. .
. . . e-mail:
[email protected] 517.522
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Abstract G. A. Akishev, Generalized Haar system and theorems of embedding into symmetrical spaces, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 319{334.
We prove Nikolski type inequalities for polynomials with respect to a generalized Haar system and the embedding conditions of some classes into symmetric spaces.
x
1.
X 0 1]
, 1) , jf(t)j 6 jg(t)j #$ 0 1] g 2 X, $ , f 2 X kf kX 6 kgkX 2) f 2 X jf(t)j jg(t)j $ , g 2 X kf kX = kgkX ( . 1]). ($ ) $ ) * kkX + 2 X. ,
# Lq , 1 6 q < +1, . , , / ( . 1]). , ) e (t) | 1 e 0 1]. 2 '(e) = kekX $ ) X, $ e | 1 e 0 1]. 3 , $ )
X ) '(t) = k t kX , $ 5 0 1]. 0
]
, 2002, 8, 0 2, . 319{334. c 2002 !"#, $% &' (
320
. .
2$ )# # '(t)
1 ) , #7 , 0 1] , 5 '(0) = 0 ( . 1, . 137]). 3 # 9- . : 1$
X X g(t), $ Z kgkX 1 sup f(t)g(t) dt < +1: 1
1
f 2X kf kX 61
0
; , X | )
, X $ 15 X 0 . , + X X ( . 1, . 138]). 3 , X | )
, $ f 2 X Z kf kX = sup f(t)g(t) dt: (1.1) 1
11
1
g2X 0 kgkX 0 61
0
.
1, '(t) | $ )
X, '(t) = ' tt t 2 (0 1] '(0) = 0 $ ) 15 X 0 ( . 1, . 144]). : '(t), t 2 0 1], 1 '(2t) : ' = tlim ' = lim '(2t) ! '(t) t! '(t) > $ ) * $ ), X(') |
$ ) '(t), t 2 0 1], # 1 < ' 6 ' < 2. > , $ Lq , 1 6 q < 1, $, '(t) = t q1 , 1 ' = ' = 2 q . , Lq , 1 6 q < +1, $ 1 0 1] f(x), $ q 1q Z (x) Zx (t) dt dx f kf kq = x x < +1 ( )
0
0
1
0
0
$ f (x) | #7 jf(x)j ( . 1]) (x) | $ 9- . , / M() 1 0 1] f(x), $ Zt kf kM = sup (t) f (x) dx < +1: 2 (n = 1 2 : : :). .75# ? fpk g = fn(t)g 0 1] $ $#7 ( . 1]): 1 (t) 1 0 1]. A n > 2, n = mk + r(pk ; 1) + s, $ mk = p p : : : pk , k = 0 1 : : :, r = 0 1 : : : mk ; 1, s = 1 2 : : : pk ; 1. B A 1
$ mlk 0 1]. 3$ t 2 B, $ B 0 1] n A, 1 1 X t = mk (t) k (t) = 0 1 : : : pk ; 1 1
+1
1
2
+1
k
=1
k
$ . 3) $ # n (t) skr (t) $#7 : (p isk+1 t r r s n(t) = kr (t) = mk exp pk+1 t 2 ( mr k rmk ) \ B 0 t 2= mk mk ]: , ) ) , 1 B #$ 0 1], # n(t) $ 1 ( mrk rmk ). , + # n (t) 1
5 $ ) , 0 1] 5 $ ) . 3 $ 5 fpn g \ ( . 1]). A pn = 2 (n = 1 2 : : :), fpn g $ ?. B C(q s : : :) $ ) 1 ) , 7 , 7 , . : $ ) ) )# A(y) B(y), , 7 # 1 ) C , C , C A(y) 6 6 B(y) 6 C A(y) $ y. : $ ) ) $ ) ) $#7 1$ . A (. 4]). 1 < < 2 9- (x), x 2 0 1], q > 0 Zx q (t) q t dt = O( (x)) x ! +0 Z tq (t)]; dt = O(;q (x)) x ! +0: 2
( )
+1
( )
+1
+1
1
1
2
1
2
0
1
1
x
B (. 4]).
9- '(x), (x), x 2 0 1].
' > > 1,
322
. .
(' x
x 2 (0 1]
(x) = x 0 x = 0 9- (x), (x) (x), x 2 0 1], 1 > 1. > X n f ; bk k En (f)X = finf bkg X (
)
(
)
1
1
k
=1
* 1 f 2 X fpk g $ * n. C ) EX () = ff 2 X(): En(f)X 6 n n = 1 2 : : :g EDX () = ff 2 X(): Emn (f)X 6 n n = 1 2 : : :g $ = fn g | $ ) ) 1 ) n # 0, n ! +1. > 7] /. 2. 3 E. 3 $ $#7 : 1 1 kTn kq 6 C(q p)n p ; q kTn kp 1 6 p < q < +1 (1.2) n P $ Tn (x) = bk k (x) fpn g, 2 6 pn 6 C , n = 1 2 : : :. k > ) $ ) fpng A. F. E 6]. H (1.2) Lq $ 5], $ ) * 7 , / $ 8]. > x 2 7 $ (1.2)
. : , $ ? . ;. J 12] $ A. 1 6 p < q < +1. f 2 Lp 1 q X n p ; Enq (f)p < +1 (1.3) 0
=1
2
n
=1
f 2 Lq
X 1q 1 q En (f)q 6 C(p q) (n + 1) p1 ; 1q En (f)p + n = 1 2 : : :: k p ; Ekq (f)p 2
k n =
+1
(1.4) H * ) (1.3) ) (1.4) K. > * 14]. > 0 < p < 1, p < q 6 +1 A $ M. F. N 1, >. J. E , ,. . )$ 16]. O 1 Ep() Lq $ 17].
323
F A $ 75 ? $ A. F. E 6], A. F. 15], N. 3 5] Lp $ , / 8]. > x 3 )
.
x
2.
> + $ $ fpng. $ ), fpng $ ) $ ) )# fpn g. , 1 v X Dv (x t) = k (x)Dk (t) x t 2 0 1]: k
1.
=1
X(') | '. !" sup kDmn (x )kX ' 6 mD n '(m;n ): 1
(
x2
)
0 1]
. O (
r r Dmn (x t) = mn x t 2 Inr = ( mn mn ) 0 ) r = 0 1 : : : mn ; 1, $ ) r+1 r r + 1 Zmn Z g(t)Dmn (x t) dt = mn g(t) dt x 2 m m +1
1
n
r mn
0
n
$ # g 2 X 0 ('), D kgkX ' 6 1. 3), ) ) (1.1), 1$
. 2. # " n = mk + 1 : : : mk , k = 1 2 : : :, 0(
)
+1
p knkX ' = mk '(m;k ): 1
(
)
. > $ n (1.1) -
knkX ' 6 pmk '(m;k ): 1
(
)
: $ ) 1 # ( t r r m '(m; ) expf2is pk+1 k+1 g t 2 mk mk ] g(t) = k k 0 t 2= mrk rmk ] ) (1.1). M
$ . 1
( )
+1 +1
324
. .
1. X(') | '. !" " Tmk (x) =
mk X
n
an n(x) x 2 0 1]
=1
1 kT k : '(m;k ) mk X ' . > ) fpng $ # x 2 0 1] Z Tmk (x) = Tmk (t)Dmk (x t) dt: kTmk k1 6
(
1
)
1
0
,+ ( . (1.1)) jTmk (x)j 6 kTmk kX ' kDmk (x )kX ' $ # x 2 0 1]. 3), ) )
1, #$ kTmk k1 6 1; kTmk kX ' : '(mk ) 3 $ . 2. X('), Y () | 1 < < < '. !" kTmk kY 6 (' )(m;k )('(m;k )); kTmk kX ' : . ,
kTmk kY = kTm k kY 6 kTm k m1k kY + kTm k m1k kY = I +I : (2.1) . I . O , Tm k (t) , ) ) ( . 1, . 89]) Z Z f (x) dx = sup jf(x)j dx (2.2) (
0(
)
(
1
1
(
)
(
)
(
)
1
)
)
1
(
0
(
]
)
(
1]
(
)
)
1
2
1
jE j E E =
0
0 1]
1 1, Zt 1 Tmk (t) 6 t Tm k (x) dx = Z jTm k (x)j dx 6 kTmk k1 6 ('(m;k )); kTmk kX ' (2.3) = 1t sup jE j t 0
1
1
(
E E =
0 1]
)
325
$ t 2 (0 mk ]. ,+ I 6 ('(m;k )); kTmk kX ' k m1k kY = (m;k )('(m;k )); kTmk kX ' : (2.4) 3) I . 3 ( . 1, . 162]) Zt '(t) sup f (x) dx 6 kf kX ' f 2 X(') 0, t 2 0 1]. N $ ), kX ; 6 k('(t)); m1k kY 6 ('(m;v )); mv1+1 m1v Y v 1 kX ; kX ; 1 ; ; ; ; 6 ('(mv )) k mv1+1 m1v kY = ('(mv )) m ; m : v v v v (2.6) 1
1
1
1
1
(
)
(0
(
]
1
1
)
(
)
2
(
0
)
1
0
1
2
(
)
(
(
1]
)
1
1
1
(
(
1]
)
1
(
+1
]
(
=0
1
)
1
1
1
1
(
+1
]
(
)
1
+1
=0
+1
=0
O , tt # (0 1] ( . 1]) 2 6 pn 6 C $ # n = 1 2 : : :, $ ), 1 1 1 1 6 (pv ; 1) m 6 (C ; 1) m : m ;m v v v v N $ ), (2.6) kX ; k('(t)); m1k kY 6 (C ; 1) (m;v )('(m;v )); : (2.7) ( )
0
+1
0
+1
+1
+1
1
1
(
(
1]
)
0
v
1
1
+1
+1
1
=0
, ) )
A B, Z dt kX ; (m; ) (m;v ) v 6 C 6 C ; ; ): t (t) '(m v v '(mv ) 1 1
1
1
1
+1 1
+1
=0
(2.8)
+1 1
1
mk
;
+1
; (2.1), (2.4), (2.5), (2.7), (2.8) $
kTmk kY 6 (m;k )('(m;k )); kTmk kX ' : 3 $ . 1. X(') = L'q1 , Y () = Lq2 , 0 < q q 6 +1, kTmk kq2 6 (' )(m;k )('(m;k )); kTmk k'q1 : 1
(
1
1
)
(
1
1
1
1
)
2
326
. .
2.
1 6 r < +1,
X(') = Lpr , Y () = Lq , 0 < p < q < +1, kTmk kq 6 (p q r)mkp ; q kTmk kpr : 1
1
.
, $ 2 p = r ) N. H. 3 5], p = r, q = | A. F. E 6] ( ) $ ) fpng), /. 2. 3 , E. 3 7] ($ $ ) fpn g). 3. 1 6 < q < +1 (t) | 9- 0 1]. !" 1 1 kTmk k 6 C(q )(ln mk ) ; q kTmk kq : . 3 # < q, , J5 )$ $ , Z 1 Zt (t) dt 6 (ln mk ) ; q kTm kq : T (x) dx (2.9) k m k t t 1 1
1
mk
;
0
: ,
A, (2.2) 1 $ ) m Z k 1 ;
0
mk 1 t 1 Z T (x) dx (t) dt 6 kT k Z (t) dt 6 mk 1 t mk t t ;
0
0
6 C( ) (m;k )kTmk k1 6 C( )kTmk kq : (2.10) 1
; (2.9) (2.10) $ 1$ . .
, $ 3 $ . F. R 13].
x
3.
EX ()
4. f 2 X() | (t) 1 < < 2. !"
t ; 1 Z f (x) dx 6 C() kf k + nX 1 E (f) ; )); E (f) ((m + mk X (t) mn X X k t k t 2 (m;n m;n ], n = 0 1 : : :. . , ) Tn(f x) | * 1 f 2 X() fpk g. , ) ) J5 )$, 1
1
(
)
+1
=0
0
1
+1
1
1
(
)
(
)
327
sup
Z
E
E t E 0 1]
t 2 (m;1
jf(x) ; Tmn (f x)j dx 6 t Emn (f)X '
'(t)
(
(3.1)
)
=
; $ n mn ]. : , 1 * 1 , $ ) nX ; kTmn (f)k1 6 kT (f)k1 + kTmk+1 (f) ; Tmk (f)k1 6 k nX ; 6 C() kf kX + ((m;k )); Emk (f)X : (3.2) 1
+1
1
1
=0
1
1
(
)
k
1
(
+1
)
=0
3 $ # 1 E 0 1] $ Z Z Z jf(x)j dx 6 jf(x) ; Tmn (f x)j dx + jTmn (f x)j dx E
E
E
(2.2) (3.1), (3.2) $ 1$ . 3 $ . .
, $ $ ) 4 M. F. N 1 9, 1]. 5. X() Y () | , 1 < < ' ' < 2. f 2 X() X 1 nX ; S(f) ((m;k )); Emk (f)X + n k 1 E (f) 1 1 < +1 (3.3) + (t) mn X mn+1 mn Y f 2 Y () kf kY 6 C( )fkf kX + S(f)g: . , 1 1 nX ; X 1 E (f) 1 m 1 (t) Q(t) = ((m;k )); Emk (f)X + '(t) mn X ' mn+1 n n k Rt $ t 2 (0 1]. .
, Q(t) t f (x) dx (0 1]. 3 # 5 f 2 X(') 1 < ' ' < 2,
4 Zt f (t) 6 1t f (x) dx 6 C()fkf kX + Q(t)g 1
1
1
(
+1
=1
)
=0
(
)
(
;
;
]
(
(
)
(
)
)
1
1
1
(
+1
=1
)
(
=0
1
0
(
0
)
)
(
;
;
]
328
. .
$ t 2 (0 1]. ,+ (3.3) $
1$ 5. . O (3.3) + $#7 : X 1 ((m;n )); Emn (f)X m 1 mn 1 < +1: (3.4) n+1 Y 1
1
(
+1
n
)
(
;
;
]
(
=0
: ), ((m;n )); Emn (f)X 6 1
1
(
+1
)
n X k
)
((m;k )); Emk (f)X 1
1
(
+1
)
=0
$ # n = 0 1 : : :, , (3.3) $ (3.4). ,1 . , ) (3.4), ) 1 X g(t) = ((m;n )); Emn (f)X mn+1 1 m 1 (t) n 1
1
(
+1
n
)
;
(
;
]
=0
t 2 (0 1], $ 1 Y (). : t 2 (0 1] ) , m; < t 6 m; . 3$ $ # g(t) Z Z g(y) X ; ~ g(y) dy > Hg(t) dy > (ln 2) ((m;k )); Emk (f)X y y k 1 1
1
+1
1
1
1
1
t
$
t 2 (m;1
1
(
+1
m
)
=0
;
m; ]. . #$ X ~ (ln 2) ((m;k )); Emk (f)X 6 Em (f)X ((m; )); + Hg(t)
1
+1
1
k
1
1
(
+1
)
(
)
1
+1
=0
$ 1$ t 2 (0 1]. 3 , $ 1$ t 2 (0 1] ~ (ln 2)Q(t) 6 g(t) + Hg(t): N $ ), 15 ?$ H~
( . 2]) (3.4) $ (3.3). 1. 1 < < ' ' < 2. f 2 X() n X sup (m;n ) ((m;k )); Emk (f)X < +1 n
f 2 M .
1
+1
1
k
+1
1
(
)
=0
. 2 Q(t) (0 1]. ,+ , ) )
A, B , (0 1] (t) ", , #
329
(t)Q (t) = (t)Q(t) 6 C
(m;1 ) n
nX ;
1
k
((m;k )); Emk (f)X + 1
1
(
+1
)
n (m;n ) ; ) X((m; )); E (f) + E (f) 6 C sup (m m mk X X n k (m;n ) n n k $ # t 2 (m;n m;n ], n = 0 1 : : :. N $ ), 5 f 2 M . 2. 1 < < < 2 1 < q < +1. f 2 X(') 1 (m; ) q X n q ; ) Emn (f)X < +1 (m n n f 2 Lq 1 X 1 (m; ) q n q (f)X q : E Emk (f)q 6 C(q ) mn ; n k (mn ) =0
1
+1
1
(
1
)
1
+1
1
(
+1
+1
)
=0
1
1
+1
1
+1
(
1
)
+1
=0
1
+1
(
1
)
+1
=
.
, $ 2 $ 8].
6. $
% "
fpn g. f 2 X(), 1 < < < 2 X 1 1 E (f) 1 1 (n; ) n X n+1 n Y (
1
n
)
(
]
=1
f 2 Y ().
(
< +1
(3.5)
)
. C # 1 X n X
1 E (f) 1 1 (t) t 2 (0 1] k X n+1 ; n n k k(k ) 5, 1 $ ) , (3.5) $ , kQ kY < +1: (3.6) N $ , * 1 tt " (0 1] $ ) n X 1 E (f) > 1 E (f) n = 1 2 : : :: k X ( n ) n X k k( k ) Q (t) = 2
(
1
=1
)
(
]
=1
2
(
)
( )
1
(
)
1
(
)
=1
N $ ), (3.6) 5 (3.5). M $ , (3.6) (3.5) + .
330
. .
3) 1 , (3.6) 5 (3.3) 5. : ), tt " (0 1] pn 6 C $ # n = 1 2 : : :, 0
( )
mX ;
1 E (f) > 1 E (f) = 1 2 : : :: ; ) n X n(n (2C (m; ) m X n m 1 O , (t) (m;k ) $ t 2 (m;k m;k ], $ kX j ; X 1 E (f) 1 E (f) + 1 E (f) > C m X (t) mk X n X ; ; n n(n ) (m ) = 1 2 : : :, $ j = mk mk + 1 : : : mk , k = 1 2 : : :. N $ ), (3.6) 5 (3.3). ( , 5 $ ), f 2 Y (). 3 $ . 7. X(), Y () | 1 < < < ' < 2. !" 1 ~EX () Y () () X((m;n )); n m 1 mn 1 < +1: n+1 1
(
1
=
)
(
1
0
;
)
+1
1
1
+1
1
+1
1
(
1
)
(
1
=1
)
(
)
+1
=1
+1
1
1
+1
n
;
(
;
]
Y (
=0
)
. : ) $ 5. :1 $ ). , ) EX () Y (). : , X 1 ((m;n )); n m 1 mn 1 = +1: (3.7) n+1 Y 1
n
1
+1
(
;
;
]
(
=0
)
N $ 10], $ ) ) fn()g $#7 : n(0) = 0 n(1) n(2) : : : n() , 1 1 n( + 1) = min n: n < 2 n : 3$ n < 12 n (3.8) n ; > 12 n (3.9) (
(
(
+1)
+1)
1
(
)
(
)
)
$ = 1 2 : : :. > (t) 0 1] $ ) f g $ ) n kX ; 1 1 1 (t) 6 1 m+1 m nk ; (t) (mn k ) nk (
+1)
1
(
=
(
)
;
;
)
(
)
1
(
+1)
331
$ # t 2 (m;n k m;n k ) k = 1 2 : : :. ,+ * (3.5) $ , X 1 ((m;n k )); n k m;n k m;n k ] = +1: (3.10) Y 1
(
1
+1)
(
)
1
k
(
1
1
(
+1)
)
(
1
(
+1)
(
)
(
=1
)
C # 1 X f (x) = (pmn (m;n )); n mn(+1) pn(+1) (x) 1
0
(
+1)
(
1
(
+1)
+
)
+1
=0
x 2 0 1]. N 7)#
2 (3.8), (3.9) f 2 X() 1 X Emn (f )X 6 Emn(s) (f )X 6 n 6 n s 6 2n s ; 6 2n 0
0
(
0
)
(
)
(
s
)
^
( )
( +1)
=
1
$ n = n(s) : : : n(s + 1) ; 1. ( , g = f 2 EX (). H mn pn+1 (t) # , + 1
0
2
0
+
mZn(j) mZn(j) t 1 Z g (x) dx > m g (x) dx > mn j jg (x)j dx = nj t Z1 = mn j (m;n k ; m;n k )((m;n k )); n k > 12 ((m;n j )); n j ;1
( )
0
0
0
( )
0
0
1
( )
k j
(
0
1
)
(
1
1
+1)
(
1
+1)
(
)
( +1)
1
( )
=
$ # t 2 (m;n j m;n j ], j = 1 2 : : :. N $ ), ?$ * (3.10) g 2= Y (). M # # E~X () Y (). 3 $ . 1
1
( +1)
( )
0
8. $
% "
fpn g 1 < < < 2. !" X 1 1 1 EX () Y () () ((n; )); n n+1 n 1
< +1:
1
(
n
]
=1
Y (
)
. : ) $ 6. N $ ) $ . , ) EX () Y (). : , X 1 ((n; )); n n+1 1 1 = +1: n Y 1
n
1
(
]
(
=1
)
C # 1 X f (x) = (pml (m;l )); n ml() pl() (x): 1
1
(
=0
)
(
)
1
(
)
+
+1
332
. .
($ ) fn g fn()g | $ ) , $ ) 7, l() = maxfk: mk < n( + 1)g: : , 1$ , $ ) 7, g = c f 2 EX (), g 2= Y (). 3 $ . . , ) Y () = Lq , X() = Lp, 1 6 p < q < 1. , pn = 2 $ n = 1 2 : : : ( . . fpn g | ?) 6 $ )
. ;. J 12], 8 > * 14]. > 2 6 pn 6 C , n = 1 2 : : :, 6 $ /. 2. 3 E. 3 7], 8 N. 3 5], A. F. 15]. 3 6 8 11]. > 2 11] : 1 ) n * . 9. 1 < q < q < +1, 1 < < 2. f 2 Lq1 1 X q (ln mn ) ; q21 Emq2n (f)q1 < +1 (3.11) (
)
1
1
1
1
0
1
2
1
1
+1
n
=1
f 2 Lq2
q12 X 1 q2 q ; 2 q 1 Emn (f)q2 6 C( q q ) (ln mk ) Emk (f)q1 n = 1 2 : : :: 1
1
2
+1
k n =
. , ) Tn(f ) | * 1 f 2 Lq1 fpng. , )# $ ) ) fn()g $#7 : 1 n(0) = 0, n(1) = 1, n( + 1) = min n: Emn (f)q1 < 21 Emn (f)q1 : 3$ Emn(+1) (f)q1 < 12 Emn (f)q1 (3.12) Emn(+1) ; (f)q1 > 21 Emn (f)q1 : (
)
1
C $
X Tmn(1) (f x) + (Tmn(+1) (f x) ; Tmn() (f x)) $ f 2 Lq1 Lq1 . , ) )
13], 3, X q12 j; q2 q ; 2 q kTmn(j) (f) ; Tmn(s) kq2 6 C(q q ) (lnmn k ) 1 Emn(k) (f)q1 : k s (3.13) 1
1
1
2
(
=
+1)
333
> * 1 (3.12) n kX ; q q (ln mn k ) ; q21 Emq2n(k) (f)q1 6 2q2 (ln mn ) ; q12 Emq2n (f)q1 : (3.14) (
+1)
1
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1
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; (3.13) (3.14) (3.11) , $ ) ) fTmn(j) (f)g Lq2 $ ) Lq2 . > Lq2 7 g 2 Lq2 , kg ; Tmn(j) (f)kq2 ! 0 j ! +1: 3 Lq2 Lq1 , kg ; Tmn(j) (f)kq1 ! 0 j ! +1: N $ ), g(x) = f(x) #$ 0 1]. ( , f 2 Lq2 . 3 $ .
#
1] . ., . ., . . . | ., 1978. 2] Sharpley R. Spaces & and interpolation // J. Funct. Anal. | 1972. | Vol. 91. | P. 479{513. 3] , - ., ./ 0 . 1 / 2 3. | ., 1958. 4] 4 . 5. 67 - 3 03 87 . | 9. 561 :6 ;. < 1036-80 9. 5] 1 0 ?7 . 6. @/ 2 3 B2 C C 4 -. | :8. 3CC.. . . 7 3. 8.-. 7. | : , 1990. 6] ,7 . :. 1 - C 0 7 3 ,D ?- / 2 CC 7 CC E // B7 2 0, C7 2 . | G2C7, 1984. | . 46{54. 7] 1 . B., 1 . C 7 CC // 0. CD. ,?. 0 3., C . | 1983. | < 9. | . 65{73. 8] :7D . :., C ;. . 1 - C, C C ??H CC E . | 9. 0661. < 3618. 9] -7 I. :. 1 - ,D ?- // . C?7. | 1975. | 1. 97, < 2. | . 230{241. 10] 3 5. . 1 - C 0 7 3 ,D ?- // . C?7. | 1977. | 1. 102, < 2. | . 195{215. 11] :7D . :. @ - 7 7 CC C, C C // 10C 37 3 -3 3. 78. KB7. C C , ?-, 0L. | ., 1995. | . 11{12. 12] ? M. . 6 ,D ?- E GD // . C?7. | 1972. | 1. 87, < 2. | . 254{274.
334
. .
13] .C 4. :. @ CC ,D ?- C C 4 7 - // 0. CD. ,?. 0 3., C . | 1987. | < 10. | . 48{58. 14] 5 D7 N. 1 - C 0 7 3 ,D ?- CC E // 0. CD. ,?. 0 3., C . | 1980. | < 4. | . 11{15. 15] MC . :. @ - 7 CC 87 , 0 3 C3 2C ,D ?- 7 CC . | 9CC.. . . 7 3. 8.-. 7. | C7 , 1988. 16] -7 I. :., 5. ., @C 23 . ? 9-7C C C Lp , 0 < p < 1 // . C?7. | 1975. | 1. 98, < 3. | . 395{415. 17] 6 0? . O., . ., :7D . :. 3C2 3 0 7Q88 B2 - C C 4 . | 9. 561 :6 ;. < 580-83 9. ) * 1999 .
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. .
512.554.5
: ,
, , ! , " , # " .
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# . '. (. )#* 1981 . - % # & !
%. ) Nk |
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& , D |
N3 N2 - 3, . .
! ! 3 =0 2( 1 2 )( 3 4 )]( 5 6 ) = 0 (
& !
" Nk Nl . , "! # " (D) " # " rt (Dn ) = + 2
" Dn = D \ Var(( ) 1 n )
# #
% D. k
x
x x
x x
x x
F
:
n
xy
zt x
:::x
Abstract
A. V. Badeev, The variety N3 N2 of commutative alternative nil-algebras of index 3 over a eld of characteristic 3, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 335{356.
A variety is called a Specht variety if every algebra in this variety has a 9nite basis of identities. In 1981 S. V. Pchelintsev de9ned the topological rank of a Specht variety. Let Nk be the variety of commutative alternative algebras over a 9eld of characteristic 3 with nilpotency class not greater than . Let D be the variety N3 N2 of nil-algebras of index 3, i.e. the commutative alternative algebras with identities 3 =0 2( 1 2 )( 3 4 )]( 5 6 ) = 0 In the paper we prove that the varieties Nk Nl are Specht varieties. Moreover, a base of the space of polylinear polynomials in the free algebra (D) is built and the topological rank rt (Dn ) = + 2 of varieties Dn = D \ Var(( ) 1 n ) is found. This implies that the topological rank of the variety D is in9nite. k
x
x x
x x
x x
:
F
n
xy
zt x
:::x
, 2002, 8, : 2, . 335{356. c 2002 !, "# $% &
336
. .
. , . . ! "# $ %2, 129] +: + - +
! . ( .) ? 2. . %3] , -
. 2 ( + .) ! ., ., !3 . (;1 1) . . 4
+- ! 5 . 6. 7- 3 %5], . 6. 9! %6], 6. . # , . :. %9]. ;. !
. ! . . , - 2, 3, i1 (4) U(F). 2 6 5 ! (xy)y = xy2 . N x u1u2 + + 5 , +- F 5 u1u2 yy = 0: H D 5 y, +-
u1u2 xy = ;u1u2 yx: 7 - F
?!
, +-
u1u2 x(1) : : : x(n) = sign()u1 u2x1 : : :xn | ! D 5 f1 : : : ng. #, + 5 !
xi xj x1 = ;xi x1xj ; xj x1xi :
?! D .
!, - U(F ) 5 D (4). ! ! !, - - (4) . x1 x2 : : : xn . # n 6 2 D - .
n X j =2
j xj xi1 xi2 : : :xin;1 = 0
- , - fi1 : : : in;1 j g = f1 2 : : : ng. # j0 > 2 3 3 xj0 ! xy
j0 xyxi1 xi2 : : :xin;1 = 0. 7 ! +
B
340
. .
! , j0 xyxi1 xi2 : : :xin;1 = 0 j0, 3 ! .
2. B . 2 7 - ,
1, !, - - f 2 x
5 ! f = u x. - f 3 x +-
+ (2) (3) f = u x = (v x) x = v x2 = 0: ! - 3 ! . O , - B %1]. @ . 3 !
2 D + !+
1 - ! !.
3. ! B D M Dn " : (1) M $ (2) M \ B 6= B. 2 G- , - M ! , M \ B 5 ! . 6 + !
B M \ B 6= B. 7+! M \ B 6= B. 9
2 , - M \ B ! , . . m
x1 x2 : : :xm 0 (mod F (2)(M)): G x1 x2 : : :xm y1 : : :yn 2 T (M), . . M ! .
3. ! -
7 ! 3 ! - , 5 , -
D ( - 3 ! -
) . 7+! F = F (D) |
D 5 . 5? . X = fx1 x2 : : : xn : : :g. 7 5 u v 2 F 2, x y z 2 F. 7 ux ux = u2x2 = 0: H u, +-
ux vx = 0: 7+! f 2 D | - 4 . 9
, - - ! 3
B.
-
341
f 2 T(B) = F (2), . . f
5 ! 3
- u v, u v 2 F 2 . 7 - + (3)
5 - !, - u, v | - 2 . 7 D 5 - .
5 !
+ F (2). - u, v 2 x +- +
u v = u1 x v1 x = 0: , X f = u v = 0: char K = 3, ! ! . - F(D) 3,
D
%1]. 5 3 ! %(x1x2 )(x3x4 )]x5 = 0: 7+! A | 5 5? . X, I | A. G - - P(I) . - A, 5? . IP Pn(I) P(I) | .
- 5 5? . Xn n. G - Zn = f1 2 : : : ng, n > 4, 'ij = x0i1 : : :x0in;4 | n ; 4, fi1 : : : in;4g = Zn n f1 2 i j g, i1 < : : : < in;4.
4.
1) % Pn(F (2)(C)) n = 4t n = 4t + 3 ) x1xi 'ij (x2xj ) 2 < i < j ) x1xi 'i3(x2x3 ) i < 3 2) n = 4t+1 n = 4t+2 Pn(F (2)(C)) & ) ) ) x1x5'54 (x2x4): 2 7 5 F = F(C), Pn = Pn (F (2)(C)).
- +- n = 4. 7
x2 y2 = (xy)(xy): 7 3 D 5, +-
(x1 x2)(x3 x4) = ;(x1 x3)(x2 x4) ; (x1 x4)(x2 x3): , n = 4 D ), ) 5 U4 . #, F + (1) 3 ! uv(pqx) = ;uvx(pq): (5)
342
. .
G , - 5 - F (2) 3 D xyw (zt), x y z t 2 X, w | . N , - + 3 ! 5 ! xyw
5 !, .
, +-
xyx(1) : : :x(n)(zt) = (;1) xyx1 : : :xn(zt)P (6) ( xyx1 : : :xn (zt) = (;1)n ztx1 : : :xn(xy) n = 0 n = 4t 4t + 3 (7) 1 n = 4t + 1 4t + 2: 7 5 (7) n = 1 2 3 4: (xy)x1 (zt) = ;(xy)%(zt)x1 ] = ;(zt)x1 (xy)P (xy)x1 x2 (zt) = (zt)x2 x1(xy) = ;(zt)x1 x2(xy)P (xy)x1 x2x3 (zt) = ;(zt)x3 x2 x1(xy) = (zt)x1 x2x3 (xy)P (xy)x1 x2x3x4 (zt) = (zt)x4 x3x2 x1(xy) = (zt)x1 x2x3 x4(xy): # n > 4 ! +5 +- + 3 . 7 !+! 5 xixj x1 = ;x1 xixj ; x1xj xi xi xj x2 = ;x2xi xj ; x2xj xi x1x2xi = ;x1 xix2 ; x2 xix1 5 (5){(7) +- +- n = 4, + !, - U(F (2)) 5 - x1xi 'ij (x2 xj ) i j > 2: (8) @ , xi xj 'xk (x1 x2) = ;xixj '(x1x2 xk ) = ;xi xj '(;x1 xk x2 ; x2 xk x1) = = xi xj '(x1xk x2) + xixj '(x2xk x1 ) = ;xi xj 'x2 (x1xk ) ; xi xj 'x1 (x2xk ) = = xi xj x2'(x1 xk ) xi xj x1'(x2 xk ) = = x2 xixj '(x1 xk ) x2xj xi'(x1 xk ) x1xi xj '(x2 xk ) x1xj xi '(x2xk ) = = x1 xk xj '(x2 xi) x1xk xi'(x2 xj ) x1xi xj '(x2 xk ) x1xj xi '(x2xk ): 7+! n > 4. M , - 4 ! D. ! , x2w x2 = 0. 6 +
C,
, +- +? 5 : 2x2w (xy) + 2(xy)w x2 = 0: + , n = 4t 4t + 3 + (7)
x2 w (xy) = (xy)w x2: G x2w (xy) = 0:
-
343
# ? + -! - wn n.
! +? +: x1xxwn ;5(x2x) = 0 n = 4t 4t + 3: H 5 , +- n = 4t 4t + 3 x1 xixk wn ;5(x2 xj ) + x1xj xk wn ;5(x2 xi) + x1 xk xiwn ;5(x2 xj ) + + x1 xk xj wn ;5(x2xi ) + x1xi xj wn ;5(x2xk ) + x1 xj xiwn ;5(x2 xk ) = 0: 7 5 ! k = 3
- ! , +-
x1 xix3 wn ;5(x2 xj ) = ;x1xj x3 wn ;5(x2xi ) ; x1x3xi wn ;5(x2xj ) ; ; x1x3 xj wn ;5(x2xi ) ; x1 xixj wn ;5(x2 x3) ; x1xj xiwn ;5(x2 x3): N , - - . D (8), - | D ), ), . . D (8) 3 D ), ). #, + (7) n = 4t + 1 4t + 2
x2wn ;4y2 + y2 wn ;4x2 = 0 - + ! !
x1xxywn ;6(x2 y) + x1yyxwn ;6 (x2 x) = 0: H + 5 , +- n = 4t + 1 4t + 2 x1 xixp xq wn ;6(x2 xj ) + x1xp xi xq wn ;6(x2 xj ) + x1xi xp xj wn ;6(x2xq ) + + x1xp xi xj wn ;6(x2xq ) + x1xq xj xiwn ;6(x2xp ) + x1 xj xq xiwn ;6(x2 xp ) + + x1xq xj xp wn ;6(x2 xi) + x1xj xq xpwn ;6(x2 xi) = 0: 7 D
j = 3, q = 4, p = 5 5 !
- ! , +-
; x1xi x5x3wn ;6(x2 x4) = x1xi x5x4wn ;6(x2 x3) + x1x5xi x4wn ;6(x2 x3) + + x1 x5xi x3wn ;6(x2x4 ) + x1x4x3 xiwn ;6(x2x5 ) + x1x3x4 xiwn ;6(x2 x5) + + x1 x4x3x5 wn ;6(x2xi ) + x1x3x4 x5wn ;6(x2xi ) = 0: 7 p = 3, q = 4 5
- ! , +-
; x1xj x4 x3wn ;6(x2xi ) = x1 xix3x4 wn ;6(x2xj ) + x1 x3xix4 wn ;6(x2xj ) + + x1xi x3xj wn ;6(x2 x4) + x1x3 xixj wn ;6(x2 x4) + x1x4 xj xi wn ;6(x2 x3) + + x1xj x4xi wn ;6(x2 x3) + x1x4 xj x3wn ;6(x2 xi) + x1xj x4x3wn ;6(x2 xi): , D x1 xi'i4 (x2x4) 3 D ){), D (8) 3 D ){) D x1xi 'i4(x2 x4).
344
. .
7. !+
+ .
D . 7 - +
! D ) ). 7 5 , - 3 . D . x1 x2 : : : xn, n = 4t 4t + 3, ! + : X X n = ij x1xi 'ij (x2 xj ) + i3x1xi 'i3(x2x3 ) = 0: 3 3
i<j
i
xi ! u1u2 xj ! v1v2 i j 6= 1 2 3 i < j
ij u1u2'ij (v1 v2) 0 (mod F1(2)) A 2 D , !
C, ! +- ij = 0. , ij = 0, i j 6= 3. 6 D +- 3 3 xj ! v1 v2 x1 ! u1u2 j 6= 2 3 xi ! u1 u2 x2 ! v1 v2 j 6= 1 3
3j = 0 i3 = 0. ! , 3 n ! . # 5 ! +
! D ){), - n = 4t + 1 4t + 2: X X n = ij x1xi 'ij (x2 xj ) + i3x1 xi'i3 (x2x3 ) + 54x1 x5'54(x2 x4) = 0: i<j
i
7 i j 6= 3 4 3 3 xi ! u1u2 xj ! v1 v2 i j 6= 1 2 3 4 i < j +- ij = 0, i j = 6 3 4. , n
X i>3
f 3ix1x3 '3i(x2 xi) + i3x1xi '3i(x2 x3)g + X +
i>4
4ix1 x4'4i(x2 xi) + 54x1x5 '54(x2x4 ) = 0:
6 ! ! 3 3 xi ! u1 u2 xm ! v1 v2 m = 2 3 4 i > 5 + (3), (5) +-
(i) i3u1u2 x1'3i (v1v2 x3) = 0 (ii) 3iv1 v2x1 '3i(u1 u2x2) + i3u1 u2'3i(v1 v2 x2) = = ; 3i v1v2 x1'3i x2(u1 u2) ; i3u1u2 '3ix2(v1 v2 ) = = ;( 3i + i3)u1 u2x1 '3ix2(v1 v2 ) = 0 (iii) 4iv1 v2x1 '4i(u1 u2x2) = 0:
-
345
! , i3 = 3i = 4i = 0, i > 5. G
n = 34x1 x3'34(x2 x4) + 35x1x3'35(x2 x5) + 43x1x4 '34(x2x3 ) + + 53x1 x5'35(x2 x3) + 45x1x4'45 (x2x5) + 54x1x5 '45(x2x4 ) = 0:
! 3 3 x3 ! u1u2 x4 ! v1 v2 x1 ! u1u2 x3 ! v1v2 +-
34u1u2 x1'34(v1 v2 x2) + 43v1 v2x1 '34(u1 u2x2) = = ;( 34 + 43)u1u2 x1'34x2 (v1 v2) = 0 43u1u2 x4'34(v1 v2 x2) + 53u1u2 x5'35(v1 v2 x2) = = ( 43 ; 53)u1 u2x4 '34(v1 v2x2 ) = 0: G 34 = ; 43, 43 = 53. - , 35 = ; 53, 45 = ; 54, 35 = 45. , 34 = 35 = 45 = ; 43 = ; 53 = ; 54. #, 3 3 x1 ! u1 u2, x2 ! v1 v2 +-
34u1u2x3 '34x4(v1 v2 ) + 35u1 u2x3 '35x5(v1 v2 ) + 43u1 u2x4'34 x3(v1 v2 ) + + 53u1u2 x5'35x3(v1 v2 ) + 45u1u2x4 '45x5(v1 v2 ) + 54u1u2x5 '45x4(v1 v2 ) = = ;( 34 ; 35 + 45 ; 43 + 53 ; 54)u1 u2x3'35 x5(v1 v2 ) = 0: N , 3, - 34 = 35 = 45 = ; 43 = ; 53 = ; 54 34 ; 35 + 45 ; 43 + 53 ; 54 = 0 | + . ! , 3 D ){) ! .
4. D
7+! F = F (D) |
D 5 . 5? . X = fx1 x2 : : : xn : : :g. 7 5 u v 2 F 2, x y z 2 F . 7 ux ux = u2x2 = 0: H u, +-
ux vx = 0: (9)
346
. .
G + (1), (3)
(ux v)x = ;ux vx ; ux2 v = 0 (10) - !
(ux v)y + (uy v)x = 0: (100) 7+! ' | + . 6 + (3), (10),
- x2 ', , +-
uvx2 = 0 (x2 ' u) = (x'x u)x = 0: 7 (2) (u x2 )x = u x3 = 0: H D
, +-
uvxy = ;uvyx (11) (xy' u)z + (xz' u)y + (yz' u)x = 0 (12) (u xy)z + (u xz)y + (u yz)x = 0: (13) 7 5 F0(2) = F (2) + 3 (2) Fp(2) +1 = Fp F: R+ - !, - 5 . 5? . F=Fp(2) +5 X. N , - P(F (2)) =
1 M
p=0
P(Fp(2)=Fp(2) +1 ):
6 D + 5 P(Fp(2)=Fp(2) +1). (2) (2) (2) O , - F0 =F1 = F (C), C D |
3 ! - . . 7 5 , Hn | D ){)
4, . . . D Pn(F0(2)=F1(2)). G - : np = x0n;p+1 : : :x0n | p, en = = x1 x5'54 x2 x4 2 Hn, enp = en;p np , Hn0 = fx1x4'4j x2 xj 2 Hn j j 2 Ng. #
- + - n . # 5 ! +? +5 .
5. % Pn(Fp(2)=Fp(2) +1 ), 1 6 p 6 n ; 4, & " ( & Enp): 1) Hnp = fbnp j b 2 Hn;pg, 0 (1k) = fb(1k) j b 2 H 0 g, k = n ; p + 1, 2) Hnp np 3) enp (1i)(2j) i j 2 f1 2g Znp , i < j, fi j g 6= f1 2g.
-
347
2 # S . n, p - 5 f1 2g Znp , | ! - Zn . 7 Xn = = fx1 x2 : : : xng ! (+ ) 5 ( ). 7 5 -, - D ij !"#$. i<j
Hnp(1i)(2j)
(14)
5 Pnp = Pn(Fp(2)=Fp(2) +1 ). N , - Pnp 5 - n (xy' zt) p (15) x y z t 2 Xn , ', p | , p p. 6 +
(11)
5 - !, -
p 5 . 7+! f | - (15). , !+ (11), (100), ', p
5 ! 5 . ! , 5 f'g f p g ( ! f g
- ) 5 ! p ! . .,
5 - !, - f p g ! ! .. 7+! D +- xi , xj , i < j, ! 5 f p g. f(1i)(2j) = f 0 np f 0 | - Xn;p = fx1 : : : xn;pg. @ , - Hn;p Pn;p (F (2)(C)). 7 D +
+ F1(2)
f 0 2 Pn;p(F (2)) = Pn;p(F (2)=F1(2) F1(2)) = Pn;p (F (2)(C)) = K(Hnp): #+ np = x0p+1 : : : x0n, +- , -
+ Fp(2) +1 - f 0 np 2 K(Hn;pnp ) = K(Hnp): (16) ! , f(1i)(2j) 2 K(Hnp) - ! f 2 K(Hnp(1i)(2j)): T 5 f'g f p g - f 5 ! p ! ., fx y z tg 5 ! +. ! ., f p g 5 + . ,
?!
(12), (13) + D f p g ! fx y z tg, +? + +-. N - , - (15) 5 K(Hnp(1i)(2j)). G (14) 5 Hnp.
+- p = 1. En1 En1 = Hn1 Hn0 1(1n) fen1 (1n)g fen1(2n)g:
348
. .
7 5 , - En1 5 Pn(F1=F2). (14) | 5? ., D - !
K(Hn1) K(Hn1(1n)) K(Hn1(2n)) K(En1): (17) Hn1 2 En1, !, - K(Hn1(1n)) K(En1) K(Hn1(2n)) K(En1): 6 + (13), (100) ! x y 2 F, u v 2 F 2 . '0 , '00 n ; 6
(ux'0 yx)y = (ux'0 y2 )x = 0 (18) 00 00 2 00 (xyy' v)x = (xyx' v)y = (x y' v)y = 0: (19) 7 5 (19) v = x2xj x, y, +-
(xn xixk '00 x2xj )x1 + (xnxk xi '00 x2 xj )x1 + g00 xn = 0 g00 | - Xn;1 . 7 + .
. # D 5 - k = 4, j 6= 4, k = 3, j = 4 + (1n). 7 +-
(xnxi x4'00 x2 xj )x1 + (xn x4xi'00 x2xj )x1 = = (x1 xi x4'00 x2 xj )xn(1n) + (x1x4 xi'00 x2xj )xn (1n) = = (x1 xi 'ij x2xj )xn(1n) (x1x4 '4j x2xj )xn (1n) (xnxi x3'00 x2 x4)x1 + (xn x3xi '00 x2x4 )x1 = = (x1 xi x3'00 x2 x4)xn(1n) + (x1 x3xi'00 x2x4 )xn(1n) = = (x1 xi 'i4 x2 x4)xn(1n) (x1 x3'34 x2x4 )xn(1n): ! , (x1xi 'ij x2xj )xn (1n) = (x1 x4'4j x2 xj )xn(1n) g00 xn (x1 xi'i4 x2x4)xn (1n) = (x1 x3'34 x2 x4)xn (1n) g00 xn: N , - + (16) g00 xn = K(Hn1) (x1 x4'4j x2xj )xn(1n) 2 K(Hn0 1(1n)) (x1 x3'34 x2 x4)xn (1n) = en1(1n): 7 +- , - (x1 xi'ij x2xj )xn (1n) 2 K(Hn0 1(1n)) K(Hn1) j 6= 4 (x1 xi'i4 x2 x4)xn (1n) 2 K(en1 (1n)) K(Hn1):
-
G,
349
K(Hn1(1n)) = Kf(x1xi 'ij x2xj )xn (1n)g En1 +-
K(Hn1(1n)) K(Hn1) K(Hn0 1(1n)) K(en1 (1n)) K(En1): (20) #, 5
. (18), (19) u = x1xi , v = xn x4 x, y, +-
(x1xi x4'0 xnxj )x2 + (x1xi xj '0 xnx4)x2 + g0 xn = 0 g0 | - Xn;1, (x1 xix3'00 xnx4)x2 + (x1x3 xi'00 xn x4)x2 + g00 x1 = 0 g00 | - Xn n fx1g. 6 5 . +.
+
+ . +. .
(x1xi x4'0 xnxj )x2 + (x1xi xj '0 xnx4)x2 = = (x1 xi x4'0 x2xj )xn (2n) + (x1 xixj '0 x2 x4)xn (2n) = = (x1 xi 'i2 x2 xj )xn(2n) (x1xi 'i4 x2 x4)xn(2n) (x1xi x3'00 xn x4)x2 + (x1 x3xi '00 xnx4 )x2 = = (x1 xi x3'00 x2 x4)xn(2n) + (x1 x3xi'00 x2x4 )xn(2n) = = (x1 xi '4i x2 x4)xn(2n) (x1 x3'34 x2x4 )xn(2n): ! , (x1 xi'i2 x2xj )xn (2n) = (x1 xi'i4 x2x4)xn (2n) g0 xn (x1 xi'4i x2x4)xn (2n) = (x1 x3'34 x2 x4)xn (2n) g00 x1: 9 . +. +-
(x1 xi'ij x2xj )xn (2n) = (x1 x3'34 x2 x4)xn(2n) g0 xn g00 x1: #, + (16) g0 xn 2 K(Hn1) g00 x1 2 K(Hn1(1n)): 9 En1 K(Hn1(2n)) = Kf(x1xi 'ij x2xj )xn(2n)g (x1x3 '34 x2x4)xn (2n) = en1(2n): , K(Hn1(2n)) K(Hn1) K(Hn1(1n)) K(en1 (2n)) K(En1): (21) 9 (20), (21) + (17). N - , En1 5 Pn1.
350
. .
# 5 +
! En1. 7+! | 3 D En1, =
X
+
ij
ij gij xn +
X j
j (xnx4'4j x2 xj )x1 + (xn x3 '34 x2x4 )x1 + (x1x3 '34 xnx4 )x2
ij j 2 K, gij xn 2 Hn1 En1. 7 5 , -
D2
= 0 ! . 3 3 x1 ! xy xn ! zt D
5+ (xyx3 '34 ztxn)x2 = 0 , + !, A 2 D ! +- = 0. N 3 3 xn ! xy x4 ! zt (xyx3 '34 ztx2)x1 = 0: G = 0.
, 3, 3 3 x4 ! xy xj ! zt +-
j (xyxn '4j ztx2)x1 = 0
+ j = 0. , X = ij gij xn = 0 - ! X ij gij = 0 F(C): G ij = 0. N - ,
! , En1 . # !
? +- + 3 +! + - + p. # p = 1
! - . 7 5 , - p > 1 ! n, - p 6 n ; 4 ( D +- + 1 6 p ; 1 6 (n ; 1) ; 4, + En;1p;1), En;1p;1 Pn;1p;1. 7 5 , - Enp Pnp .
-
N , - Enp p > 1 Enp = fenp(1r)(2n)g En;1p;1xn : r !"#$. r m + 1. + % ,
HK m (MV).
. - , HK r (AL G) = HK r (AL G). 8 , A]X = A c-dimG X = c-dimG X 6 m. 8 &
7, 1] HK r (BL G) = 0 r > m+1 X B. 8 # , HK r (AL G) = 0 r > m + 1 X A. 3. X |
c-dimG X 6 m, $
fAj gnj -
Z p
=1
. 1. fAj gki k 6 m + 2 ! " # k
$, % F(Aj1 : : : Aj ) (m ; r)- #% , r | ' " #. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #% ( % , ). i
=1
k
i
=1
k
1 -
# HK ; HK : : : HK m ( < 5 6). 7. X |
, c-dimG X 6 m HK m (XL G) = 0,
A X HK r (AL G) = 0 r > m. + % ,
HK m; (MV).
. 8 & 6 # # # r = m. 8
O , 7, 1], 11] HK m (XL G) HK m (XL G) ! HK m (AL G) HK m (AL G) < A X, , , . . HK m (AL G) = 0 A X. 4. X |
, c-dimG X 6 m HK m (XL G) = 0, $
fAj gnj 1
0
1
=1
. 1. fAj gki k 6 m + 1 ! " # k
$, % F(Aj1 : : : Aj ) (m ; 1 ; r)- #% , r | ' " #. i
k
=1
376
C. .
2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%.
1 # HK ; HK : : : HK m; ( < 5 7). 2. 1
m-
(m+1)-
Qm , ' @Qm S m , , 3 m+2 #
#/ # m+1 ( ). H , # , #, $ % & . I &' # , , 2 . 1 G ; m- # X, HK r (XL G) = 0 r 6= m HK m (XL G) 6= 0. 8. X |
, c-dimG X 6 m X % G ; m-",
HK m; HK m (MV).
. 1 # A 2 HK m; , . . HK r (AL G)r = 0 r 6 m ; 1. 8 & 6 HK (AL G) r > m+1. - X & & G ; m- , HK m (AL G) = 0, . . A 2 HK m . 5. X |
c-dimG X 6 m. $
fAj gnj i
=1
k
1
0
1
+1
+1
1
1
=1
. 1. fAj gki k 6 m + 1 ! " # k
$, % F(Aj1 : : : Aj ) (m ; 1 ; r)- #% , r | ' " #. ,'(mS
Aj $ (m + 2)
% i G ; m-". 2. fAj gki ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. i
=1
k
+2
i
=1
i
=1
k
. 8 & 2 1 k ,
S
fAj gki . Aj i (m ; 1)-2 . - . (m + 2) G;m- , . 2 . H.
#/ (m ; 1)-2 . (m+2) , i
i
=1
=1
377
2
,
, &
# #/ m. 8 # , & 7
Sk fAj gki
' k 6 m + 2 . Aj 2 . i 1
3 / # . 3. G , n 6 m+1, ,
#, .
G ; m- . 4. - 1 , , , ,
, ( ) . ,
. # 2 / & . - , , " # # , 2 . i
i
=1
=1
. - % $ $
Rm % , %
% ' m + 1 .
. - & & (m + 1)
, k 6 m + 1 & k . 8 / , 2 , , (m ; k)-2 . 1
4 / # . I , &'
' O {O {E 12] O 13], 1]. 6. k + 1 $ $ (
) (k ; 1)- #% '( k
$ ' %, $ ' % , %
$ # % '( ( ' % % ), #% .
. 8
4 & & 1 2 ( m = k ; 1). 8 # , '
(;1)-2 , . . , 2 . 1
3 ( HK 1 2 ) / # . R 14] ' O {O {E O. # # , # R . H # ,
&' # # R .
378
C. .
4.
Hl : : :Hm
(MV), Hl % ,
Pkr 1 6 k 6 m ; l + 1 0 6 r 6 k ; 1, % Pkk; = Hm ;k Pkr \Hl k; ;r = Hm;r r 6 k ; 2.
fAj gnj . 1. fAj gki k 6 m ; l + 1 ! " # k
$, % F (Aj1 : : : Aj ) Pkr , r | ' " #. 2. fAj gki ! " # F(x : : : xk ) ' r 6 m ; l F(Aj1 : : : Aj ) Hm;r . 1
+1
+
2
=1
i
=1
k
i
=1
1
k
. - Hm;r Pkr , 1 2 2 2 ) 1 . 1 ) 2. 1 2 k,
& 1 2. 1 k = 1 & , & Aj 2 P = Hm . B
fAj gkj , 2 6 k 6 m ; l + 1. 8 &
& F(Aj1 : : : Aj ) 2 Pkr . @ r = k ; 1, < . 1 # r 6 k ; 2. 8 &
fAj gkj & 1 2 # Hl : : : Hl k; . 8 # , & 2 2 F(Aj1 : : : Aj ) Hl k; ;r . 1 & & F (Aj1 : : : Aj ) 2 Pkr \ Hl k; ;r = Hm;r . F 2 2 4 & . 21. $
fAj gnj
X % m > ;1 . 1. fAj gki k 6 m + 2 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; rg. 2. fAj gki ! " # F (x : : : xk) ' r 6 m + 1 F (Aj1 : : : Aj ) (m ; r)- #% .
. H : PpKkr = fA: A | , X, HK (XL G) = 0 p = k ; 2 ; r : : : m ; rg 0 6 r 6 k ; 1 6 m + 1. 1
4 # HK ; HK : : : HK m , 1 2. 1
2 / # . 1 0
=1
k
=1
+
2
k
+
2
+
k
2
=1
i
=1
k
i
1
=1
k
1
0
379
61.
'( k + 1 $ $ (
) (k ; 1)- k $ ' %, $ ' %
.
. 8
3 & & 1 2 ( m = k ; 1). 8 # , '
. (k ; 1)-2 . 1
6 / # . H & 1
62 . m
% $ $ R k 6 m + 1 ' k ' % p 2 fk + 1 : : : m + 1g '( ' $ p (p ; 2)- % , $ ' % , %
$ # % '( ( ' % % ), #% .
8 I {1 15, . 159] Y Rm
(m ; 1)-
# , Rm n Y , . . PKm Rm
. O
, / , / 6 R 14], k = m. 31. X |
c-dimG X 6 m, $
fAj gnj +1
+2 0
2
=1
. 1. fAj gki k 6 m + 2 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; rg. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%. i
=1
k
i
=1
k
. 8 & 2 1 1 1
3. 1
3 / # . I '#& 4 41. X |
, c-dimnG X 6 m m K H (XL G) = 0, $
fAj gj =1
. 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg. i
k
=1
380
C. .
2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%. 51. X |
c-dimG X 6 m, $
fAj gnj i
=1
k
=1
. 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ mS p 2 fk ; 2 ; r : : : m ; 1 ; rg. ,'( Aj $ (m + 2) i
% G ; m-". 2. fAj gki ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. i
=1
k
+2
i
=1
i
=1
k
. 8 & 2 1 k , 1
S
fAj gki . Aj i (m ; 1)-2 . 1
5 / # . O
,
m-
m + 2 #
# m + 1. H 5 ,
m+2 #
m+1. C &' . i
i
=1
=1
1
9. + '
m- ' m- . 52. $
fAj gnj m- ' X . 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg. . m + 2 ' X . 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%.
. 8 & 9 X
& m-
& , G ; m- . S , /
& 1 5 , & & . =1
i
k
i
=1
k
1
=1
381
53.
$
fAj gnj m- ' X , % G ; m-", . 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg. 2. fAj gik ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. =1
i
=1
k
i
=1
k
. 8 & 9 /
& 1 5 , & & . 5. O '#& 2 , , . H 2
. H # . &' # . - # # E {3 # , . 1 , / # , &' 1{4,
. 1 5{9 . H A X #, & x y 2 A X ' / , &' , , A. 1
54.
$ $ $
fAj gnj
m- ' X ,
" " S m , . 1. fAj gik k 6 m+1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) (k ; 2 ; r)- . 2. fAj gki ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. =1
i
=1
i
=1
k
k
. J2 k ( & 2) 1 , k 6 m+1 & k
. C , , 2 . 8 # , 2
# # #. -
O 16] &
, , 2 . 1
2 / # . 6. 3 /
# #& E {3 , -
382
C. .
8 {8 17] ( , # , 2 ). 3 , # , 2 & ,
. 3 #
(
# E {3 ) ' '#& . 7. - / # 2 # , 2 . , # # ' , 2 ' . .
. 7. m- ' X $
fAj gnj fAj gki k 6 m + 1 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg m + 2
/ ' X . /
/ ' X .
. 8 & 5 .
2 , , # X. 71. X | #%
, c-dimG X 6 m $
fAj gnj fAj gki k 6 m+2 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; rg. =1
i
=1
k
2
=1
i
=1
k
% , ' $
, % ,
.
. 8 & 3 '
, 1
'
. 2 , , . 8. #
X, 5 7 # , 2 Z . H
, 2 &' # , , 7 7 & . 3 3 , 4 5 # # $ % . E # # /# ? 3 6 , 6 5 # # /#
, .
4. 8 &' 2 ' # . 2
2
1
1
1
1
1
2
4
383
x
3. !
) U #, X Y - , Y X # & & . 1 X q- # (q- # | X 2 C q ), S p - p 6 q. U #, X # Y - , x 2 X < Ux ' # Vx , Y Vx Ux # & & . 1 X q- # ( q- # | X 2 LC q ), # S p - p 6 q. " #$. n- Qn n ; B : : : Bn , %
j , j - nT ! , Bj . Bj 6= ?. 1
1
+1
+1
j
=1
. 1 # jT Bj = ?. - x 2 Qn n
+1
=1
' j, dist(x Bj ) > 0. 1 x 2 Qn f(x) 2 Qn &' 2 : 8 9n > < dist(x Bj ) > = x 7! f(x) = > nP : : dist(x Bi ) > j i n -
fBj gj ,
2 f(x) &, . . f(x) 2 @Qn x 2 Qn. & x 2 jn; j- 2 f(x) &, . . f(x) 2 jn; . A , 2 @Qn f . C # 2. 5.
X
$
fAj gm j , % ' k 6 m+1 % ' $ k S m;k - , '( $
mS Aj S m - . % $
+1
+1
=1
=1
+1
=1
1
1
+2
=1
+2
j
. =1
. r = 1 : : : m+2T # Pr | , & Aj . J2 j6 r mS n 6 m f : Qmn ! Aj , Qmn =
+2
+1
j
+1
=1
384
C. .
n-
(m+1)-
, (q ; 1)-
hj : : : jq i ( , T &' q /)
& f(hj : : : jq i) Aj . j 6 j1:::j T J , n = 0 f(hri) = Pr 2 Aj . 1
1
=
q
j6 r =
B # q-
hj : : : jq jq i. 1 & 2 , fT ,
& f(@ hj : : : jq jq i) Aj . j 6 j1 :::j +1 - 2 q-
(q ; 1)-
, # 2 # . 1 ' , , # q-
. mS 1 # # f : Qmm ! Aj | , # j . 8 & # mS F : Qm ! Aj . E j Bj = F ; (Aj ) & Qm . 1 ,
Bj #, & / j. 8 # ,
W ,
& . -
& Aj . 6.
X
$ $
fAj gm j , % ' k 6 m + 2 '( ' $ k S k; - . % $ 1
1
+1
+1
=
q
+2
+1
=1
+2
+1
=1
1
+1
+2
=1
2
.
. j = 1 : : : m + 2 # Pj | , & Aj . J2 mS n 6 m + 1 f : Qmn ! Aj , j (qS; 1)-
hj : : : jq i
& f(hj : : : jq i) Aj . +2
+1
=1
1
1
j j1 :::jq =
J , n = 0 f(hj i) = Pj 2 Aj . B # q-
hj : : : jq jq i. 1 & 2 , fS ,
& f(@ hj : : : jq jq i) Aj . j j1 :::j +1 - 2 q-
(q ; 1)-
, # 2 # . 1 ' , , # q-
. 1
1
+1
+1
=
q
385
mS 1 # # f : Qm ! Aj | , # j . E Bj = f ; (Aj ) & Qm , &'
&
W 18, . 215]. 8 mT mT # , Aj = f Bj 6= ?. +2
+1
=1
1
+1
+2
j
+2
j
=1
7.
=1
Rm
fAj gnj , n > m, % % ' $ k k 6 n+1 S n;k - . % $ . +2
=1
. 1 # 5, n
S f : Qnn ! Aj , (q ; 1)-
j T A. hj : : : jq i
& f(hj : : : jq i) j j 6 j1 :::j 1 F : Qn ! Rm. 8
B 19] ' & &' , F( ) \ F( ) 6= ?. nT - F ( ) \ F ( ) = f( ) \ f( ) Aj , j . 8.
$ % r 6 m . 1. 0 m 6 2r. 2.
% n > m + 1 $
fAj gnj Rm, % ' n + 1 ; r 6 k 6 n + 1 % ' $ k S n;k - , % $ +2
+1
=1
1
1
=
q
+1
1
2
1
2
+2
1
2
1
2
=1
+2
=1
. 3.
% n > m + 1 $ $ r- $ fAj gnj Rm, % ' n + 1 ; r 6 k 6 n + 1 % ' $ k (r ; 1)- , % $ . +2
=1
. J 2 1 ) 2. 1 #n
S 5, f : Qnr ! Aj , j (qT; 1)-
hj : : : jq i
& f(hj : : : jq i) Aj . j 6 j1:::j 8 '<
3 O {; 20, 1.3 n = n+1, s = r+1, j = 2, p = 2, m = n;m;1, l = m], 21, N = n+1, s = r+1, j = 2, q = 2, k = n ; m ; 1, M = Rm] ' & &' nT , f( ) \ f( ) 6= ?. - f( ) \ f( ) Aj , j . J 2 2 ) 3 . +2
+1
=1
1
=
1
q
+2
1
2
1
2
1
2
=1
386
C. .
J 2 3 ) 1. @
m > 2r + 1,
C< {1 , Qnr Rm. U #, , Qnr
. (F #, Rm n + 2 ' .) 3 Aj #
k = n + 2L
