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L
Q
L Q
L + Q
L1Q
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L Q
L Q
L + Q L1Q
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M[a, f](x)
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Iw[a,f] Iw[a,a M]
R1az Iw[az ,fz] R1az
Iw[az,az Mz]
L1
[a, f
] Q
[a, f
]
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f a
f fP={f(x), x P}
P
RaP[f
P] f
P
aP
aP
fP
P
P
fP
RaP[f
P] a
P
fP
a
P
aP
aP
fP
aP
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aP
aP
fP
RaP
[f
P
]
P
(Pz)zR
Pz ={(y, z) R3, y R2} z R f R3 fz
(y
)=f
(y, z
), y R2 Pz
a
f R3
Pz
Raz[fz](s, )= R
fz(t + s)e 0 az(t++s)ddt s R, S1, S1 ={ R2, = 1} R2
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P
=
0 1
1 0
(s,
) R S1
R2 Raz[fz] z
az fz
M[a, f](x) x R3
M[a, f](x)= S2
0f(x + t)e t0 a(x+)ddtd, S2 ={ R3,= 1}.
Pz
Raz[azMz[a, f]](s, )=
0 az(t+s
)Mz[a, f](t+s
)e
0 az(t++s)d
dt s R, S1
.
M[a, f] Pz
a f R3
Raz
[fz
](s,
) s, z R, S1
Raz
[az Mz
[a, f
]](s,
) s, z R, S1.
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(a, f)
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f Rn
f
(x
)dx<
Rn
f
(x
)2dx< f
Ff()= f()= Rn
eixf(x)dx. f
Ff, =f,F S(Rn), f f Rn Sm R Sm ={x Rm+1,x = 1}
Ff(, )= Rn eixf(x, )dx a.e. Sm.
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f
f
F1f(x)= f(x)= 1
(2
)n
Rneixf(x)dx
f
x Rn
f nxif(x)
F( xj
f)()= ijf() j {1, 2,..n}.
f
g
f g(y)= Rnf(y x)g(x)dx F
(f g
)(
)= F
(f
) F
(g
)(
).
f
Rnf(x)2dx=(2)n
Rnf()2d.
Rn
C0(Rn)={f Rn R, f },C0()={f C0(Rn), (f)={x Rn, f(x) 0} },
C0( Sm)={f Rn Sm R, f (f) Sm} m N.
C(Rn)={f Rn R, f }C(Rn Sm)={f Rn Sm R, f }
C0(Rn)={f C(Rn), f }C0(Rn Sm)={f C(Rn Sm), f }
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Rn
C()={f C(Rn), (f) }C( Sm)={f C(Rn Sm), (f) }
C0
(
)=
{f C
(
), f
}C0( Sm)={f C( Sm), f } Lp Lp(Rn) Lp(Rn) p[1, ]
fLp(Rn) =Rnf(x)pdx1p p[1, ), f = xRn f(x).
Rn, p[1, ] Lp
(
)=
{f Lp
(Rn
), f
(x
)= 0 x
}.
f
K K L2(K) L2(K) Lp Rn p[1, )
C0()Lp(Rn) Lp() p[1, ).
S
(Rn
)=
{f C
(Rn
), sup
xf
< , Nm, m N
}.
j 0 S(Rn) xj 0 Rn
S(Rn) S(Rn) Hs s R f S(Rn) f
f
Hs
(Rn
)=
f
s=
Rn f
(
)2
(1 +
2
)sd
12
.
Rn
Hs(Rn)={f S(Rn),fHs(Rn)
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f(, ) Hs(Rn) a.e. Sm Hs(Rn Sm)
fHs(RnSm) =Smf(, )2sd12 ,
Rn
Hs(Rn Sm)={f S(Rn Sm),fHs(RnSm) 0
H
s(Rn) Hs(RSm) Hs(Rn) Hs(RSm) , Hs(R Sm) f, g= Smf(, ), g(, )d f(, ) Hs(Rn), g(, ) Hs(Rn) Sm f, g CfHs(Rn)gHs(Rn) f Hs(Rn), g Hs(Rn), f, g CfHs(RnSm)gHs(RnSm) f Hs(Rn Sm), g Hs(Rn Sm).
f, g
Sm
f
(,
), g
(,
)d
CSmf(, )Hs(Rn)g(, )Hs(Rn)d
C
Smf(, )2
Hs(Rn)d
Smg(, )2
Hs(Rn)d
=CfHs(RnSm)gHs(RnSm).
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0 < 1 f Rn
f
C(Rn) = sup
x f
(x
)+ sup
xy f
(x
) f
(y
)x y ,
fC(Rn) = supxy
f(x) f(y)x y , Rn
C(Rn)={f C0(Rn),fC(Rn)
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Hf(x)= f P.V. 1x
=1
lim0+
x
f
(t
)x tdt +
x+
f
(t
)x tdt
=
1
lim0+
f(x + t) f(x t)t
dt.
Lp(R) f Lp(R), p > 1
HfLp(R) CpfLp(R)
f L2(R)
F(Hf)()= isgn()f ,
sgn()=
1 >0
0 =0
1
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F1x ()=
R
eix1
xdx
= R
cos
(x
)xdx i
R
sin
(x
)xdx
=0 iR
sin(x)(x)(dx)=i sgn().
Hs(R) f Hs(R) s R Hf Hs(R)
f
Hs(R) =
Hf
Hs(R).
Hs(R) HfHs(R) =
R
(1 + 2)s Hf()2d=
R
(1 + 2)s isgn()f()2d=
R
(1 + 2)sf()2d=fHs(R).
f H1(R) d
dxHf=H
d
dxf R.
L2
F( d
dxHf)()= i(i) sgn()
f()= isgn()if()=F(H d
dxf)().
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f S(Rn) Rf[, ]=
x=f(x)dl(x), Sn1, R,
x = ni=1 xii Rn dl
(x
)
n 1 {x Rn, x = }
f S
(Rn
)
F(Rf)(, )= f(), Sn1, R.
F(Rf)(, )= R
eis[Rf](s, )ds=
R
eisx=
f(x)dl(x)ds=
R
x=
ei(x)f
(x
)dl
(x
)ds
= Rn
ei(x)f(x)dx =F(f)().
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f S(Rn)
jRf(, )= R[xjf](, ) j {1, 2,...n}.
F(jRf(, )) = jiF(Rf(, ))=jif(), R, Sn1.
F(R[xjf](, )) =xjf()=i
(
)j
f
(
)=ijf(), R, S
n1
.
f S(Rn) 2Rf(, )= R[f](, ).
R R g(, ) C0(R Sn1) Rg(x)=
Sn1g(x , )d.
Rf,gL2(RSn1) =f, RgL2(Rn) . f C0(Rn) g C0(R Sn1)
Rf,g
L2(RSn1) =
Sn1
R
Rf
(,
)g
(,
)dd
= Sn1 R x= f(x)g(, )dl(x)dd= Sn1 Rnf(x)g(x , )dxd= Rnf(x) Sn1g(x , )ddx=f, RgL2(Rn) .
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KRn f L2(K) Rf(, ) L2(R)
Rf(, )L2(R) C(K)fL2(Rn) Sn1, C(K) K g C0(Rn) (g) K
1K(x)= 1 x K0 x K ,
Rg
(s,
)2
L2(R
)
=
R x=s
g
(x
)dl
(x
)2
ds
= R
x=s
g(x)1K(x)dl(x)2 ds
R
x=s
(g(x))2dl(x)x=s
12K(x)dl(x)ds
(K)n1Rx=s
(g(x))2dl(x)ds= (K)n1
R2(g(x))2dx
=
(K
)n1
g
2L2(R2)
,
C0(K) L2(K) f L2(K) K Rn f Hk(K), k 0
Rf(, )Hk(R) C(K)fHk(Rn) Sn1, C(K) K
L2
(R
)
g C0(K) isRg(, )L2(R) =R[xig](, )L2(R)
C(K)xigL2(Rn)C(K)gH1(Rn).
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sRg(, )2L2(R) = R
sRg(s, )2ds=
R
n
i=1
2i
sRg
(s,
)2ds
=ni=1RisRg(s, )2ds
=
n
i=1
isRg(, )2L2(R)
n
i=1
C(K)g2H1(Rn)nC(K)g2H1(Rn).
k =1
f Hk(K) g Hk+1(K) xig Hk(K) R[xig](, )Hk(R) C(K)xigHk(Rn) C(K)gHk+1(Rn) k N
f C0(Rn)
f(x)= cn RHn1 Rf
(x
) n
Rn1 Rf(x) n cn =
1
2(2)1n (1)(n2)2 (1)(n1)2 ,
Rf
f C0
f(x)= 1(2)nRn eixf()d =
=
f(x)= 1(2)n 0 Sn1 exf()n1dd
[0, ] R
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n
n1
f(x)=1
2(2)nRSn1 eix
f()n1
dd
=1
2(2)nRSn1 eix 1in1F[n1 Rf](, )dd=
in1
2(2)n1Sn1 12R eixF[n1 Rf](, )=
in1
2(2)n1Sn1 n1 Rf( x, )d=cnR
n1 Rf(x). n
sgn(x)
f(x)= 12(2)nRSn1 eixf() sgn()n1dd
=1
2(2)nRSn1 eixiF[H(Rf)](, )n1dd=
1
2(2)nRSn1eix
i
in1
F
[n1 H
(Rf
)](,
)dd
=in2
2(2)n1 RHn1 Rf(x)
a C0(Rn) x Rn
Sn1
(Ba)(x, )= 0
a(x + t)dt x Rn, Sn1.
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R> 0 x (a),x R xBa = a
Ba = 0 x > R
Ba = R0[a] x < R g(t) = a(x + t) d
dtg(t) = xa(x + t)
a
a(x + t)t= = 0 a
x > R
a
a C(Rn) K Ba(x, ) C(RnSn1) BaC C(K)aC C(K)
Ba(x, ) Ba(y, ) 0a(x + t) a(y + t)dt
0
a
(x + t
) a
(y + t
)(1K
(x + t
)+ 1K
(y + t
))dt
0 aC x +t y t(1K(x + t) + 1K(y + t))dt=aC x y
0(1K(x + t) + 1K(y + t))dt
2aC (K)x y. a C0(Rn) K
Ba(x, ) CKa x Rn, Sn1.
Ba(x, )= 0
a(x + t)dta
01K(x + t)dt
(K)a.
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a H1(R2) C0(R2) K
Ba
(x,
) C
a
H1(R) x R
2, S1.
x= s +
Ba(x, )= 0
a(s + ( + t))dt
R
a(s + t)dt=R[ a ](s, ).
H1(R) L(R) Ba(x, ) CR[ a ](, )H1(R) x R2, S1
Ba(x, ) C(K) a H1(R2) x R2, S1 a H1(Rn) aH1(R2) Ba(x, ) C(K)aH1(R2), x R2, S1.
a, f C0(R2)
Raf(s, )= R
f(s + t)e(Ba)(s+t,)dt, s R, S1. f(x)
s e(Ba)(s+t,)
s + t
Raf(s, )= x=s
f(x)e(Ba)(x,)dl(x), s R, S1 Raf(s, )= Raf(s, ) Raf(s, )=Raf(s, )
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a, f C1(R2)
f
(x
)=
1
4
S1eBa(x,
)
(ehHehRaf
)(x ,
)d,
h(s, )= 12(I+ iH)Ra(s, )
Raf
Raf Gg(s, )= g(s, )
f(x)= 14
S1
eBa(x,)(ehHehGRaf)(x , )d.
f L2(R2) w C0(R2 S1)
Iwf(s, )= Rw(s
+ t,)f(s
+ t)dt. f Iw(, ) L2(R)
K R2 f L2(K) w C0(Rn Sn1) KR2
Iwf(s, )= 1K(s)Iwf(s, ) s R, Sn1. R> 0 KB
(0, R
) f
(x
)= 1B(0,R)
(x
)f
(x
)
Iwf(s, )= Rw(s + t,)f(s + t)1B(0,R)(s + t)dt, s> R t R s + t B(0, R)
Iwf(s, )= 1[R,R](s)Iwf(s, ).
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0 k < 12 k +12 < 1 f Hk(R2) w(x, ) C(R2 S1)
IwfHk+12(RS1) CwC(R2S1)fHk(R2).
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u R3 S2 R
x R3 S2
xu(x, )= 0 x R3, S2 xu(x, )
u(x + s,)= u(x, ), s R, x R3, S2
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xu(x, )= a(x)u(x, ), a
(x
) R3 R
u(x, )= Ce0 a(x+t)dt C R x R3 {, 1, 2} R3 x s, 1, 2 R x= s + 1
1 + 2
2
xu(x, )= su(s + 11 + 22 , )=s
Ce
0
a((s+t)+1
1+2
2)dt
=s Ces a(t+11+22)dt=Ces a(t+11+22)dt (a(s + 11 + 22))= Ce0 a(x+t)dta(x)= u(x, )a(x).
xu(x, ) + a(x)u(x, )= f(x) x R3, S2,limt
u(x t,)= 0 x R3, S2, f R3 R
f
a
R3 R f(x + t)dt< x R3, S2
u(x, )= 0
f(x + t)e0t a(x+s)dsdt.
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x= + {w R3, w = 0}, R xu(x, )= u( + , )
xu
(x,
)=
0
f
(( + t
) +
)e
0
t a((+s)+)dsdt
=
f(t + )e 0ta((+s)+)dsdt=
f(t + )e t a(s+)dsdt
0g(t, )dt= g(, ) +
0g(t, )dt
xu(x, )= f( + )e a(s+)ds +
f(t + )e t a(s+)ds(a( + ))dt=f(x) a(x)u(x, ).
x= + ()= u( + , )
() + a( + )()= f( + )lim
()= 0,
s(x,,) x R3 S2
S2
x R3 S2
xu
(x,
)+ a
(x
)u
(x,
)+
S2u
(x,
)s
(x,,
)d =f
(x
)+
S2u
(x,
)s
(x, ,
)d
limt
u(x t,)= 0.
S2u(x, )s(x,,)d
(x, ) S2u(x, )s(x, , )d
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x
x
x
x R3
(x, ) s(x,,) =s(x, )
xu(x, ) + a(x)u(x, ) +S2
u(x, )s(x, )d =f(x) +S2
u(x, )s(x, )d xu(x, ) + a(x)u(x, ) + u(x, )
S2s(x, )d =f(x) +
S2u(x, )s(x, )d
xu
(x,
)+ aT
(x
)u
(x,
)= f
(x
)+
S2u
(x,
)s
(x,
)d x R3, S2
aT(x)= a(x) +S2 s(x, )d. ui(x, ) i
u(x, )= i=0
ui(x, ),
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xu0(x, ) + aT(x)u0(x, )= f(x) x R3, S2 xui
(x,
)+ aT
(x
)ui
(x,
)=
S2
s
(x,
)ui1
(x,
)d i 1, x R3, S2
limt
ui(x t,)= 0, i 0, x R3, S2
s(x, ) = s(x)k( ) s(x) C Cs(x)=a(x) k( )= 14 [0, 1]
aT(x)= a(x) +S2
s(x)k( )d =s(x)(C+ 1) C=
(4
(1 + C
))1
xu0(x, ) + aT(x)u0(x, )= f(x) x R3, S2 xui(x, ) + aT(x)ui(x, )= CaT(x)
S2ui1(x, )d i 1, x R3, S2
limt
ui(x t,)= 0. i 0, x R3, S2
f a D > 0, R f(x+ t)dt 0
yBaz(y, )= tBaz(t + s, )= az(t + s),
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Baz(y, )= 0 y > R Baz(t + s, )= 0 t> R, Baz(y, )= R0[az](y , ) y < R
Baz
(t + s,
)= R0
[az
](s,
)= R az
( + s
)d t< R.
Raz+az fz(s, )= R
eB[az+az](t+s,)fz(t + s)dt,
d
d=0
Raz+az fz(s, )= R
eBaz(t+s,)fz(t + s) (B[a](t + s, ))dt.
eBaz(t+s)fz(t + s) = tB[eBaz(,)fz()](t + s, ),
dd =0Raz+az fz(s, )= R tB[eBaz(,)fz()](t + s, )B[az](t + s, )dt,
d
d=0
Raz+az fz(s, )= R
B[eBaz(,)fz()](t + s, )tB[az](t + s, )dt+ B[eBaz(,)fz()](t + s, )B[az](t + s, )t=
t=
d
d=0
Raz+az fz(s, )= R
B[eBaz(,)fz()](t + s, )az(t + s)dt
R
eBaz(+s,)fz( + s)d
R
az(t + s)dt=
R
B[eBaz(,)fz()](t + s, ) R
eBaz(+s,)fz( + s)d az(t + s)dt.
B[eBaz(,)fz()](t + s, ) R
eBaz(+s,)fz( + s)d
=
0eBaz(t++s
,)fz(t + + s)d R
eBaz(+s,)fz( + s)d
= 0
eBaz(t++s,)fz(t + + s)d,
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d
d=0
Raz+az fz(s, )= R
0
eBaz(t++s,)fz(t + + s)d az(t + s)dt,
w
[az, fz
](y,
) = 0 eBaz(y+,)fz
(y +
)d,
d
d=0
Raz+az fz(s, )= Iw[az,fz]az(s, ).
a f (a,f) DRaz[fz](az, fz)= Iw[az ,fz][az] + Raz[fz].
Raz[az Mz[a, f]] a()a a
a(Raz(az Mz[a, f]))a = Iw[az ,az Mz[a,f]az + Raz(az Mz[a, f]) + Raz(az aMz[a, f]a). a f
(a,f
)
DRaz(az Mz[a, f])(a,f)=Iw[az ,az Mz[a,f]]az + Raz(az Mz[a, f])+ Raz(az aMz[a, f]a) + Raz(az M[a,f]).
aM[a, f]a
a, f
a
f
a = a a
f= f f
M=Mz[a, f] Az0 Raz(fz) Iw[az ,fz][az] + Raz[fz],
Az1 Raz(az M) Iw[az ,az Mz]az + Raz(az Mz) + Raz(az a Mza) + Raz(az Mz[a,f]).
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Pz
R1az
R1azAz0 Raz(fz)R1az Iw[az,fz][az] + fz,R1azAz1 Raz(az M)R1az Iw[az,az Mz]az + az Mz + az a Mza + az Mz[a,f]. K R (f) (a) (a) (f) K3 =
K K K (fz) (az) (az) (fz)K2 =K K zR zKc, fz, az, az, fz =0
R1az
z K K K K
(y)= 1, y K2 1 Az0 = R
1az
Iw[az ,fz][az] + fz,Az1 = R
1az
Iw[az ,az Mz]az + az Mz + az a Mza + az Mz[a,f].
Az0 Az1 A
z0 A
z1
R1azAz0
fz Az0
R1azAz1 az
Mz Az1.
K2 () zRLz[a, f], Qz[a, f] L2(K3) L2(K3) L2(K2) L2(K2),
Lz[a, f](a,f)= R1az Iw[az,fz][az] + fzaz Mz
,Qz[a, f](a,f)= 0R1az Iw[az,az Mz][az] + (az a Mza) + (az Mz[a,f]) .
a,f L2
(K3
), K3 R3 az, fz
z R
az, fz
L
2
(K2
) Lz, Qz z R a,f L2(K3)
(Lz + Qz)[a, f](a,f)= Az0Az1
zR.
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a,f
a,f L2(K3) L2(K) L2(K)
L
[a, f
], Q
[a, f
] L2
(K3
) L2
(K3
) L2
(K; L2
(K2
) L2
(K2
)),
L[a, f](a,f)=Lz[a, f](a,f)zR ,Q[a, f](a,f)=Qz[a, f](a,f)zR .
(L + Q)[a, f](a,f)= Az0Az1
zR
.
L Q
(L + Q)[a, f] a, f C0(K3) C>0 M[a, f](x) C x K3
L[a, f]
L1[a, f] L2(K; L2(K2) L2(K2)) L2(K; L2(K2) L2(K2)) L2(K3) L2(K3)
L1
[a, f
] g
h=
hz
Mz, gz R
1az
Iw[az,fz]
[hz
Mz
]zK
C>0 M[a, f](x) C x K3 f0 f0 a f M[a, f]
3.5
a, f C0(K3) Mz(y), az(y) fz(y) zK, y K2 L1z
[a, f
]
L1z[a, f] L2(K2) L2(K2) L2(K2) L2(K2)
L1z[a, f] gzhz =hz Mz, gz R1az Iw[az ,fz][hz Mz] gz, hz L2(K2) M[a, f](x) C>0 x K3 M z K az, fz az, fz L2(K2)
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L1z[a, f]Lz[a, f](a,f)= L1z[a, f] R1az Iw[az ,fz][az] + fzaz Mz =
az
Mz
Mz, fz +
R1az Iw[aZ,fz]
[aZ
]
R1az Iw[az ,fz]
[az
Mz
Mz
]=(az, fz)zR=(a,f)
L1[a, f] gh=L1z gzhz
zK
,
L1[a, f]L[a, f](a,f)=(az, fz)zK=(a,f). z R
f 0 f0 f >0 A
f(x) 1A(x) f2
x K3
M[a, f](x)= S2
0f(x + t)et0 a(x+s)dsdtd
ediam(K3)a
S2
0f(x + t)dtd
ediam(K3)a
f
2 S2
0
1A
(x + t
)dtd
ediam(K3)a
f2
minxK3
S2
01A(x + t)dtd
>0.
x
a= 0 f C0
(K3
) C>0 M
(x
) C x K3 Q
[a, f
]= 0
(L + Q
)[a, f
]
L[a, f] Q[a, f]= 0 a= 0 az =0
az a Mza = 0,
az Mz[a,f]= 0,
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w[az, az Mz]= w[0, 0]= 0, Iw[0,0]az(s, )=
R
w[0, 0](s + t,)az(s + t)dt= 0. Qz
[a, f
] Qz
[a, f
]= 0 Q
[a, f
]=
Qz[a, f]zK=0 K, K R K K a H52(K3) f C(K3) > 12 f0 f0 C0(K2) K L1 Q
L1[a, f] L2(K; L2(K2) L2(K2)) L2(K; L2(K2) L2(K2))Q[a, f] L2(K3) L2(K3) L2(K; L2(K2) L2(K2))
D >0
(L + Q
)[a, f
] L2
(K3
) L2
(K3
)
a H52
(K3
)
a
H52
(R3
)
< D
(L + Q)1[a, f]= L1[a, f] k=0
(QL1)[a, f]k .
a f K
H52
(K3
) H52
(K3
) C
(K3
)
C
(K
3
) I
w[a,f] Iw[a,a M]
Rn f C() z R fz(y) =f(y, z), y Rn1 z ={y Rn1,(y, z) } fz C(z) fzC(Rn1) fC(Rn),fzC(Rn1) fC(Rn).
fz f fz(y1) fz(y2) =f(y1, z) f(y2, z)fC (y1, z) (y2, z)fC y1 y2.
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RN f1, f2 C() f1 f2 C() f1 f2C(Rn) 2f1C(Rn)f2C(Rn),
f1 f2
C(Rn) 3
f1
C(Rn)
f2
C(Rn).
f1f2(x) f1f2(y)=f1(x)(f2(x) f2(y)) f2(y)(f1(y) f1(x))f1f2C x y + f2f1C x y2f1C f2C x y,
f1f2 f1f2 f1C f2C,
f1f2C(Rn) 3f1C(Rn)f2C(Rn). K R a, f C(K3) M[a, f] C(R3)
M[a, f]C(R3) C(K)eC(K)a(1 + aC(R3))fC(R3), C
(K
)
K
a f
M[a, f](x)= S2
0f(x + t)e t0 a(x+s)dsdtd.
at,(x)= e t0 a(x+s)ds at,(x) at,(y) eC(K)a t
0a(x + s) + a(y + s)ds
eC(K)a
t
0 a
C
x y
1K3
(x + s
)+ 1K3
(y + s
)ds
C(K)eC(K)a aC x y, at,(x) eC(K)a
at,C(R3) C(K)eC(K)a(1 + aC(R)).
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ft,(x) = f(x + t) f
M[a,f](x) M[a, f](y)
S2
0 ft,at,
(x
) ft,at,
(y
) (1K3
(x + t
)+ 1K3
(y + t
))dtd
2S2
0fC at,C(1K3(x + t) + 1K3(y + t))dtdx y
C(K)eC(K)a(1 + aC)fC S2
0(1K3(x + t) +1K3(y + t))dtdx y
C(K)eC(K)a(1 + aC)fC x y, M
M[a, f](x) C(K)eC(K)a f C(K)eC(K)a fC,
M[a, f]C(R3) C(K)eC(K)a(1 + aC)fC.
K R a, f C(K3) w[az, az Mz], w[az, fz] C(R2 S1) z K
w[az, fz]C(R2S1) C(K)eC(K)a(1 + aC(R3))fC(R3),
w
[az, az Mz
]C(R2S1) C
(K
)eC(
K)a
(1 +
a
C(R3)
)2
a
C(R3)
f
C(R3),
C(K) K w[az, fz]
w[az, fz](y, )= 0
e
0 az(y+t+)dfz(y + t)dt y R2, S1.
w[a, f]C(R2S1) 2C(K)eC(K)az (1 + azC)fz C,
3.6
w[a, f]C(R2S1) C(K)eC(K)a(1 + aC)fC. w[az, az Mz]= w[az, az Mz[a, f]]
w[a, a M]C(RS1) C(K)eC(K)a(1 + aC)a MC(R3),
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w[a, a M]C(RS1) C(K)eC(K)a(1 + aC)aC MCC(K)eC(K)a(1 + aC)2aC fC.
K R a, f C(K3) > 12 z K
Iw[az ,fz], Iw[az ,az Mz[a,f]] L2(K2)H12(R S1),
az L2(K2)Iw[az ,fz][az]H12(RS1) C(K)eC(K)a 1 + aC(R3) fC(R3)az L2(R2),Iw[az ,az Mz[a,f]][az]H12(RS1) C(K)eC(K)a 1 + aC(R3)2 aC(R3)fC(R3)azL2(R2).
w[az,fz], w[az, az Mz[a,
f]] C
(R2
S1
) az
Iw[az ,fz][az]H12(RS1) w[az, fz]C(R2S1)azL2(R2)C(K)eC(K)a 1 + aC(R3) fC(R3)az L2(R2),
Iw[az ,az Mz[a,f]][az]H12(RS1) w[az, az Mz[a, f]]C(R2S1)azL2(R2)C(K)eC(K)a(1 + aC(R3))2aC(R3)fC(R3)az L2(R2).
R1
azIw[az,fz]
R1
azIw[az,az Mz]
KR a C(K3) z K kz(y, ) = eBaz(y,), y R2, S1 kzC(R2S1) C(K)eC(K)a 1 + aC(R3) z K C(K) K
kz(x, ) kz(y, ) C(K)eC(K)az azC(R2)x y, kz(, ) eC(K)az az a
kzC(R2S1) C(K)eC(K)a(1 + aC(R3)) z K.
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f H12(R) g Hs(R), s> 1 f g H12(R) f gH12(R) CgHs(R)fH12(R).
f gL2(R) fL2(R)gL CfH12(R)gHs(R), C Hs(R) L(R)
12f g() L2(R)
12f g()2 = 12 [f g]()2=
Rf
( t
)g
(t
)dt
2
= R
f( t) 1(1 + ts)(1 + ts)g(t)dt2
R
f( t) 1(1 + ts)2
dtR
(1 + ts)g(t)2 dt=g2Hs(R)
R
f( t)2 (1 + ts)2 . L2
12f g2L(R)
= R
12f g()2 dg2Hs(R)
RR
f( t)2 (1 + ts)2 dtd( y + t)=g2Hs(R)
RR
f(y)2y + t(1 + ts)2 dtdyg2Hs(R)
R
f(y)2 R
y + t(
1 +
t
s
)2
dtdyg2Hs(R)CR f(y)2(y + 1)dyCg2Hs(R)
R
f(y)2(y 12 + 1)2dyCg2Hs(R)f2H12(R),
(1 + ts)2 t(1 + ts)2 s> 1
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K R a H52(K3) hz(s, ) =12(I+ iH)Raz(s, ) K (s)W2, + (s)W2,2 C(K) C(K) K
(
)ehz(,) H2
(R
) z K
()ehz(,)H2(R) C(K)eC(K)a 1 + aH52(R3)2 S1, z K.
a H52(K3) az H2(K2) azH2(R2) CaH52(R3) Raz(, ) H2(R) Raz(, )H2(R) C(K)azH2(R2) C(K)aH52(R3) S1, z K,
Hk(Rn) hz(s, )= 12(I+ iH)Raz(s, )
hz(, )H2(R) C(K)aH52(R3) S1, z K. ehz(s,) (s)
()ehz(,) H2(R)
(s), s(s), 2s(s) C(K) s R,L2(R), sL2(R), 2sL2(R) C(K),
ehz(s,)
= e12Raz(s,) eC(
K)a s R, S1,
shz(s, )shz(, )H1(R) hz(, )H2(R) C(K)aH52(R3) s R, S1. L L2(R) (s)ehz(s,)
()ehz(,)L2(R) eC(K)a L2(R) C(K)eC(K)a,s ()ehz(,) L2(R) s()ehz(,)L2(R) + ()ehz(,)shz(, )L2(R)C(K)eC(K)a sL2(R) + shz(, )L2(R)C(K)eC(K)a 1 + aH52(R3) ,
2s
(
)ehz(,)
L2(R)
2s
(
)ehz(,)
L2(R) + 2
s
(
)ehz(,)shz
(,
)L2(R)
+ ()ehz(,) (shz(, ))2 L2(R) + ()ehz(,)2shz(, )L2(R)C(K)eC(K)a 2sL2 + shz(, )L2+(shz(, ))2L2 + 2shz(, )L2C(K)eC(K)a 1 + aH52(R3)2 .
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f gL2 fL gL2 f2L2 fL fL2
()ehz(,)H2(R) C(K)eC(K) 1 + aH52(R3)2 S1, z K. h
K R a H52(K3) J H12(R S1) C0(R2), C0(R) K R1az [J]L2(R2) C(K)eC(K)a 1 + aH52(R3)5 JH12(RS1) z K.
J=J z K R1az
G
hz(s, ) = 12(I + iH)Raz(s, ) I H s
L2
(R2
) R1az
[J
] g C0
(R2
)
R1az [J], gL2(R2)=
1
4 1
Se(Baz)(y,
) ehzHehz J (y , )d, gL2(R2)
=1
4 1
Se(Baz)(y,
) ehzHehz J (y , )d,gL2(R2)
,
g =g gL2 gL2 (g) K g g(y)= B(0,R)(y)g(y) R> 0 K2 (g)B
(0, R
) B(0,R) C
0
(R2
) B(0,R)
(y
)= 1y B
(0, R
)
y(g(y)B(0,R)(y)) = yg(y)B(0,R)(y) +
0g(y)yB(0,R)(y).
R1az [J], gL2(R2) = 14 S1 B(0,R)(y)e(Baz)(y,) ehzHehz J (y , )d, g(y)L2(R2) , B(0,R) C
0
(R
)
B(0,R)(y)= (y ) y R2, S1, Fz(s, )= (s)ehzHehz J(s, )
R1az [J], gL2(R2) = 14 R2 S1 eBaz(y,)Fz(y , )d g(y)dy.
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R2
R1az [J], gL2(R2) = 14 S1 Ry=s yg(y)e(Baz)(y,)dl(y)Fz(s, )dsd=
4y=s
(g
(y
)e(Baz)(y,
)
)dl
(y
)
y=sg
(y
) ye
(Baz)(y,)dl
(y
), Fz
L2(RS1),
L2(R2) Cy=s
y(g(y)e(Baz)(y,))dl(y), FzL2(RS1)
+ C
y=sg(y) ye(Baz)(y,)dl(y), Fz
L2(RS1)
.
yeBaz(y,) =eBaz(y,
)B
[ az
](y,
).
C(K) K
yeBaz(y,) C(K)eC(K)az B[ az](y, )C(K)eC(K)az azH1(R2)C(K)eC(K)az azH2(R2) y R2, S1.
S1 Ry=sg
(y
) ye
Baz(y,)dl
(y
)Fz
(s,
)dsd
S1Ry=s
g(y) yeBaz(y,)dl(y)Fz(s, )dsdC(K)eC(K)az azH2(R2)
S1Ry=s
g(y)dl(y)Fz(s, )dsdC(K)eC(K)az azH2(R2) Rg(s, ), Fz(s, )L2(RS1)C(K)eC(K)az azH2(R2)Rg(s, )L2(RS1)Fz(s, )L2(RS1)C(K)eC(K)az azH2(R2)gL2(R2)Fz(s, )L2(RS1),
y
y=s
(g(y)e(Baz)(y,))dl(y), FzL2(RS1)
=sy=s
g(y)e(Baz)(y,)dl(y), FzL2(RS1)
.
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kz(y, )= e(Baz)(y,)
Hs
s
y=s
g
(y
)e(Baz)(y,
)dl
(y
), Fz
L2(RS1)=
sIkz g(
s,
), F
(s,
) L2(RS1)
sIkz gH12(RS1)FH12(RS1)Ikz gH12(RS1)FH12(RS1).
a H52(R) a C(K3) > 12 kz C(R2 S1)
Ikz gH12(RS1) C(K)kzC(R2S1)gL2(R2)C
(K
)eC(
K)a
1 +
a
C(R3)
g
L2(R2).
g
aC(R3) aH52(R3)
y=s (g(y)e(Baz)(y,))dl(y) C(K)eC(K)a 1 + aH52(R3) gL2(R2).
R1az J , g C(K)eC(K)a 1 + aH52(R2) gL2(R2)FzH12(RS1). Fz(s, )= (s)ehz(s,)Hehz(s,)(s)J(s, ) ()ehz(,)
H2(R) Fz(s, )2H12(RS1) =
S1()ehz(,)Hehz(,)()J(, )2
H12(R)d
S1()ehz(,)2H2(R)Hehz(,)()J(, )2H12(R)d
=
S1
(
)ehz(,)
2H2(R)
ehz(,)
(
)J
(,
)2
H
12
(R
)
d
C(K)eC(K)a 1 + aH52(R2)4S1ehz(,)(s)J(, )2
H12(R)d
C(K)eC(K)a 1 + aH52(R2)8S1J(, )2
H12(R)d
=C(K)eC(K)a 1 + aH52(R2)8 J2H12(RS1),
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R1az J,gL2(R2) C(K)eC(K)a 1 + aH52(R3)5 JH12(RS1)gL2(R2)C
(K
)eC(
K)a
1 +
a
H52(R3)
5
J
H12(RS1)
g
L2(R2),
C(K) K z g g C0(R2) g L2(R2) z K z K
R1az JL2(R2) eC(K)a 1 + aH52(R3)5 JH12(RS1) z K. K R a H52(K3) aH52(R3) < D f C(K3) > 12 C0(K2) K
R1
azIw[az ,fz]L(L2(K2),L2(K2)) C(
K, D)fC(R3) z
K
R1az Iw[az ,az Mz[a,f]]L(L2(K2),L2(K2)) C(K, D)aH52(R3)fC(R3) z K, C(K, D) K D D
az L2(K2) Iw[az,fz], Iw[az ,az Mz[a,f]] H12(RS1) B R
Iw[az ,
fz][az
](s,
)= 1B
(s
)Iw[az,
fz][az
](s,
) s, zR, S1
, Iw[az ,az Mz[a,f]][az](s, )= 1B(s)Iw[az,az Mz[a,f]][az](s, ) s, z K, S1,
R1az Iw[az,fz][az]L2(R2) C(K)eC(K)a 1 + aH52(R3)5 Iw[az,fz][az]H12(RS1),C(K)eC(K)a 1 + aH52(R3)5 1 + aC(R3) fC(R3)azL2(R2),
R1az Iw[az,az Mz[a,f]][az]L2(R2) C(K)eC(K)a 1 + aH52(R3)5 Iw[az,az Mz][az]H12(RS1),C
(K
)eC(
K)a
1 +
a
H52(R3)
5
1 +
a
C(R3)
2
a
C(R3)
f
C(R3)
az
L2(R2).
aC(R3) CaH52(R3) >12 a aH52(R3) aH52(R3 D
R1az Iw[az ,fz][az]L2(R2) C(K, D)fC(R3)azL2(R2) z K,R1az Iw[az ,az Mz[a,f]][az]L2(R2) C(K, D)aH52(R3)azL2(R2) z K.
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K2
() K2 C(K, D)
D
L1[a, f] Q[a, f]
KR a H52(K3) f C(K3) > 12 C0(K2) K aH52(R3)
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az a Mz[]az a Mza(y) =az(y)
S2
0f((y, z) + t)et0 a((y,z)+)d t
0a((y, z) + s)dsdtd
a
1K3
(y, z
)S2
0 f
((y, z
)+ t
)e
t0 a((y,z)+)ddt
R a
((y, z
)+ s
)dsd.
K3 (y, z) ,t,s [ (K3), (K3)] = J(K) az a Mza(y) C(K)aeC(K)a f1K3(y, z)
S2J(K)
a((y, z) + s)dsd, a CaH52(R3) C D
f fC(R3)
az a Mza
(y
) C
(K, D
)a
H52(R3)
f
C(R3)1K3
(y, z
)S2J(
K)
a
((y, z
)+ s
)dsd,
L2
az a Mza2L2(R2) C(K, D)a252f2C1K(z)K2
S2J(K)
a((y, z) + s)dsd2 dyC(K, D)a252f2C1K(z)
K2S2J(K)
(a((y, z) + s))2 dsddy. z
R
az a Mza
2L2(R2) C
(K, D
)a
252
f
2C
K
K
2S2J(
K) (
a
((y, z
)+ s
))2
dsddydz
=C(K, D)a252f2C S2J(K)
KK2(a((y, z) + s))2 dydzdsd
=C(K, D)a252f2C S2J(K)
a2L2(R2)dsdC(K, D)a2
H52(R3)f2C(R3)a2L2(R2).
az Mz[a, ]
az Mz[a,f](y)=a(y, z)S2 0 f((y, z) + t)e 0 a((y,z)+)ddtdeC(
K)a a1K3(y, z)S2J(K)
f((y, z) + t)dtdC(K, D)aH52(R3)1K3(y, z)
S2J(K)
f((y, z) + t)dtd.
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L2 y
az Mz[a,f]2L2(R2) C(K, D)a2521K(z)K2
S2J(K)
f((y, z) + t)dtd2 dyC
(K, D
)a
2
52
1 K(
z
)K2 S2 J(K) (f
((y, z
)+ t
))2
dtddy,
zR
R
az Mz[a,f]2L2(R2)dz C(K, D)a252KK2S2J(K)
(f((y, z) + t))2 dtddydz=C(K, D)a252
S2J(K)
KK2(f((y, z) + t))2 dydzdtd
C(K, D)a2H52(R3)
fL2(R3).
Q
Q[a, f](a,f)L2(R;L2(R2)L2(R2)) = R
Qz[a, f](a,f)2L2(R2)L2(R2)dz
R
R1az Iw[az,az Mz][az]2L2(R2) + az a Mza2L2(R2) + az Mz[a,f]2L2(R2)dzC(K, D)a2
H52(R3)f2C(R3)a2L2(R3) + f2L2(R3) a,f L2(K3),
Q
[a, f
] C
(K, D
)a
H52(R3)
(1 +
f
C(R3)
).
Q L2(K, L2(K2) L2(K2)), z K Qz[a, f](a,f) = 0 a,f L2(K2) Qz[a, f](a,f)L2(K2)2 z K R a C0(K3), f C(K3) f 0 f0
M
[a, f
](x
) C
(K
)eC(
K)a ff
C(R3)
3
f
x K
3.
f
x K3 f(x)=f > 0 A={x K3, f(x) f
2},
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f
f(x) f(y)fC x y y Ac
f
2
f
C
x y
y Ac
R =f2fC
1
(x, Ac). fC 0 f f0 B(x, R) A
f(x) f2
x B(x, R)f(x) f
2 1B(x,R)(x) x K3.
M x K3
M[a, f](x)= S2
0f(x + t)et0 a(x+s)dsdtd
eC(K)a
S2
0f(x + t)dtd
eC(K)a
f2
S2
01B(x,R)(x + t)dtd,
t > (K3) (x + t) B(x, R)x K3, S2 B(x, R) K3
M[a, f](x) eC(K)a f2
S2
(K3)
01B(x,R)(x + t)dtd
=eC(K)a
f2
S2
(K3)
01B(x,R)(x + t)t2
t2dtd
eC(K)a
f2 (K3)2S2 (
K3)
01B(x,R)(x + t)t2dtd
=eC(K)a
f2
(K3
)2R31B(x,R)(x)dx
=eC(K)a
f
2 (K3)2 4R3C(K)eC(K)a ffC(R3)
3 f x K3.
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K R a H52(K3) f C(K3) > 12 f0, f0 C0(K2)
K aH52(R3) < D
L1
[a, f
] L2
(K2 K
) L2
(K2 K
) L2
(K; L2
(K2
) L2
(K2
)),
L1[a, f] gh=L1z[a, f] gh
zR
,
L1z
L1z[a, f] L2(K2 K) L2(K2 K) L2(K2) L2(K2)L1z
[a, f
] g
h
=
hz
Mz, gz R
1az
Iw[az ,fz]
[hz
Mz
],
L1[a, f] 1 + C(K, D)(1 + fC(R3)) f3C(R3)f3+1 , g, h L2(K3) gz, hz L2(K3) z R
L1z
[a, f
] g
h
2
(L2(R2))2
hz
Mz
2L2(R2) +
gz
2L2(R2) +
R1a Iw[a,f]
[hz
Mz
]2L2(R2).
R1az Iw[az ,fz][hz Mz]L2(R2) C(K, D)fC(R3)hz MzL2(R2) z K, z K zR
R1az Iw[az ,fz][hz Mz]L2(R2) C(K, D)fC(R3)hz MzL2(R2) z R.
hz MzL2(R2) C(K)eC(K)a fC(R3)f 3
1f hzL2(R2)C(K, D)fC(R3)f
3
1f hzL2(R2),
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z R
L1z[a, f] gh 2(L2(R2))2 gz 2L2(R2) + 1 + C(K, D)fC hz MzL2(R2)gz 2L2(R2) + C(K, D)(1 + fC)
f
3C
f3+1 hzL2(R2). R gz2L2(R2) =R R2g(y, z)2dydz =gL2(R3) hz L1[a, f] zL1[a, f] g
h L2(R;L2(R2))=
R
L1z[a, f] gh 2(L2(R2))2dz
R
gz2L2(R2) + C(K, D)(1 + fC)f3C
f
3+1
hzL2(R2)dzgL2(R3) + C(K, D)(1 + fC)f3Cf3+1 hL2(R3),
L1[a, f] 1 + C(K, D)(1 + fC(R3)) f3C(R3)f3+1 . L1
[a, f
] L2
(K; L2
(K2
)
L2
(K2
)) L2
(K3
) z K gz, hz, fz 0
L1z[a, f] z K gz, hz, L2(K2) L1z[a, f] K2 K2
K R f C(K3) > 12 f0, f0 C0(K2) K D> 0 (L+Q)[a, f] L2(K3)L2(K3) a H52(K3), aH52(R3) 0 aH52(R3) D (QL1)[a, f] < 1.
aH52(R3) D
Q
[a, f
] C
(K, D
)a
H52(R3)
(1 +
f
C(R3)
),
L1[a, f] 1 + C(K, D)(1 + fC(R3))
f3
C(R3
)f3+1C(K, D)(1 + fC(R3))1 +
f3C(R3)f3+1
, C(K, D)> 0 K D
(QL1)[a, f]Q[a, f]L1[a, f]C(
K, D)aH52(R3)(1 +
fC(R3))
2
1 + f
3)
C(R3)
f3+1 . C(K, D)> 0 D D > 0
D 12 K R a,f L2(K3) F[a + a, f + f] F
F H52(K3) C(K3) Y =(Y1 Y2)zR (Y1 Y2)zR
F
[a + a, f+ f
]= F
[a, f
]+ DF
[a, f
](a,f
)+ o
(a,f
).
K R K K C(K2) K
(Y1 Y2)zR R1a[(f, g)z]=R1az fz, R1az gzzR)
R1aF[a + a, f + f] F[a, f]= R1aDF[a, f](a,f) + R1a[o(a,f)]=(L + Q)[a, f](a,f) + R1a[o(a,f)],
L Q f 0, f
0
a
H52
(R3
)
=(L + Q)1[a, f] R1a[o(a,f)] (L + Q)1[a, f] R1aF[a + a, f+ f] F[a, f]=(az, fz)zR +
L1[a, f] k=0
(QL1)[a, f]k R1aF[a + a, f+ f] F[a, f]=(az, fz)zR + .
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(a,f)L2(R3)L2(R3) 0 R1a o(a+f)L2(R;L2(R2)L2(R2))
0 a,f L2(K3)
DF[a, f] DF[a, f]1 =L1[a, f]
k=0
(QL1)[a, f]k R1a .
a, f
F[a, f] F KR a, f H52(K3) GK L2(K3) H52(K3) a, f L2(R3) GK(a, f) =(GK(a), GK(f))
F L2(K3) L2(K3) Y =(Y1, Y2)zRF(a, f)= F[GK(a, f)] F[a, f].
a, f L2
(K3
) F
[GK
(a, f
)]=
F[a, f] F GK(a, f) =(a, f)
F
DF[GK(a, f)] (DF[GK(a, f)])1 =L1
k=0
(QL1)k[GK(a, f)]R1GK(a). F
DF(a, f)= DF[GK(a, f)] GKDF(a, f)1 =G1K (DF[GK(a, f)])1
=G1KL1
k=0
(QL1)k[GK(a, f)]R1GK(a),
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(an+1, fn+1
)=
(an, fn
) G1KL
1
k=0
(QL1
)k
[GK
(a, f
)]R1GK(a)F
(GK
(an, fn
)).
GK(an) GK(fn) n
GK GK(a) = a a
I+
(
)54
a= a x K3
a=
0
x
K
3
,
> 0 ()p
a f R3
zR
Az0 =Raz[fz], z R
Az1 =CRaz[az Mz[a, f]].
a0
R3
f0
a1 ai fi
i N
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i N fi ai
A0
fi
(y, z
)= R1
aiz
[Az0
](y
) y R2, z R.
ai =a
A0
f
i N ai ai+1
A1 fi M[ai, fi] ai+1
ai+1(y, z)= R1aiz [ 1CAz1](y)M[ai, fi](y, z) y R2, zR.
ai =a fi =f A1 ai+1 =a
ai, fi 0 f > 0 M[ai, fi](y, z) C>0 y R2, zR ai fi ai ai+1
T[ai](y, z)= R1aiz [ 1CAz1](y)Mai,R1
aiz[Az0]
zR
(y, z) y R2, z R.
T
C
[1, 1
]
[1, 1
]
[1, 1
]
128 128 128 2128 2128 2128 2 2
128 ==H= 128 H = ={(2j 1)128 1, i {1, 2, ..128}}, ={2j128, j {0, 1, 2, ..127}}
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0 0.1
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0 0.2
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L2
L2
z
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L2
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L2
L2
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az fz
= =H =128
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C
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F
[a, f
](s,,z
)=
Raz
[fz
](s,
)CRaz[
azMz[a, f
]](s,
) s, zR, S1
F
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