn
e2ix |f(x)| e2ixh 1 dx
|f(x)|
e2ixh 1
2 |f(x)| L1(Rn).
f
f L1(R)
f L1(R)
f() = 2if()
(xf)() = (1/2i)(f)()
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f, g L1(R)
f()g()d=
f()g()d
f()g()d=
f(x)e2ixdx
g()d
e2ixg()df(x)dx=
f(x)
g(x)dx
g L1(R)
g(0) = 1
t > 0
Ag,tf(x) =
f()g(t)e2ixd
Rn
f L1(Rn)
f() = Rn
f(x)e2ixdx
f L1(R)
Ag,tf
Ag,t =
Rn
f()g(t)e2ixd= (f t)(x)
(x) =g(x) t(x) = (1/tn)(x/t)
Rn
L1(Rn)
= 1
t(x) = (1/t
n)(x/t)
f L(Rn)
(f t)(x)
t0 f(x)
f Lp(Rn)
p [1, )
(f t)
Lp f
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= 1
t = 1
(f t)(x) f(x) =
Rn
f(x y)t(y)dy f(x) =
Rn
[f(x y) f(x)] t(y)dy
> 0 >0 |f(x y) f(x)| < |y| <
|f t(x) f(x)|
Rn
|f(x y) f(x)| |t(y)| dy
=
|y|/t
(z)dz
t 0
f t fp
Rn
f( y) f()p |t(y)| dy=
Rn
wp(y) |t(y)| dy
wp(z) := f( y) f()p wp(tz) 0 t 0
wp 2 fp L
1(Rn).
f (Lp(Rn) L1(Rn))
g L1(Rn)
Rng = 1
Ag,tf f
t 0
Lp
1 p <
p=
Ag,tf(x) f(x)
Agtf=f t g(x) =(x)
1 p <
f L1 Lp
f(x) =limt0Rn
f()et||2e2ixd Lp
p=
f(x) =limt0Rn
f()et||2e2ixd
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g(0) =limt0
Rn
g()et||2
d
g L1(R)
g(0) =
Rn
g()d
f L1(Rn)
f L1(Rn)
f(x) =
Rn
f()e2ixd
g(x) =e|x|2
g
g(t) t0 g(0)
f(x) =limt0
Rn
f()et||2e2ixd en L1(Rn)
f() L1(Rn)
f()et||
2
e2ix
f()
L1(Rn)
f(x) =
Rn
f()e2ixd.
f(x) =
Rn
f()e2ixd
L2(Rn)
f (L1(Rn) L2(Rn))
f L2(Rn)
f2
= f2
g(x) =f(x)
f L1(Rn)
f
f(x) = g(x)
g
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g=f=f
h
f g
h= f g=fg=ff= f2 .
f L1(Rn) g L1(Rn)
h L1(Rn)
h= f g
|h(x)| = |(f g)(x)| =
Rn
f(x y)g(y)dy
Rn
|f(x y)g(y)| dy= f g1 f2 g2
h
|h(x0+ k) h(x0)| =Rn
f(x0+ k y)g(y)dy Rn
f(x0 y)g(y)dy
=
Rn
(f(x0+ k y) f(x0 y)) g(y)dy
Rn
|f(x0+ k y) f(x0 y)| |g(y)| dy
f(x0+ k ) f(x0 )2 g2k0 0
h L1
(Rn
)
h
h(x) =limt0
Rn
h()et||2e2ixd
h(0) =limt0
Rn
h()et||2d
h 0
f
0
h Rn
limt0h()et||2d= Rn
h()d
h L1(Rn)
h(x) =
Rn
h()e2ixd
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h(0) =
Rn
h()d= Rn
f()2 d= f22
h(0) = (f g)(0) =Rn
f(y)g(y)dy=Rn
f(y)f(y)dy=Rn
|f(y)|2 dy = f22
f2
= f2 f L2(Rn)
X
K
K = R
C
X
X
K
X X (x, y) x + y X
K X
X X
K X
Tx : XX Tx(y) =x +y
x X
O(x) x + O(0) {x + U=Tx(U) :U O(0)}
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p: X R
x, y X
K
p(x) 0
p(x) = ||p(x)
p(x + y) p(x) +p(y)
X
K
F
X
x=y
p F
p(x y) = 0
dz,p : X R
+
dz,p(x) =p(x z) X
(X, F)
X
dz,p
X
K
(X, F)
F
X
= (xi)iI X
xi x X (X, F) p(xi x) R p F
K
C X
tC+ (1 t)C C
0 t 1
B X
B B
K
|| 1
E X
s > 0
E tV
t > s
X
X
X
X
d
X
F
X
F
X
d
X
X
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{Vn}
Vn+1+ Vn+2+ Vn+3+ Vn+4 Vn n= 1, 2, . . .
D
r
r=
n=1
cn(r)2n
ci(r) r D 0 r 1
A(r) =X
r 1
r D
A(r) =c1(r)V1+ c2(r)V2+
f(x) = nf{r: x A(r)} x X y
d(x, y) =f(x y)
d
A(r) + A(s) A(r+ s) (x X, y X).
A(s)
A(r) A(r) + A(t r) A(t) (si r < t.)
{A(r)}
f(x + y) f(x) + f(y) (x X, y X)
0
r
s
D
f(x)< r, f(y) < s y r+ s < f(x) + f(y) + .
x A(r)
y A(s)
x + y A(r+ s)
f(x + y) r+ s < f(x) + f(y) +
A(r)
Vi f(x) =f(x) x A(r) x A(r)
f(0) = 0
x = 0
x / Vn= A(2
n)
n
f(x) 2n >0
X
B(0) = {x: f(x)< } =r
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< 2n
B(0) Vn {B(0)}
d
X
r+ s 1
A(r+ s) = X
r+ s
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D =
x1
1
xn
n || =1+ + n
|| = 0, Df=f
f
Rn
C()
Df C()
f
{x: f(x) = 0}
K
Rn
DK
f C(Rn)
K
C()
DK
C
()
K
Ki i = 1, 2, 3, . . . Ki Koi+1 =
i=1,2,...
Ki
Ki= Bi(x) x
pN C
() N = 1, 2, 3, . . .
pN(f) = max {|Df(x)| :x KN, || N} .
f = g
f g = 0
D(f g) = 0
N = ||
pN(f g) = 0
C()
x f f(x) fi C()
f
pN(fi f) 0 N = 1, 2, . . .
(fi f)(x) 0 fi(x) f(x) DK
x
K
DK C()
VN = {f C() :pN(f)< 1/N} N= 1, 2, . . .
{fi} C() N fi fj VN i, j
|Dfi D
fj| < 1/N KN || N Dfi
g
fi(x) g0(x)
g0 C() g = Dg0 fi g0 C()
C()
DK
DK
Rn
K
DK DK
D()
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Rn
K K DK pN
W D()
DK W
K
K
+ W
D()
W
D()
D()
D()
V
K DK D() DK
D()
E DK K
MN < E
N MN para N= 0, 1, 2,
{i} D() {i} DK K
limi,j
i
j
N
= 0 para N= 0, 1, 2,
i 0 D() K
i Di
i 0
D()
D()
i 0 D() i 0
DK D()
D
D() D()
D()
D()
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D()
D()
K
C <
|| CN
DK
u
Rn
x Rn
(xu)(y) =u(y x) y u(y) =u(y) para y Rn
(xu)(y) = u(y x) =u(x y).
u
v
Rn
u
v
(u v)(x) =
Rn
u(y)(xv)(y)dy
(u )(x) =u(x
) con u D(Rn), D(Rn), x Rn
(xu).v=
u.(xv)
u
v
xu u D
(xu)() =u(x) para D, x Rn
u D
D
D
x(u ) = (xu) = u (x) x Rn
u C
D(u ) = (Du) = u (D)
u ( ) = (u )
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y Rn
(x(u ))(y) = (u )(y x) =u(yx)
((xu) )(y) = (xu)(y) =u(yx)
(u (x))(y) =u(y (x)) =u(yx)
yx= yx
(x) =x
u
x (D) = (1)||D(x)
(u (D))(x) = ((Du) )(x).
e
Rn
r =r1(0 re) (r >0).
r(u ) =u (r)
r 0
r De n De e
x( (r)) x (De) en n
x Rn
limr0(u (r))(x) = (u (De))(x)
De(u ) =u (De)
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xf exf
exf xf
(f g) =
f
g
>0
h(x) =f(x/)
h(t) =nf(t)
x et
xf(t) = (xf) et= f xex= f et(x)et= ex(t)f(t)
exf(t) = exf et = f e(tx)= (xf)(t)
f
f C(Rn)
sup||NsupxRn(1 + |x|2)N|(Df)(x)| <
N= 0, 1, 2,
n
P Df Rn
P
n
n
g n
f P f
f gf
f Df
n n
f n
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P(D)f=PfP f=P(D)f
n n
f, g C()
D(f g) =
c
Df
Dg
n
n {fi} n
xDfi(x)
g i
g(x) = x
Dg00(x) fi g00 = = 0 xDfi = x
0D0fi = fi
n
f n Df n
D(P f) =
c
DP
Df
D(gf) =
c
Dg
Df
P f
gf n
f n P f n
(P(D)f) et= f P(D)et = f P(t)et= P(t) [f et]
Rn
(P(D)f)(t) =P(t)f(t)
t= (t1, , tn) t = (t1+ , t2, , tn) = 0
f(t) f(t)i
=
Rn
x1f(x)eix1 1
ix1 eixtdmn(t)
x1f L
1
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1
i
t1f=
Rn
x1f(x)eixtdmn(t).
P(x) =x1
f n g(x) = (1)
||xf(x)
g n
g = Df P Df =P g = (P(D)g) P(D)g L1(Rn)
f n
fi f n fi f L1(Rn) fi(t) f(t) t Rn.
ff n n
n R
n
n(x) =e 1
2|x|2
n n n= n
n(0) =
Rn
ndmn
n n 1
y + xy= 0
1
1/1
1(0) = 1
1(0) = R
1dm1 = (2)1/2
e(1/2)x2
dx= 1
1= 1
n(x) =1(x1) 1(xn) (x Rn)
n(t) =1(t1) 1(tn) (t Rn)
n = n n n(0) = ndmn
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g n
g(x) =
Rn
gexdmn (x Rn)
n n
f L1(Rn),f L1(Rn)
f0(x) =
Rn
f exdmn
f(x) =f0(x)
f, g L1(Rn)
Rn
Rn
f(x)g(y)eixydmn(x)dmn(y)
Rn
Rn
fgdmn= Rn
Rn
fgdmn
g n n f(x) =(x/) >0
Rn
g(t)n(t)dmn(t) = Rn
y
g(y)dmn(y)
Rn
g
t
(t)dmn(t) = Rn
t
g(t)dmn(y)
g
t
g(0)
y
(0)
g(0)Rn
dmn= (0) Rn
gdmn (g, n)
n x = 0
g(x) = (xg)) (0) =
Rn
xgdmn=
Rn
gexdmn.
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2.
g=g
n g = 0 g= 0
2g= g
g(x) =g(x)
4g= g
:n n
1
1 = 3
g n
Rn
f0gdmn= Rn
fgdmn (g n)
g
n D(R) n Rn
(f0 f)dmn= 0
D(Rn)
f0 f = 0 ctp
f n
g n
f g n
f g=fg
f g=fg
(f g) = f g
f g f g (fg) = 2f 2g= fg= (f g) = 2(f g)
1
2
f g n fg n
n n
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: L2(Rn) L2(Rn)
f=f para todaf n
f =f
n L1 L2
n
L2
L1
L2
f
f L1
f =f
L1 L2
L2
f
f
f L2(Rn)
f
g n
Rn
f g= Rn
g(x)dmn(x) Rnf(t)eixtdmn(t) =
Rnf(t)dmn(t)
Rng(x)e
ixt
dmn(x)
g(t)
Rn
fgdmn=
Rn
fgdmn (f, g n)
g= f
f2=f2
(f n)
n
L2
(Rn
)
n
L1
(Rn
)
f f
n L
2(Rn) n
f f
: L2(Rn) L2(Rn)
L2(Rn)
f, g L2(Rn)
L2
n D(Rn))
D(Rn)
n
D(Rn)
n
D(Rn)
n
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f n, D(R
n)
= 1
Rn
fr(x) =f(x)(rx) (x Rn, r >0)
fr D(R
n)
P
P(x)D(f fr)(x) =P(x)
c(Df)(x)r||D[1 ](rx)
D[1](rx) = 0
|x| 1/r
f n P.D
f C0(Rn)
Rn
r 0
fr f n
K
Rn
DK n
i : D(Rn) n
n
uL = L i
uL D(Rn)
L1 L2 n D(Rn) n
n n
u D(Rn)
n
uL
Rn
n
Rn
Rn
(1 + |x|2)kd(x)<
k
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g n
Du
(Du)(f) = (1)||u(Df)
(P u)(f) =u(P f)
(gu)(f) =u(gf)
u
n
u() =u() ( n)
n n n u n
u
u
n n
u n
(P(D)u) =PuP u= P(D)u
W
n 1, , k n
u n: |u(i)|
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u() =u() ( n, u n)
u V
u W
u=u
n
n
4u= u
u n 1 = 3
(P(D) u) () = (P(D) u) =u P(D) =u P =u (P ) = (Pu) ()
(P(D)u) () =u (P(D) ) =u (P(D) ) =u P = (P u) = P u ()
n
w= (1, 0, , 0) Rn, n
(x) = (x + w) (x)
(x Rn), >0,
0 x1 n
0 x1 n
x1
n
0 0 en n
(y) =eiy1 1
iy1 (y R
n, > 0)
P.D() =
cP.(D).(D)
D(y)
y21 si || = 0
|y1| si || = 1
||1 si || >1
Rn
0
n
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u n n
(u ) (x) =u
x
(x Rn)
x n x R
n
n
u C(Rn)
D (u ) = (Du) = u (D)
u
u =u
(u ) = u ( )
n
u =u
D(u ) =u (D)
w 0
(u ) =u
w 0
.
D(u ) =u (D) = (1, 0, , 0).
pN(f) f n
1 + |x + y|2 2(1 + |x|2)(1 + |y|2) para x, y Rn
pN(xf) 2N(1 + |x|2)NpN(f) para x R
n, f n.
u
n pN n
N
C <
|u(f)| CpN(f) para f n.
|(u )(x)| =u(x) 2N.CpN()(1 + |x|2)N,
u
n D(R
n)
K
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(u )() = (u )() = Rn
(u )(x)(x)dmn(x)
= K
u (x)x dmn(x) =u K
(x)xdmn(x)
u
=u =u
u
() = u ()
n
(u )() =u
((u ) ) (0) = (u ( ))(0).
x
u
= u= u.
u=u
L2
L2
C
Cn
f
f
(a1, , an)
gi() =f(a1, , ai1, ai+ , ai+1, , an)
g1, , gn C
Cn
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Cn
z = (z1, , zn) zk C zk = xk +iyk
x= (x1, , xn) y= (y1, , yn) z = x + iy
x= Re(z)
y= I m(z)
z
Rn z Cn Im(z) = 0
|z| = (|z1|2 + + |zn|
2)1/2
|Im(z)| = (y21+ + y2n)
1/2
z =z11 znn
z t= z1t1+ + zntn
ez(t) =eizt
t Rn
Cn
Rn
n
n= 1
Pk f z C
n
k
f(z) = 0 Pn P0 Pi 1 i n
a1, , ai gi
C
Pi1
rB
{x Rn : |x| r}
D(Rn)
rB
f(z) = Rn
(t)eiztdmn
(t)
N <
|f(z)| N(1 + |z|)Ner|Im(z)| (z Cn, N= 0, 1, 2, )
D(Rn)
rB
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t rB
eiz.t =ey.t e|y||t| er|Im(z)|.
(28) L1(Rn) z Cn f Cn f f
zf(z) =
Rn
(D)(t)eiz.tdmn(t).
|z| |f(z)| D1 er|Im(z)|
f
(t) =
Rn
f(x)eit.xdmn(x) con t Rn.
(1 + |x|)Nf(x) L1(Rn)
N
C(Rn)
f(+ i,z2, , zn)ei[t1(+i)+t2z2++tnzn]d
t1, , tn z1, , zn
(+ i)
= 1
f
= 0
= 1
(t) =
Rn
f(x + iy)eit.(x+iy)dmn(x)
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y Rn
t Rn
t = 0
y= t|t| >0
t.y= |t| , |y| =
f(x + iy)eit.(x+iy) N(1 + |x|)Ne(r|t|)
|(t)| Ne(r|t|)
Rn
(1 + |x|)Ndmn(x)
N
|t| > r
(t) = 0
rB
u D(Rn)
f(z) =u(ez) (z Cn)
f
Rn
<
|f(z)| (1 + |z|)Ner|Im(z)| (z Cn)
Cn
u D(Rn)
u
Cn
u(z) = u(ez) z Cn
u
Rn
Rn
L2
K
|f|2 dmn<
K
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u D()
L2
L2
u() =
gdmn
D()
Df
L2
gdmn= (1)||
f Ddmn
D()
C(p)()
Df || p
Dfi (/xi)
k
n > 0, p 0
r > p+ (n/2)
Rn Dkif L
2
1 i n, 0 k r
f0 C(p)() f0(x) = f(x) x
:
gik L
2
gikdmn= (1)k
f Dkidmn con D(),
1 i n, 0 k r.
K
D()
= 1
K
F
Rn
F(x) = (x)f(x) si x
0 si x /
F (L2 L1)(Rn)
D(f g) =
c(Df)(Dg)
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f
g C()
Dri =
r
s=0
rs(Drsi )(Dsi f) =r
s=0
rs(Drsi )gis
0 , Dri F = 0 0
Dri F Rn
L2(Rn)
1 i n
(Drsi )gis L
2()
F, Dr1F, , D
rnF
Rn
F2 dmn<
Rn
y2ri
F(y)2 dmn(y)< (1 i n).
(1 + |y|)2r
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F = F R
n
yF(y) L1
|| p
F C(p)(Rn).
f
F
f=F
F C
(p)(Rn)
f
F =F
f0
f0(x) =F(x) si x .
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