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Analisi Armonica Tor Vergata
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|anzn| = |z|/R#A ( $* $ # A O $5( *$( $ $ T( #U K $ ( 5Q $ $
w ( 5 1B ( $ ( B $ *$( $# A1= ( 1 R z K 1 |z| |w| < R# 7 |anzn| |anwn| ($* $ 6$ ( $* z ( ( O T ##%4U#
4"
limn
anan+1
1 6$ B $$ (! ' T( #U (1= 6$ #
9
)
6 ' 2
n=1 zn
n=1 z
n/n
n=1 zn/n2 1 $
( $$ 1 6$ ( B 5 1# $ ' ( T
*( 5 1U B ! $ ' ( $ ( ## 2 1= (# 25$ $6$ 1= J $
1/n2# 2
n=1 z
n/n z = 1 J 1/n $$ $ *(
1 T ##.# z = 1 2'( $ 1 1 $ z = eit t = 2k Tk ZU#
6 ) ( n
n! zn.
4
P$ an = n! 6$
R = limn+
|an||an+1|
= limn+
n!
(n+ 1)!= lim
n+n+ 1 = 0.
A z = 0#
+ $ %
' f(z) =
n=0 an (z z0)n
R > 0 z1 BS(z1) 5
BR(z0)
|z1 z0|+ S = R . T#U
k N
bk =n=k
(n
k
)an (z1 z0)nk
5
BS(z1) f 5
z1. z BS(z1)
f(z) =
k=0
bk (z z1)k . T#U
1#
+
E 1 6$ $# z BS(z1) |z z1 < S# A w = z0 + |z z1|+ |z1 z0| 1
|w z0| < S + |z1 z0| R
5 T#U# A $ T#U 1
n=0
k=0
|u(n, k)| =n=0
|an| (|z z1|+ |z1 z0|)n =n=0
|an| (w z0)n ,
6$ ( 1= |w z0| B * $ R# 7'' 6$ ( $ 6$ T #U ' 5 $ n k!
T##U
/ *
n=0 an zn
n=1 n an zn1
) K! +
! 5 +
6 1 ))?7) + /
)
1#
,
1# $ 2 #,#4 ( ##,#
'
(
f(x) =
n=0 an (xx0)n 5
+ ) x |x x0 < R. x
x0
f(t) dt =n=0
an
xx0
(t x0)n dt =n=0
ann + 1
(x x0)n+1 .
1# 2 $* [x0, x] ( T #,#%U# $ $ 6$ ( T ##U#
+
f(x) =
n=0 an x
n
(r, r)) ' x = r limxr f(x)
n=0 an r
n) @+
5
$
+ 5 )
1# $ x = r $
R $$( R r#
n=0 an x
n =n=0 an
(xr
)nrn ( 6$ 5
[0, r]# A1= xr 1 $
*$( x
rB $*
B ; # A
n=0 an rn
B # P$ ' $* 6$ ( $* 7' T##3 (ii)U! $ B $ *$( $ [0, r] (r, r]#
6 $ ( n
2n log(1 1
n
)zn.
,
> 0# '
) an = 2n log(1 1
n
)
( B
R = limn+
|an||an+1|
= 2
P$ $ |z| < 2 |z| > 2# |z| = 2 $ > 1!* z = 2 ei T RU 2'( T( #U $ 0 x = 1 T1 B U 8 * 1 B 6$ $ $ x = 1 12n log(1 1n
)2nein
1n ( $ *
1n
T( #U# 7 $ $ 12n log(1 1n
)2nein
1n 12n2 > 12n n B $J # P$ 1 |z| = 2 ( *#
6 $ ( n
2n zn.
'
) P$ an = 2n# P$
R =1
limn+
n
|an|=
1
limn+
n2n=
1
limn+
2
nn
= 1
( B R = 1 6$ |z| < 1 |z| > 1# 1 |z| = 1 1B (#
,%
6 $ ( n
(nn cosn) zn.
'
# A1= an = nn cosn ( B
R = limn+
|an||an+1|
= limn+
|nn cos n||n+ 1
n + 1 cos(n+ 1)|
= limn+
|1nncosn|
|1 + 1n
n+1n
cos(n+ 1)|= 1 .
P$ $ |z| < 1 |z| > 1# 1 |z| = 1 1B (#
6 ' $ ( n
(2x)n
n+ 3n
x#'
# P$ an = 1n+3n (
(n
zn
n+ 3n
B
R = limn+
|an||an+1|
= 1.
P$ $ |x| < 1/2 |x| > 1/2# * T( #U 1 x = 1/2 # ) 2'( T( #U 1 x = 1/2 #
6 ) A k > 0
n=1
xnk
,
$$ (0, 1)X ) $* (0, 1)X '
) ( 6$ T#+U B k 6$ $ $$ (0, 1)#
*$( $ ' $ 5 (0, 1] 1 $ x = 1 !6$ $* $5 (0, 1) $ # A ( m (0, 1) 1 1 (0, 1] $* # K 1 ($* $ k #
# $ !
6 ' 6$ 5 [2, 2] n=1
xn
n2
$*X
6 ' # (( O 1
k=1
ek|x|
k5
$* $ [0, 1]#
%# 1 ( $ TU
Sn =
nk=1
ek|x|
k5
,,
* $ $
)$10#
'
.
Nk=M+1
ek|x|
k5
M,N $J #U
6 ' # (( O 1
k=1
arctan(x k)k3
$* $ [0, 1]#
%# 1 ( $ TU
Sn =
nk=1
arctan(x k)k3
* $ $
)$10#
T$!
Nk=M+1
arctan(x k)k3
M,N $J #U
6 ' # (( O 1
k=1
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k2
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,.
%# 1 ( $ TU
Sn =n
k=1
e(xk)2
k2
* $ $
)$10#
T$!
Nk=M+1
e(xk)2
k2
M,N $J #U
6 '' # (( O 1
k=1
cosxk
k4
$* $ [0, 1]#
%# 1 ( $ TU
Sn =n
k=1
cosxk
k4
* $ $
)$10#
T$!
Nk=M+1
cosxk
k4
M,N $J #U
6 ') 6$ 5 [0, 4] n=1
(x)n
n3
$*X
,-
6 '$ 6$ 5 (1,+) n=1
(log(x+ 1))n
n4
$*X
6 '* 6$ 5 [1, 1] n=1
x2n
n3
$*X
6 ' 6$ 5 [1, 1] n=1
x2n
n3
$*X
6 ' (( O 1 n=1
arctan(x n)n3
$* $ $ R#
6 ' 6$ R n=1
e(xn)2
n2
$*X
6 ' 6$ R n=1
xn
n4
$*X
,3
6 ' 6$ R n=1
arctan(x n)n2
$*X
6 ' 6$ R n=1
ex2+n
n2
$*X
6 '' 6$ R n=1
cos(x n)n3
$*X
6 ') 6$ R n=1
arctan(x n)n3
$*X
6 '$ 6$ R n=1
sin(x+ n)
n3
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6 '* 6$ R n=1
x2n
n2
$*X
,4
6 ' 6$ R
n=1
xn
n2
$*X
6 ' 6$ R
n=1
e(xn)2
n2
$*X
6 ' 6$ 5 [a, a] a R
n=1
xn
n3
$*X
6 ' $
*$( fn(x) = cos xn R+' 6$ R $*#
6 ' 6$ [a, b] R
n=1
(x+ n)3
n5
$*X
6 ' $ ( $* $
*$ ( fn(x) = (
x)n 5 [0, 1]#
6 '' $ ( $* $
*$ ( fn(x) =
sin(nx)n2
#
,+
6 ') 6$ 5 [2, 2] n=1
xn
n3
$*X
6 '$ 6$ 5 [2, 2] n=1
xn
n2
$*X
6 '* 6$ 5 [0, 4]
n=1
(x)n
n3
$*X
6 ' 6$ 5 (0,+) n=1
enx2
n2
$*X
6 ' f(x) = 1/(1 + (x)2)# 2 $
*$(fn(x) = f(x/n) $$ $ RX ) $* $
X ) $* $ $ RX '
) V
1 fn(x) 1 $$ x# 2 ( B $* $R 1= n 1 fn 1 = supxR 1 fn(x) = 1 1limx 1 fn(x) = 1 limx fn = 1# 1 5 $ B $ 1 $ $ $ R 5! 6$ O $ *$( $ $ T( #U 1= $ *$( 1 fn $5 $ $ #
.
A 6$ $ $;
(( $ fn!
(('' $ #) ( $* $ # E R B $ $ [a, b] a < b 6$ ( 5( $ # ) a < 0 < b# E 1 *$( 1 fn(x) = 1 11+x2/n2 B 1 1
1+x2= x
2
1+x2 6$ B $ *$( ;
[0, b] ; [a, 0]# P$ 1 fn Q1 5 [0, b] 1 $ b 5 [a, 0] 1 $ a #P$ *$( 1 fn 5 [a, b] B 8 * $ 1 $ a bH *$( 1 fn *$( f B 6$ $ $ B $ 8 * $ a b# ( $ 1 8 $ b# 7 fn 1L[a, b] = 1 fn(b) = 1 11b2/n2 ( n 6$ 1 ( $* [a, b] fn *$( # 5 [a, b] B $ $ R+ R ( $*$ ( 1 5#
6 ' f(x) = 1/(1+x2) fn(x) = f(x1/n)# fn $*
# $ H
%# $ $ R#
'
) V 1 1 $$ *$( fn B f 1 1/n ( n# A 6$ $ ( $*
1 1
(fn f)(x) =1
1 + (x+ 1n)2
11 + x2
=1
n
2x+ 1n
(1 + (x+ 1n)2)(1 + x2)
.
P$ 1 11+(x+ 1
n)2
1 ''
|fn(x) f(x)| 1
n
2|x|+ 1n
1 + x2 a B
+ 5
$ O 9L 6' )7)
1# 2 *$( f(x) = (1x)a B Df(x) = a (1x)a1 B D2f(x) = a (a 1) (1 x)a2 = a (1 a) (1x)a2 $ n B Dnf(x) = a (1 a) (2 a) (n 1 a) (1 x)an# $ ! B 1
Dnf(0) = (1)n n!(an
); T#3U
B 1 a < 0 x < 1 $ f !6$ (
.4
$
(1 + x)a+1 = 1 +
n=0
a+ 1
n + 1
(a
n
)xn+1 = 1 +
n=0
(a+ 1
n+ 1
)xn+1
= 1 +
n=1
(a+ 1
n
)xn =
n=0
(a + 1
n
)xn .
'' 5 T#-U $ 8 $ a < 1#
5$ $ a R#
' p
.1 $ p ' x = {x1, x2, . . . , xn, . . . }
), p 1
xp =(
j=1
|xj |p) 1
p
.
' xp < x p)
$ $!
p = 1
x1 =j=1
|xj | .
p = 2
x2 =(
j=1
|xj |2) 1
2
.
4" $ 2 2 B K #%#%
x, y =n
i=1
xiyi.
P .+
*
x, x =n
i=1
xixi =
ni=1
|xi|2 = x22 .
6 $ 7 1 *$( R x |x|p p 1# )1 $ ( $; $
$
( #-##
.1 $ ' x = {x1, x2, x3, . . . , xn, . . . } )' p = +
x = supjN
|xj |.
' x < x l)
( $ * ( p# 25
1 $ 1 $ 1 $ #
/ $' ' 1 p < a b > 0
(a+ b)p ap + bp .
5 p > 1
5
! p = 1) 0 < p < 1
(a+ b)p ap + bp .
1# p = 1 B 1 5$ 5$$(#$ p = 1 p > 0# A t 0
(t) = (a+ tb)p ap tpbp .
7 (0) = 0 (t) = pb(a + tb)p1 ptp1bp $ t > 0 1= (t) = 0 '' (a + tb)p1 = tp1bp1! 1= ab t 6$ 6$ a + tb = tb $# (0) =pbap1 > 0 T6$ B 1 p < 1
-
5$ p < 1 *$( t tp1 ( t 0+U# A; 1 t 0 6$ (1) > (0) = 0#A1= (1) = (a + b)p ap bp 6$ 1 $$( #K 0 < p < 1 $ $ 0 < t 1
limt0+
(t) = limt0+
pb(a+ tb)p1 ptp1bp = pbap1 pbp limt0+
tp1 = .
P$ < 0 (0, 1] $$( #
6 $) 1 p 1 5$ 2#3#. B $ $$( M 1 $ A( #.##
'
. $ *$( $ [0, 1] 1 $ a $ 5 b 5 #
$$ + # p ' 0 p < q p q)
1# 25$ 6$ 1 0 p < q C = Cp,q > 0 1 $
{an} $$(
{an}p Cp,q {an}q .A 6$ $$( C = 1
{an}p {an}q . T#4U
7($ 5$ 1 q# $ 2 #3#. 1
q 1 $
{an} $
N 1 (
Nn=1
|an|)q
N
n=1
|an|q .
A 5 $ N {an}1 {an}q#P$ 1 lq#
P -
E 0 p < q# 1 {an}p {an}q B 1( n=1
|an|p)1/p
(
n=1
|an|q)1/q
.
A bn = |an|p# 7 {bn} 1 5$ $$( (
n=1
bn
)1/p
( n=1
bq/pn
)1/q,
Bn=1
bn (
n=1
bq/pn
)p/q.
P$ $$( B {bn}1 {bn}q/p! 6$ $ $( B ( 1q/p > 1#
6 $* ( ' V $
a = {a0, a1, 0, 0, . . . } $ 1 $ # 1
1 5$ A( #3#3 $$(aq aq 0 p < q 6$ $$( #3#. $ * * $ $ &$ $
K #3##
4" $ 7" p8 $ T)
#-#U 1 ( p ( $ p > 0 T ( p 1 6$ {xn}p = (
n=0 |xn|p)
1/p
$$( 6$ B $ H p < 1 B W * * $ $ K #3#U# A p = 2 $ ( T * U $ 5/( ,#,#U#
4" $ 7# !
p ) ( V p 1 $
$ $ T * C2U!
x = (x1, x2, 0, 0, . . . )
-%
2 ( V B 1$ ( $ 6$ B $ ( p# E V B * C26$ p $ ( * C2# $ * * $ 6$ ( p# )
( $ ! 6$ # 5( ( B $
$ # T ( #%#U 1 $ $ ( $
6$ 6$ 1 6$ * $ (
$ 5H 8 $$( T#4U $ 1 * $ p B 5 * $ q p < q#
* 6$ *! V T= R2U *$ p p#
p = 1! 56$( *( $ B
x1 = |x1|+ |x2|.
2 * T U $ B
|x1|+ |x2| 1.
K 6$ '' x1 > 0 x2 > 0# P$
|x1|+ |x2| 1 x1 + x2 1 x2 1 x1 ; $ $!
K 6$ x1 < 0 x2 > 0 6$
|x1|+ |x2| 1 x1 + x2 1 x2 1 + x1 $ $ $!
K ( 6$ x1 < 0 x2 < 0 $
|x1|+ |x2| 1 x1 x2 1 x2 1 x1 $ B $!
P -
&$ #!
R2
1
&$ #,!
R2
1
K ( 6$ x1 > 0 x2 < 0 $
|x1|+ |x2| 1 x1 x2 1 x2 1 + x1 $ $!
6$ 1 5 $B 6$ $ $!
-,
&$ #.!
R2
1
&$ #-!
R2
1
p = 2! 6$ 56$( *( $
x2 =(|x1|2 + |x2|2
) 12 .
A $ B
|x1|2 + |x2|2 1 x21 + x22 1
B 1 $ 5#
P -.
&$ #3! R2
1
&$ #4! R2
2
p = +x = sup
j=1,2|xj | = max {|x1|, |x2|} .
6$ $ B
x 1.
--
&$ #+! R2
1 < p < 2! $ p $ 1 B $ 6$ 2# 9 * $
p! 6$ * B $ $( $$(
(|x1|p + |x2|p)1p (|x1|r + |x2|r)
1r
p < r# 2 6$ $$( (# '
!$ 5( B 6$ 0 0 = > 0 x y f 5 |f(y) f(x)| < ) C
Rn Cn
|f(y) f(x)| 6);7)T
5 ) '
8
! !
x x f 5 f(x) !
5 x)
2 ( $* $ ( #4#% $; *$
$#
.1 * - % 1 f R $ $ 5 = f R+
(t) = sup{|f(x) f(y)| : |x y| < t} .
4" * 2 *$( $ $ B 1 $ # $ ( #4#% 1 f B$* $ > 0 > 0 1(t) < 0 < t < limt0+ (t) = 0#
6 *' (i) 2 *$( x B $* $#
(ii) 2 *$( x2 B $* $ B $ $ R#
(iii) 2 *$( (ii) 1 x 5 ( B ( $J $* $! *$(
x B
$* $#
(iv) 2 *$( tan(x) B $* $ 1$
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2, 2)H
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$ ( B $ $ $# 2 ( ( $ ( $ $ ( 5 f # Jk $ $1(($$ # 7 Ik ( *$( f
Ik = {x : f(x) Jk} = f1(Jk) .
&$ #!
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10n+1,
1
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10n+1
],
B $' 2'$ $#'
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$ $$(
1
10n+1+
1
10n+2);) g m2 W
m1(W ) =
W
g(t) dm2(t) Rg(t)W (t) dm2(t)
W 5 W 61 ))?7) '
g 5 )M
)
(ii) ' m1 5
m2 ms m2
3+
g m2
E
m1(E) = ms(E) +
E
g(t) dm2(t) .
(iii) '
f1(x) = m1(, x) fs(x) = ms(, x) f 1 = g
f s = 0 + +)
.1 * g
$ )>)A CK0> m1 m2
g =dm1dm2
.
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m 6 + Rn ! 1 )>)>7 "K9J4 g 6
1 )>)=7) +
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d/dm = min{g, 0})
= + ) 1 ( || 6
m7 || = + + )
' 5
m = Re +i Im || = |Re|+ | Im|)
' Rn
.
= ||(Rn) .
$ )
4" V * 1 m2 B $ $ m1B $ $ m2 g = dm1/dm2 L1(m2) gL1(m2) =
|g|dm2
|g| d|m2| = m1# E
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K $ $' 6$ $!
4
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6 ' E $ $ $ *$( f B 1 $$ 2'$ f #
6 ) 2 *$( 1 $ Q B '
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Nn=N1 fnn
fn = f(xn) 6$1 xn Jn# B $ 8! ( f 5 Jn In
C(I) LP (I)
1 f L 6$ $6$ $ $ $ (# E
1 5 *
$ *$( f $ $ A B
ess infA f = sup{ infxA\O
f(x) : O A, m(O) = 0} T#,+U
5 $ (# 7 In = [ess infxAnj f(x) ess supxAnj f(x)) B 5 * * $ ( f Jn $; $ Zn Jn $ $ $ 6$ f
! f $ $ $ $ $ 1= $ f $ 6$ ( 2'$ $ $6$ *
6$ ' xn Jn \Zn# ) 6$ $ *$( B $ *$( 1 f < 6$ $6$! * x Jn f(x) fn = f(xn) ' 5 In $1(( 1
k< # P$ 5$ 1 $
5( *$( T $$( T#,3U T#,4UU#) 5( *$( 1 B 8 # A K #+# > 0 n = N 1. . . , N $ On Jn 1m(On \ Jn) < /(2N + 2)# E = Nn=N1On \ Jn# 7 m(E) < # 2 #+# (i) n $ * $' $ Anj j = 1, 2, . . . 1 On = j=1Anj # 71 Anj $ $ n B ' 1 $ An0j01 $ An1j1 n0 n1 1= On (! 6$ $;
$ On0 On1 1B $ $ T( /(N+1) 1=
On0 On1 = (On0 On1) \ (Jn0 Jn1) = (On0 \ Jn0) (On1 \ Jn1) ,
1 Jn $#E $ $ *$( 1 f $* $ $ # ) 6$ 5( * 8 '' $# *$( Anj nj *$( =
Nn=N1
j=1 fnjnj fnj = f(xnj)
6$1 xnj Jn T $ 8 xnj Jn \ Zn#U
,
$ 6$ x AnjJn = Anj\E ' f(x) fnj = f(xnj) 5 In $1(( 1k < ! 6$
|f(x) fnj | < T#.U
x Anj \ E# A; *$( T$ $' U $* f *$ 5E $ #
A 5$ $ ( || < |f | T || < |f |U
*$( T *$( U 1 Jn Anj \ E 6$ $6$ $ * 6$ f # 7 6$
B $J cnj = ess infIn |f(x)| T $ T#,+U! cnj $$( T#.U cnj |f(x)| x Anj \ E Jn# A; *$( TU =
Nn=N
j=1 cnjnj * $$( 1# 7
*$( 5$ $ =N
n=N
j=1 tnn
tn = ess infxJn |f(x)# V $( 1 B B f #
/ * L Lp ( Lp p ' > 0 K > 0
f Lp p < fK |fK | K f fKp < )
1# A1= f Lp $ E R *$ 6$ 1
R\E |f |p < p/2# A fK $$ ( *$ E#
$ 1 f # A n > 0
fn(x) = f(x) |f(x)| n f(x) = n T nU f(x) > nT f(x) < nU# 7 |fn| n 6$ x 1fn(x) f(x) n # |f fn| |f | 6$ 1 *$( t tp Tt 0U B $ 1 |f fn|p |f |p# A )( T #+#.%U 1
limn
f fnp = limn
|f fn|p = 0 .
C(I) LP (I) .
A 6$1 K ''( E|f fK |p < p/2# 7
R|f fK |p =
E
|f fK |p +R\E
|f fK |p 0 s p < )1# P$ $ B $( 5$ ( 2 #+# T( $ $' $ U#
* 1) ! Lp
7 6$ ( $ ( *$( $ $ ( 1 (#. ,#-# $ ( ' ( #%#
V B $ ( 1 T B $ 1
limvv0
T (v) = T (v0)
T Tv 1 T (v)U#
LP +
' V,W T : V W 5 T 5 C > 0 v V
TvW CvV .
1! ( 6$ J $; ( *$ $ ' 8 A( ,#-# $#
.1 ' V,W T : V W 5 T
T = inf {C > 0 : TvW CvV v V }
.1 ' X )
X ( $ X
X )
.1 2
#
T Lp T (f) R f Lp R T (f) 0 f Lp 6
5 7)
/ ' (i) M T Lp T = ReT + i ImT ReT ImT ) ' T 5 ReT ImT )
(ii) M T Lp
T = T+T T+ T ) ' T 5 T+
T)
1# A *$( T (ReT )(f) = ReT (f) $ *$( f # V 1 ReT B $ *$( |(ReT )(f)| |T (f)| Tfp $ $ ReT H ( ImT # f B T (f) =T (Re f)+iT (Im f) = (ReT )f+i(ImT )f ?; T ReT+i ImT$ *$( H 1= f Lp B '( $*$( f = Re f + Im f 1 T = ReT + i ImT # P$ (i)#
A *$( T f T+(f) =sup {T (h) : 0 h f}# T+ B $ *$( ! * f 0 1
,
T+(f) T+(0) = 0# E 1 1 f 0 T (f) = T+(f)# f f 9( $ f = f+ f f+ = max f, 0 f = min f, 0# E0 f+ |f | f# P$ |T+(f)| |T+(f+)|+|T+(f)| =|T (f+)| + |T (f)| 2|T (f)| T+ 2T# f T (f) = T (f+) T (f) = T+(f) T(f) $ T = T+ T#P$ (ii)#
B $ 6$ (# / 1 $ Lp Lq 1 p < q B 5
$ p# K p = 2 1 $ ( F'L2 B ( L2 $ * 1 $ ( F' ( ( ,#-#,#
+ ) % # Lp Lq ' I R 1 p < f Lp(I)) ' q 1/p + 1/q = 1 61 )
LP ,
)( ; #+#.# A1= T B $ $ 1 (A) =n (An) 6$ B $ $ T( #+#U# )
$ 2'$ mH 1 m(A) =0 A B *$( $ Lp 6$ (A) = 0# A 6$ K>0 #+#%3! $ *$( g L1(I) g 0 1
T (A) =
g(x)A(x) dx
$' A# h L(I) $ *$( # A1= m(I) < h Lp(I)#A 2 #4#% $ $
*$( hn 0
1 x I hn(x) h(x) T ( B ; x 6$ n U hn $ *$( T( #+#,U B $ '( *$( 1 An $# $ )( ; #+#. limn hn hp = 0 1= g B 1 ghn ghp g hn hp 0# A1= T B $
T (h) =
g(x)h(x) dx .
A K > 0 IK = {x : g(x) K} h = gq1IK # A1= p q $ T( #-#,U ( T#+U 1
p(q 1) = p/(p 1) = 1(p 1)/p =
1
1/q= q .
A; 1
IK
g(x)q = T (h) T(
IK
g(q1)p dx) 1
p
= T(
IK
gq dx
)1 1q
.
IK
g(x)q (IK
g(x)q) 1
q
T .
,%
A1= g B K * IK = {x : g(x) K} I 6$
(IK
g(x)q) 1
q gq T# P$ g Lq(I)# P$ (#
.1 $ ) 1 X 6 X 7 '$ X X )
4" * ,
g X
f X g, f Tg(f))4" A ( X = Lp# X '$# * f X g X (
f g, f f $ *$( $ $ X $$ f $ #+#-#
A8 $ B $ (
! R%, )# , ( .S#
.1 7 I 5
)
4" $ Lp #+#- $ 1 1 p < Lp B I# L B I! $ $ B L1 B 8 $ '$ B 8 L
# ( #%#
, 0 2
.1 - ' m1 X m2 Y ) $ m1m2 X Y 5 E1 E2 6 Ei 5 + mi i = 1, 27
m1 m2(E1 E2) = m1(E1)m2(E2).
,
4" 1 E X Y x X y Y Ex = {y : (x, y) E} Ey = {x : (x, y) E})D Ex Ey ( E)
f 5 XY
fx(y) = f(x, y) f y(x) = f(x, y) fx : Y C f y : X C
)
$ B $ 1 T R%% )# 3 . -SU#
/ ' f 5 X Y m1 m2 fx f y m1 m2
) E X Y
(x) = m2(Ex) (y) = m1(Ey) m1
m2
X dm1 =
Y dm2) g 5 R
(x, y) g(x y) 5 R2)
'
. '' 6$
$ 5$ (# ' ( #+#3 1 > 0 5 E :={(x, y) : g(x, y) > } B $'# A B := {t R : g(t) > }#P$ B $' 1= g B $ *$( $'T $ ( #+#3U# E E = {(x, y) : x+y B}# 1 6$ B $' $ * 1B ( T$ *$( $U $ $' TBU# ! 6$ * B $ 1 $(
$ 1 (#
:( 2 #%# 1 $!
+ 2
(i) ' m1 X m2 Y f : X Y R+
m1m2)
(x) =
Yfx dm2 (y) =
Xfy dm1 6
))? !
(7)
' f 0 5 m1 5 m2
*$ $( 6
7.
,,
X
dm1 =
XY
fd(m1 m2) =Y
dm2 . T#.%U
(ii) 2 ' f L1(m1 m2) fx L1(m2) + x X 6 m17 f y L1(m1) 6m27+ y Y L1(m1) L1(m2) );)
1#
(i) ) *$( 1 ST S ' $ m1 T 6$ m2# 6$ $ B $ 2#%## A '( $ $ *$( m1 m2$'#A ! f 0 B $ *$( m1 m2 $'# A ) #4#. $ $
sn *$ ( $ X Y m1 m2$' 1 f$$ ;! 0 s1(x, y) s2(x, y) . . . limn sn(x, y) = f(x, y) x y# n n ( 5$
sn# 7 1Xn(x)dm1(x) =
XY sn(x, y)d(m1 m2) =
Yn(y)dm2(y)# A
x *$( n n T )( ; #+#.U 6$ T#.%U $ $ ( )( ;#
(ii) A ' *$( 1 6$ '( $ *$( T U# 5
*$( f B 9( $ *$( f+ f T ( 2 #+#.U f+ f |f |# + (
f+ f# E (i) $ 1 + L1(m1)#E 1 fx = (f+)x (f)x (i) *$( (f+)x (f)x 1 x 1 + TU# A1= + L1(m1) 6$
m16$
$6$# A 6$ x 1 (x) = +(x) (x) 6$ L1(m1)# A; *$ $( T#.%U
,.
f+ + f ! ( f # 2( *$ B #
' '
f 5 (x) =Y|f |xdm2) ' 5 5
Xdm1 < f L1((m1
m2))
1#
,-
6 * T 5 $ (0, 1) (1, 0)# E 1 T
$ x = 0 y = 0 x+ y = 1# P$ *$ $(
T
f(x, y) dx dy =
10
( 1x0
f(x, y) dy
)dx =
10
( 1y0
f(x, y) dx
)dy .
6 V 5 $ (0, 0, 1)(0, 1, 0) (1, 0, 0)# E 1 *
V
$ x = 0y = 0 z = 0 x+y+z = 1# & z = z0 * 0 1 (
z = z0 B z = z0 0 y 1 z0 0 x 1 y z0#
P$ *$ $( V
f(x, y, z) dx dy dz =
10
( 1z0
( 1yz0
f(x, y, z) dx
)dy
)dz; .
*$ $( 5 ( T z y xU# ) 5 $ V *$( 1/(x2+y2+z2) R# A 6$ 5 B X ) ' 5 B 1/(x+ y + z)X
" ,3
6 5( ((
V C ' z = 0 5 1 $ (0, 0, 1)# 1 *$ $(
C
f(x, y, z) dx dy dz =
10
( 1z0
( 1y2z2
1y2z2f(x, y, z) dx
)dy
)dz; .
6 5( (( * S 5 1# 1 *$ $(
S
f(x, y, z) dx dy dz =
10
( 1z21z2
( 1y2z2
1y2z2f(x, y, z) dx
)dy
)dz; .
+ ) 3
'' J ' 1 1
$
6$ *( $ ( $ * $ $ $
$ 6$#
Rn B $ ( ( B ( #% 8 # 2 (
$ B #
.1 ' x0 A Rn 6
/
A7)' f : A C x x0 V C U Rn x0
f(U A \ {x0} V ) ,-
limxx0 f(x) = > 0 > 0 x A 0 < x x0 < |f(x) | < )' x0 A 6 + f 5 x0 f 5 x0 limxx0 f(x) = f(x0))
,4
A 5( n = 2# $ $ B
!
' f A R2(x0, y0) A) ' f (x, y) (x0, y0)
g(t) = f(x(t), y(t)) f + (x(t), y(t)) 6 0 t 17
(x0, y0) 6
x(t) x0 y(t) y0 t 17 limt1 g(t) = )
Rn)
1# 2 ( $ * 1
1 (x(t), y(t)) (x0, y0) t 1 '' 1 $6$ $ U (x0, y0) t $J $(x(t), y(t))
U # 2 $ #
6 ) (
5( 8 $ $ T$ $ 1 $ (x0, y0)# 2
1 6$ 5( ' $ f $ *$( $ A R2 (x0, y0) A#
(i)lim
(x,y)(x0,y0)f(x) = lim
xx0( limyy0
f(x, y)) = ,
$ ' 5 $ #
P$ $ ' ) 1= $ $ $ 6$ B 6$ (x, y) (x, y0) (x, y0) (x0, y0)#
(ii)lim
(x,y)(x0,y0)f(x) = lim
t0f(x0 + ta, y0 + tb)) =