Tese-Campos

8/18/2019 Tese-Campos

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P mCoh,p


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p

p

l1 l2 2 l2 l∞

2

p

p∗ 1/p + 1 /p ∗= 1


13/94

p

p

p


14/94

p


15/94

•

K

R

C

K = R C

• A n ∈ N An n

A

• N n N

a j1 ,...,j n ∈ l1(F ; Nn)

a j1 ,...,j n ∈ l1(F )

• E F

K L (E ; F )

E F.

• L(E 1,...,E n ; F ) n

E 1 × · · · × E n F E 1 = · · · = E n L(n E ; F )

• n E

F P (n E ; F )

Ls (n E ; F ) n E n

F P̆ Ls (n E ; F ) P (x) = P̆ (x,...,x )

• E F K

u E F

Graf u = {(x, y) ∈ E × F ; y = u (x)} .


16/94

u Graf u E × F

• u v u ◦ v uv.

M (u1,...,u n )

n

M (u1,...,u n )(x1,...,x n ) := M (u1(x1),...,u n (xn )) M n

u i i = 1,...,n

• E,F,G,E i , G i ,H,...

E

· ; · E

sup

BE {x ∈ E ; x ≤ 1} .

• E E .

• u E F u (E )


17/94

p

E 1 ≤ p ≤ ∞ (x i)∞i=1 E

p (|| xi || )∞i=1 ∈ l p

p l p(E )

|| (x i)∞i=1 || p = (∞i=1 || x i ||

p)

1/p

, se 1 ≤ p < ∞sup

i|| xi || , se p = ∞ .


18/94

(x i)∞i=1 E p (ϕ(x i))∞i=1 ∈ l p

ϕ ∈ E p lw p (E )

|| (xi)∞i=1 || w,p =

supψ∈B

E

( ∞i=1 |ψ(x i)| p)1/p , se 1 ≤ p < ∞

supi

sup||ϕ||≤ 1

|ϕ(xi)| , se p = ∞ .

(xi)∞i=1 E p

∞i=1 ϕi(x i) (ϕi)

∞i=1 ∈

lw p∗(E ) 1

p + 1 p∗ = 1

p E

l p E

∞i=1 ϕi(x i)

∞i=1 |ϕi(x i)|

(xi)∞i=1 E ∞i=1 ϕi(xi)

(ϕi)∞i=1 ∈ lw p∗(E ) ∞i=1 |ϕi(xi)| (ϕi)∞i=1 ∈ lw p∗(E )

∞i=1 ϕi(xi) (ϕi)

∞i=1 ∈

lw p∗(E )

ψ j =

ϕ j , se ϕ j (x j ) ≥ 0

−ϕ j , se ϕ j (x j ) < 0

ψ j = ϕ j e− iθ j θ j ϕ j (x j )

∞ j =1 ψ j (x j )

∞

j =1

|ϕ j (x j )| =∞

j =1

ψ j (x j )


19/94

p = 1

q = p∗ l p E

|| (xi)∞i=1 || C,p = sup|| (ϕi )∞i =1 || w,p ∗≤ 1

∞

i=1|ϕi(xi)| .

(l p(E )) = l p∗(E ) 1 p + 1 p∗ = 1

1 ≤ p ≤ ∞

i) l p E ⊂ l p(E ) ⊂ lw p (E )

ii ) p = 1, l p E = l p(E ) p = ∞ , l p(E ) = lw p (E )

l p(E ) ⊂ lw p (E ) l∞ (E ) = lw∞ (E )

(l p(E )) = l p∗(E ) (x i) ∈ l p E

|| (x i)∞i=1 || p = sup(ϕi )∞i =1 ∈B l p∗ ( E )

∞

i=1

|ϕi(xi)| ≤ sup(ϕi )∞i =1 ∈B l w

p∗ ( E )

∞

i=1

|ϕi(xi)| = || (xi)∞i=1 || C,p ,

l p E ⊂ l p(E )

p = 1 p∗= ∞ l∞ (E ) = lw∞ (E )

|| (xi)∞i=1 || p = || (x i)∞i=1 || C,p l p E = l p(E )

T ∈ L(E ; F )

T s : l p (E ) → l p(F ) denido por (xi)∞i=1 → (T (xi))

∞i=1 ,

T s

(xi)∞i=1 ∈ l p(E )

T s ((x i)∞i=1 ) p

= (T (x i))∞i=1 p ≤ T (xi)∞i=1 p .

l p(E ) l p F

1 < p ≤ ∞ T ∈ L(E ; F ) p

(T (xi))∞i=1 ∈ l p F sempre que (xi)∞i=1 ∈ l p(E ) ,


20/94

T : l p (E ) → l p F ; (x i)∞i=1 → (T (xi))

∞i=1

D p(E ; F )

p D1(E ; F ) = L(E ; F )

p

T ∈ L(E ; F )

1 p +

1 p∗ = 1

i) T p

ii ) C > 0

∞

i=1

|ϕi(T (x i)) | ≤ C || (x i)∞i=1 || p|| (ϕi)∞i=1 || w,p ∗ ,

(xi)∞i=1 ∈ l p(E ) (ϕi)∞i=1 ∈ lw p∗(F )

iii ) C > 0

m

i=1

|ϕi(T (x i)) | ≤ C || (x i)mi=1 || p|| (ϕi)mi=1 || w,p ∗ ,

m ∈N , xi ∈ E, ϕi ∈ F , i = 1,...,m

(i) ⇒ (ii ) T p

T : lw p∗(F ) × l p(E ) → l1

((ϕi)∞i=1 , (x i)∞i=1 ) → (ϕi(T (x i)))

∞i=1 ,

T

(ϕk , xk) → (ϕ, x) em lw p∗(F ) × l p(E )

e T (ϕk , xk) → y em l1 ,


21/94

T (ϕ, x) = y

y = limk→∞

T (ϕk , xk) = limk→∞

ϕki (T (x

ki ))

∞i=1 ,

limk→∞

ϕki (T (x

ki )) = yi , para todo i ∈N .

limk→∞

ϕki (T (x

ki )) = ϕi(T (x i)) , para todo i ∈N .

ϕi(T (x i)) = yi , para todo i ∈N

, T (ϕ, x) = y T T ∞

i=1

|ϕi(T (xi)) | = T ((ϕi)∞i=1 , (xi)∞i=1 )l1

≤ T ||(xi)∞i=1 || p|| (ϕi)∞i=1 || w,p ∗ .

(iii ) ⇒ (ii ) (x i)∞i=1 ∈ l p(E ) (ϕi)∞i=1 ∈ lw p∗(F )

∞

i=1

|ϕi(T (x i)) | = supm

m

i=1

|ϕi(T (xi)) |

≤ supm

(C || (xi)mi=1 || p|| (ϕi)mi=1 || w,p ∗)

= C || (xi)∞i=1 || p|| (ϕi)∞i=1 || w,p ∗.

(ii ) ⇒ (i) (ii ) ⇒ (iii )

C D p(E ; F ) d p(·)

T = d p(T )

T 1, T 2 ∈ D p(E ; F ) λ ∈ K T 1 + λT 2 m ∈ N , xi ∈ E, ϕi ∈ F , i =

1,...,n m

i=1

|ϕi((T 1 + λT 2)(x i)) | ≤m

i=1

|ϕi(T 1(xi)) | +m

i=1

|ϕi(λT 2(xi)) |

=m

i=1

|ϕi(T 1(xi)) | + |λ |m

i=1

|ϕi(T 2(xi)) |

≤ (d p(T 1) + |λ |d p(T 2)) || (x i)mi=1 || p|| (ϕi)mi=1 || w,p ∗ ,


22/94

T 1 + λT 2 ∈ D p(E ; F ) D p(E ; F ) L(E ; F )

d p(·) d p(λT ) ≤ | λ |d p(T )

d p(T ) = 0 m = 1 |ϕ(T (X )) | = 0 ϕ ∈ F

x ∈ E ||T (x)|| = 0 x ∈ E

T = 0

T = sup|| (x i )∞i =1 || p , || (ϕi )

∞i =1 || w,p ∗≤ 1

|| (ϕi(T (x i))∞i=1 || l1

= sup|| (x i )∞i =1 || p , || (ϕi )

∞i =1 || w,p ∗≤ 1

∞

i=1

|ϕi(T (x i)) |

≤ sup|| (x i )∞i =1 || p , || (ϕi )

∞i =1 || w,p ∗≤ 1

(d p(T ) || (xi)∞i=1 || p || (ϕi)∞i=1 || w,p ∗) = d p(T ) ,

T ≤ d p(T ) T = d p(T )

T ∈ D p(E ; F )

T = T

T = d p(T ) ) T = sup|| (x i )∞i =1 || p ≤ 1 T ((x i)

∞i=1 )

C,p= sup

|| (x i )∞i =1 || p ≤ 1|| (T (xi))∞i=1 )|| C,p

= sup|| (x i )∞i =1 || p ≤ 1

sup|| (ϕi )∞i =1 || w,p ∗≤ 1

∞

i=1

|ϕi(T (xi)) |

= sup|| (x i )∞i =1 || p , || (ϕi )

∞i =1 || w,p ∗≤ 1

|| (ϕi(T (xi))) ∞i=1 || l1

= T .

p

p∗

p

E idE : E → E p dim E < ∞


23/94

T

: F → E T : E → F

idE : E → E p id

E : E → E

p∗

E E

dim E < ∞ id

E : E → E p∗

idE : E → E p

p1 ≤ p2

D p2 (E ; F ) ⊂ D p1 (E ; F ).

p

1 < p ≤ ∞ E j , F j = 1,...,n T ∈ L(E 1,...,E n ; F ) p

T x(1)i ,...,x(n )i

∞

i=1∈ l p F sempre que x

( j )i

∞

i=1∈ lnp (E j ) , j = 1, ..., n ,

T : lnp (E 1) × · · · × lnp (E n ) → l p F x(1)i ∞i=1 , ..., x(n )i ∞i=1 → T x(1)i ,...,x (n )i ∞i=1

n p

LCoh,p (E 1,...,E n ; F ) LCoh,p (E 1,...,E n ; F )

L(E 1,...,E n ; F )


24/94

T ∈ L(E 1,...,E n ; F ) 1 p + 1 p∗ = 1

i) T p

ii ) C > 0

∞

i=1

ϕi T x(1)i ,...,x

(n )i ≤

≤ C ∞

i=1

x(1)inp

1/np

... ∞

i=1

x(n )inp

1/np

|| (ϕi)∞i=1 || w,p ∗ ,

x( j )i∞

i=1 ∈ lnp (E j ) , j = 1,...,n (ϕi)∞i=1 ∈ lw p∗(F )

iii ) C > 0

m

i=1

ϕi T x(1)i ,...,x

(n )i ≤

≤ C m

i=1

x(1)inp

1/np

... m

i=1

x(n )inp

1/np

|| (ϕi)mi=1 || w,p ∗ ,

m ∈N , x( j )i ∈ E j , ϕi ∈ F , i = 1,... ,m , j = 1,...,n

(i) ⇒ (ii ) T p

T : lw p∗(F ) × lnp (E 1) × · · · × lnp (E n ) −→ l1

(ϕi)∞i=1 , x(1)i

∞

i=1, ..., x(n )i

∞

i=1 −→ ϕi T x(1)i ,...,x

(n )i

∞

i=1

(n + 1)

T

∞

i=1

ϕi T x(1)i ,...,x

(n )i = T (ϕi)

∞i=1 , x

(1)i

∞

i=1, ..., x(n )i

∞

i=1 l1

≤ T x(1) np · · · x(n )

np ||(ϕi)∞i=1 || w,p ∗ .

(ii ) ⇒ (i) (ii ) ⇒ (iii )


25/94

(iii ) ⇒ (ii )

C ||T || Coh,p LCoh,p (E 1,...,E n ; F ) T = ||T || Coh,p

1 < p ≤ ∞ E, F n P ∈ P (n E ; F ) p

(P (x i))∞i=1 ∈ l p F sempre que (x i)∞i=1 ∈ lnp (E ) .

n p

P Coh,p (n E ; F ) P Coh,p (n E ; F )

P (n E ; F )

P ∈ P (n E ; F ) p ̌P ∈ Ls (n E ; F ) p

P̌ p (x i)∞i=1 ∈ lnp (E )

(P (x i))∞i=1 = ( P̌ (xi ,...,x i))∞i=1 ∈ l p F .

P p

x0 = 0

n!2n P̌ x(1)i ,...,x(n )i =

εj = ± 1

ε1 · · · εn P ε1x(1)i + · · · + εn x(n )i ,

i ∈ N x( j )i∞

i=1∈ lnp (E ) j = 1,...,n P

p

P ε1x(1)i + · · · + εn x

(n )i

∞

i=1∈ l p F ,

ε j

P̌ x(1)i ,...,x (n )

i

∞

i=1∈ l

pF

̌P p


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P ∈ P (n E ; F ) 1 p + 1 p∗ = 1

i) P p

ii ) C > 0

∞

i=1

|ϕi(P (xi)) | ≤ C ∞

i=1

|| xi || np1/p

|| (ϕi)∞i=1 || w,p ∗ ,

(xi)∞i=1 ∈ lnp (E ) (ϕi)∞i=1 ∈ lw p∗(F )

iii ) C > 0

m

i=1

|ϕi(P (xi)) | ≤ C m

i=1

|| xi || np1/p

|| (ϕi)mi=1

|| w,p ∗ ,

m ∈N , xi ∈ E, ϕi ∈ F , i = 1,...,m

C ||P || Coh,p P Coh,p (n E ; F )

(i) ⇒ (ii ) : P p

P̌ p

∞

i=1

|ϕi(P (xi)) | =∞

i=1

|ϕi( P̌ (x i ,...,x i)) |

≤ C ∞

i=1

|| xi || np1/p

|| (ϕi)|| w,p ∗ .

(ii ) ⇒ (i) (ii ) ⇒ (iii )

(iii ) ⇒ (ii ) (xi)∞i=1 ∈ lnp (E ) (ϕi)∞i=1 ∈ lw p∗(F )

∞

i=1

|ϕi(P (xi)) | = supm

m

i=1

|ϕi( P̌ (xi ,...,x i)) |

≤ supm

C m

i=1

|| x i || np1/p

|| (ϕi)mi=1 || w,p ∗

= C ∞

i=1

|| xi || np1/p

|| (ϕi)∞i=1 || w,p ∗ .

C

P Coh,p (n E ; F )


27/94

p

p

1 < p ≤ ∞ E j , F j = 1,...,n n

T : E 1 × ... × E n → F p

C > 0 x( j )1 ,...,x

( j )m ∈ E j ϕ1,...,ϕm ∈ F m ∈N

m

i=1

ϕi T x(1)i ,...,x

(n )i ≤ C

m

i=1

n

j =1

x( j )i p

1/p

|| (ϕi)mi=1 || w,p ∗ .

m

i=1

x(1)i ... x(n )i

p1/p

≤ m

i=1

x(1)inp

1/np

... m

i=1

x(n )inp

1/np

,

⇒ ⇒

X 1,...,X n , Y E 1,...,E r H

X 1 × · · · × X n Y K 1,...,K t

G1,...,G t

R j : K j × E 1 × · · · × E r × G j → [0, ∞ ) , j = 1, ..., t ,

S : H × E 1 × · · · × E r × G1 × · · · × Gt → [0, ∞ )

1.

x(l)

∈ E l

b ∈ G j

( j, l ) ∈ {1,...,t }×{ 1,...,r }

(R j )x (1) ,...,x ( r ) ,b : K j → [0, ∞ ) ,


28/94

(R j )x (1) ,...,x ( r ) ,b(ϕ) = R j ϕ, x(1) ,...,x (r ) , b

2.

R j ϕ, x(1) ,...,x (r ) , η j b( j ) ≤ η j R j ϕ, x(1) ,...,x (r ) , b( j )

S f, x (1) ,...,x (r ) , α 1b(1) ,...,α t b(t ) ≥ α1...α t S f, x (1) ,...,x (r ) , b(1) ,...,b(t )

ϕ ∈ K j , x(l) ∈ E l l = 1,...,r 0 ≤ η j , α j ≤ 1, b( j ) ∈

G j j = 1,...,t f ∈ H

0 < p1,...,p t , p0 < ∞

1 p0 =

1 p1 + · · · +

1 pt

f : X 1 × · · · × X n → Y H R1,...,R t S ( p1,...,p t ) C > 0

m

i=1

S f, x (1)i ,...,x(r )i , b

(1)i ,...,b

(t )i

p01/p 0

≤ C t

k=1

supϕ∈K k

m

i=1

Rk ϕ, x(1)i ,...,x

(r )i , b

(k)i

pk1/p k

x(s)1 ,...,x (s)m ∈ E s , b(s)1 ,...,b(s)m ∈ Gl , m ∈N (s, l ) ∈ {1,...,r } × { 1,...,t }

f ∈ H R1,...,R t S ( p1,...,p t ) C > 0 µk K k , k = 1,...,t

S f, x (1) ,...,x (r ) , b(1) ,...,b(t ) ≤ C t

k=1 K k Rk ϕ, x(1) ,...,x (r ) , b(k) pk dµk 1/p k , x(l) ∈ E l, l = 1,...,r b(k) ∈ Gk k = 1,...,t

1 < p ≤ ∞ 1/p + 1 /p ∗= 1 T ∈ L(X 1,...,X n ; Y )


29/94

i) C > 0

m

i=1

ϕi T x(1)i ,...,x

(n )i ≤ C

m

i=1

n

j =1

x( j )i p

1/p

|| (ϕi)mi=1 || w,p ∗,

m ∈N , x( j )i ∈ X j , ϕi ∈ Y , i = 1,... ,m , j = 1,...,n

ii ) C > 0

m

i=1

ϕi T x(1)i ,...,x

(n )i ≤ C

n

j =1

m

i=1

x( j )inp

1/np

|| (ϕi)mi=1 || w,p ∗ ,

m ∈N , x( j )i ∈ X j , ϕi ∈ Y , i = 1,... ,m , j = 1,...,n

iii ) C > 0 µ BY

|ϕ(T (x1,...,x n )) | ≤ C || x1|| ... ||xn || B Y |ψ(ϕ)| p∗dµ(ψ)1/p ∗

,

x j ∈ X j , ϕ ∈ Y , j = 1,...,n

(i) ⇒ (ii ) :

(ii ) ⇒ (iii ) :

t = n + 1 r = 1

E 1 = {0}

K k = {0} , k = 1,...,n K n +1 = BY Gk = X k , k = 1,...,n, e Gn +1 = Y

H = L(X 1,...,X n ; Y )

p0 = 1, pk = np, k = 1,...,n e pn +1 = p∗

S T, 0, x (1) ,...,x (n ) ,ϕ = ϕ T x(1) ,...,x (n )

Rk γ, 0, x (k) = x(k) , k = 1,...,n

Rn +1 (ψ, 0,ϕ) = |ψ(ϕ)|

m

i=1

S T, x(1)i ,...,x(r )i , b

(1)i ,...,b

(t )i

p01/p 0

= m

i=1

S T, 0, x (1)i ,...,x(n )i ,ϕi

p01/p 0

=m

i=1

ϕi T x(1)i ,...,x

(n )i ,


30/94

n +1

k=1

supϕ∈K k

m

i=1

Rk ϕ, x(1)i ,...,x

(r )i , b

(k)i

pk1/p k

= supψ∈K n +1

m

i=1

Rn+1 (ψ, 0,ϕi) pn +11/p n +1

·n

k=1

supγ ∈K k

m

i=1

Rk γ, 0, x(k)i

pk1/p k

= supψ∈B

Y

m

i=1

|ψ (ϕi) | p∗

1/p ∗

·n

k=1

m

i=1

x(k)inp

1/np

= || (ϕi)mi=1 || w,p ∗ ·n

k=1

m

i=1

x(k)inp

1/np

.

T R1,...,R t S ( p1,...,p t )

C > 0

µk K k , k = 1,...,t

S (T, x1,...,x r , b1,...,bt) ≤ C t

k=1 K k Rk(ϕ, x1,...,x r , bk) pk dµk 1/p k ,

|ϕ(T (x1,...,x n )) | ≤ C n

k=1 K k || xk ||np dµk

1/np

B Y |ψ(ϕ)| p∗dµ(ψ)

1/p ∗

≤ C || x1|| ... || xn || B Y |ψ(ϕ)| p∗dµ(ψ)1/p ∗

.

(iii ) ⇒ (i) : m ∈ N 1 ≤ i ≤ m

ϕi T x(1)i ,...,x

(n )i ≤ C x

(1)i ... x

(n )i

B Y |ψ(ϕi)|

p∗

dµ(ψ)

1/p ∗

.


31/94

m

i=1

ϕi T x(1)i ,...,x

(n )i

≤ C m

i=1

x(1)i ... x(n )i B Y |ψ(ϕi)| p∗dµ(ψ)

1/p ∗

≤ C m

i=1

x(1)i ... x(n )i

p1/p m

i=1 B Y |ψ(ϕi)| p∗dµ(ψ)1/p ∗

= C m

i=1

x(1)i ... x(n )i

p1/p

B Y m

i=1

|ψ(ϕi)| p∗

dµ(ψ)1/p ∗

≤ C m

i=1x(1)i ... x(n )i

p1/p

supψ∈B

Y

m

i=1|ψ(ϕi)| p∗

1/p ∗

= C m

i=1

x(1)i ... x(n )i

p1/p

|| (ϕi)mi=1 || w,p ∗ .

f : X 1 × · · · × X n → Y H

0 < p∗,u ,s ,p 1,...,p t− 1, q 1,...,q t− 1 < ∞ ,

1u

= 1 p1

+ · · · + 1 pt− 1

+ 1 p∗

e 1

s =

1q 1

+ · · · + 1q t− 1

+ 1 p∗

.

Rk( x 1 ,...,x r ,b ) (·) x1,...,x r , b 1 ≤ k ≤ t − 1

i) f R1,...,R t S ( p1,...,p t− 1, p∗)

ii ) f R1,...,R t S (q 1,...,q t− 1, p∗)

f R1,...,R t S ( p1,...,p t− 1, p∗)

C

µi K i i = 1,...,t

S (f, x 1,...,x r , b1,...,bt) ≤ C t

i=1 K i R i(ϕ, x1,...,x r , bi) pi dµi1/p i

.


32/94

f R1,...,R t S ( p1,...,p t− 1, p∗)

C µ K t

S (f, x 1,...,x r , b1,...,bt ) ≤ C

t− 1

i=1R i(ϕ, x1,...,x r , bi) · K t R

t (ϕ, x1,...,x r , bt ) p∗

dµ

1/p ∗

,

ϕ ∈ K i

K i R i(ϕ, x1,...,x r , bi) pi dµi 1/p i = R i(ϕ, x1,...,x r , bi) , i = 1,...,t − 1. f R1,...,R t S (q 1,...,q t− 1, p∗)

C

µ K t

S (f, x 1,...,x r , b1,...,bt ) ≤ C t− 1

i=1

R i(ϕ, x1,...,x r , bi) · K t R t (ϕ, x1,...,x r , bt ) p∗dµ 1/p∗

,

p∗∈ (1, ∞ )

Γ = (r, q ) ∈ [1, ∞ ) × (1, ∞ ) : 1r

= 1q

+ 1 p∗

.

C r,q (E ; F ) T ∈ L(E ; F )

C > 0

m

j =1

|ϕi (T (xi)) | r1/r

≤ C (x i)mi=1 q (ϕi)mi=1 w,p ∗

m xi ∈ E ϕi ∈ F i = 1,...,m

C r 1 ,q1 (E ; F ) = C r 2 ,q2 (E ; F )

(r 1, q 1) , (r 2.q 2) ∈ Γ.

C r,q (E ; F ) = C 1,p(E ; F ) = D p(E ; F ) ,

1 = 1/p + 1 /p ∗ (r, q ) ∈ Γ

C 1,q

(E ; F )

p < q


33/94

1 < p < ∞ 1 = 1/p + 1 /p ∗ p < q C 1,q(E ; F ) = {0}

E = {0} F = {0} E = {0}

F = {0}

(λ i)∞i=1 = ( α iβ i)∞i=1 /∈ l1 (α i)∞i=1 ∈ l p∗ (β i)∞i=1 ∈ lq

T ∈ C 1,q(E ; F ) 0 = x ∈ E 0 = ϕ ∈ F m ∈N

m

i=1

|ϕ(T (λ ix)) | =m

i=1

|ϕ(T (α iβ ix)) | =m

i=1

|(α iϕ)(T (β ix)) |

≤ C || (β ix)mi=1 || q || (α iϕ)mi=1 || w,p∗ .

|ϕ(T (x)) |m

i=1

|λ i | ≤ C || x|||| (β i)mi=1 || q supψ∈B

F

m

i=1

|ψ(α iϕ)| p∗

1/p ∗

= C || x|||| (β i)mi=1 || q supψ∈B

F

|ψ(ϕ)| m

i=1

|α i | p∗

1/p ∗

= C || x|||| (β i)mi=1 || q ||ϕ|| m

i=1

|α i | p∗

1/p ∗

.

sup ||ϕ||≤ 1 sup|| x ||≤ 1

|| T ||m

i=1

|λ i | ≤ C || (β i)mi=1 || q m

i=1

|α i | p∗

1/p ∗

,

T = 0 (λ i)∞i=1 ∈ l1


34/94

p

I L

E F, I (E ; F ) := L (E ; F ) ∩ I

i) I (E, F ) L (E ; F )

ii )

u ∈ L(E ; F )

v ∈ I (F, G )

t ∈ L(G, H ) , tvu ∈ I (E ; H ) .


35/94

( I , · I ) I · I : I → [0, ∞ )

i) · I

I (E ; F ) E F

ii ) idK I = 1, idK : K → K idK (x) = x;

iii ) u ∈ L(E, F ) v ∈ I (F ; G) t ∈ L(G; H )

tvu I ≤ t v I u .

I (E ; F ) I · I ,

I

( I , · I ) t ≤ t I t ∈ I .

ϕ ∈ E ϕ = ϕ I ϕ ϕ ∈ I

ϕ I = idK ϕ I ≤ idK I ϕ = ϕ .

E 1,...,E n A ∈ L(E 1,...,E n ; F ) m ∈N ϕ( j )i ∈ E j bi ∈ F i = 1,...,m j = 1,...,n,

A (x1,...,x n ) =m

i=1

ϕ(1)i (x1) · · · ϕ

(n )i (xn ) bi .

Lf (E 1,...,E n ; F ) Lf (E 1,...,E n ; F ) L (E 1,...,E n ; F ) .


36/94

M

n ∈ N E 1,...,E n F M (E 1,...,E n ; F ) :=

L (E 1,...,E n ; F ) ∩ M

i) M (E 1,...,E n ; F ) L (E 1,...,E n ; F ) n

ii ) A ∈ M (E 1,...,E n ; F ) , u j ∈ L(G j , E j ) j = 1,...,n t ∈ L(F ; H ) , tA (u1,...,u n ) ∈ M (G1,...,G n ; H ) .

M n

M n =E 1 ,...,E n ,F

M (E 1,...,E n ; F )

n

(M , · M ) M · M : M −→ [0, ∞ ) ,

i) · M M (E 1,...,E n ; F ) E 1,...,E n F n ∈N

ii ) idK n M = 1 idK n : K n −→ K idK n (x1,...,x n ) =x1 · · · xn n ∈N

iii ) M ∈ M (E 1,...,E n ; F ) u j ∈ L(G j , E j ) j = 1,...,n t ∈L (F ; H ) ,

tM (u1,...,u n ) M ≤ t M M u1 · · · un .

n M n

n M (E 1,...,E n ; F )

· M M

M n .

M n M n K


37/94

Lf (E 1,...,E n ; F )

M ∈ Lf (E 1,...,E n ; F )

M (x1,...,x n ) =m

i=1ϕ(1)i (x1) · · · ϕ(n )i (xn ) bi

u j ∈ L(G j ; E j ) j = 1,...,n t ∈ L(F ; H ) .

tM (u1,...,u n ) (x1,...,x n ) = tM (u1 (x1) ,...,u n (xn ))

= t m

i=1

ϕ(1)i (u1 (x1)) · · · ϕ

(n )i (un (xn )) bi

=

m

i=1 t ϕ(1)

i (u1 (x1)) · · · ϕ(n )

i (un (xn )) bi

= m

i=1

ϕ(1)i (u1 (x1)) · · · ϕ

(n )i (un (xn )) t (bi)

= m

i=1

ϕ(1)i u1 (x1) · · · ϕ

(n )i un (xn ) t (bi) ,

ϕ( j )i u j ∈ G j t (bi) ∈ H

tM (u1,...,u n ) ∈ Lf (G1,...,G n ; H ) .

(M , · M ) M ≤ M M M M

M ∈ M (E 1,...,E n ; F ) , ϕ ∈ F x j ∈ E j j =

1,...,n j = 1,...,n, R j : K →E j , R j (λ) = λx j

R j = x j

ϕM (R1,...,R n ) (λ1,...,λ n ) = ϕM (λ1x1,...,λ n xn ) = λ1 · · · λn (ϕM ) (x1,...,x n ) .

ϕM (R1,...,R n ) = (ϕM ) (x1,...,x n ) idK n ,


38/94

|(ϕM ) (x1,...,x n )| = |(ϕM ) (x1,...,x n )| idK n M

= (ϕM ) (x1,...,x n ) idK n M

= ϕM (R1,...,R n ) M

≤ ϕ M M R1 · · · Rn .

M (x1,...,x n ) = supϕ ≤ 1

|(ϕM ) (x1,...,x n )|

≤ supϕ ≤ 1

ϕ M M R1 · · · Rn

= M M x1 · · · xn ,

M ≤ M M .

P (n

E ; F )

P (x) = ϕ(x)n

b

ϕ ∈ E b ∈ F P f (n E ; F )

n P A (n E ; F ) P f (n E ; F )

P (n E ; F ) , P A (n E ; F )

U n ∈ N E

F

U (n

E ; F ) := P (n

E ; F ) ∩ U

i) U (n E ; F ) P (n E ; F ) n

ii ) u ∈ L(G; E ) P ∈ U (n E ; F ) t ∈ L(F ; H ) , tP u ∈ U (n G; H ) .

n ∈N

U n :=

E,F

U (n E ; F )

n


39/94

( U , · U ) U · U : U −→ [0, ∞ ) ,

i) · U U (n E ; F ) E F n ∈N

ii ) idnK U = 1 idnK : K −→ K idnK (x) = xn

iii ) u ∈ L(G, E ) , P ∈ U (n E ; F ) t ∈ L(F ; H ) , tP u U ≤t P U u

n .

U (n E ; F ) · U , U

U n .

n P n

M

P M = P ∈ P n ; P̆ ∈ M, n ∈N ,

P P M := P̆ M , M

M P M

E, F P 1, P 2 ∈ P M (n E ; F ) k ∈K .

P̆ 1, P̆ 2 ∈ M (n E ; F ) M (n E ; F ) P̆ 1 + k P̆ 2 ∈ M (n E ; F )

(P 1 + kP 2)∨ = P̆ 1 + k P̆ 2 ∈ M (n E ; F ) ,

P 1 + kP 2 ∈ P M (n E ; F ) .

P n

P̆ P ∈ P M (n E ; F ) .

u ∈ L(G, E ) P ∈ P M (n E ; F ) t ∈ L(F ; H )

tP̆ (u,...,u ) (x,...,x ) = tP̆ (u (x) ,...,u (x)) = tP u (x) = ( tP u )∨ (x,...,x ) ,


40/94

(tP u )∨ = tP̆ (u,...,u ) ∈ M (n E ; F ) ,

tP u ∈ P M

(n G; H )

· P M P M (n E ; F )

idnK (idnK )∨ = idK n

idnK P M = idK n M = 1.

t ∈ L(F ; H ) P ∈ P M (n E ; F ) u ∈ L(G, E )

tP u P M = tP̆ (u,...,u ) M≤ t P̆

Mu n = t P P M u

n .

M ∨ : P M (n E ; F ) −→ M (n E ; F ) : P −→ P̆

P M (n E ; F )

∨ (P M (n E ; F )) M (n E ; F ) .

limk→∞

∨ (P k) = A ∈ M (n E ; F )

M P̆ k − AM

→ 0 · ≤ · M P̆ k − A → 0

Ls (n E ; F ) L (n E ; F ) A ∈ Ls (n E ; F )

Â ∈ P M (n E ; F ) Â∨

= A ∈ M (n E ; F ) . A = ∨ Â ∈ ∨ (P M (n E ; F ))

∨ (P M (n E ; F )) P M

D p

p

(D p, d p)


41/94

0 = T ∈ L(E ; F ) ϕi ∈ F , i = 1,...,m

|| (ϕiT )mi=1 || w,p ∗ = supy∈B E

|| (ϕi(T (y))) mi=1 || p∗

= ||T || supy∈B E

ϕi T (y)|| T ||

m

i=1 p∗

≤ || T || suph∈B F

|| (ϕi(h))mi=1 || p∗

= ||T |||| (ϕi)mi=1 || w,p ∗ ,

1 < p ≤ ∞ (D p, d p)

D p(E ; F )

d p(·) E F

D p(E ; F )

Lf (E ; F ) ⊂ D p(E ; F ) E F

T : E → F : T (x) = ψ(x)y, com ψ ∈ E e y ∈ F,

D p(E ; F ) m ∈N , xi ∈ E, ϕi ∈ F , i = 1,...,m

m

i=1

|ϕi(T (x i)) | =m

i=1

|ϕi(ψ(x i)y)| =m

i=1

|ψ(xi)ϕi(y)|

≤ || ψ||m

i=1|| xi ||| ϕi(y)|

≤ || ψ|| m

i=1

|| xi || p1/p m

i=1

|ϕi(y)| p∗

1/p ∗

= ||ψ|||| y|||| (x i)mi=1 || p m

i=1

|ϕi(y)| p∗

|| y|| p∗1/p ∗

≤ || ψ|||| y|||| (xi)mi=1 || p supw∈B F

m

i=1

|ϕi(w)| p∗

1/p ∗

= C || (x i)mi=1 || p || (ϕi)mi=1 || w,p ∗ ,


42/94

T ∈ D p(E ; F )

A1 ∈ L(E 0; E ) T ∈ D p(E ; F ) A2 ∈ L(F ; F 0)

m ∈N xi ∈ E 0 ϕi ∈ F 0 i = 1,...,m m

i=1

|ϕi((A2T A1)(x i)) | =m

i=1

|(ϕiA2)(T (A1(xi)) |

≤ d p(T )|| (A1xi)mi=1 || p || (ϕiA2)mi=1 || w,p ∗

≤ d p(T )|| A1|||| (x i)mi=1 || p || A2|||| (ϕi)mi=1 || w,p ∗

= ||A2|| d p(T ) || A1|||| (x i)mi=1 || p || (ϕi)mi=1 || w,p ∗

= C || (xi)mi=1 || p || (ϕi)mi=1 || w,p ∗

A2T A1 ∈ D p(E 0; F 0)

d p(A2T A1) ≤ || A2|| d p(T ) || A1|| .

d p(idK ) = 1 idK

p lwq (K ) lq(K )

m = 1

|ϕ(idK

(x)) | = |ϕ(x)| ≤ || x|||| ϕ|| = || (x, 0, 0,...)|| p || (ϕ, 0, 0,...)|| w,p∗

, d p(idK ) ≤ 1 x,ϕ ∈K ||ϕ|| = 1

|| x|| = || (x, 0, 0,...)|| C,p = || (idK (x), 0, 0,...)|| C,p ≤ d p(idK ) || x|||| ϕ|| ,

1 ≤ d p(idK ) d p(idK ) = 1

1 < p ≤ ∞ (D p, d p)

(T n )∞n =1 (D p(E ; F ), d p)

||·|| ≤ d p(·) (T n )∞n =1 L(E ; F )

limn →∞

T n = T ∈ L(E ; F ).

T ∈ D p(E ; F )

T n p

T n : l p(E ) → l p F

n T n = d p(T n ) ( T n )

∞n =1

L(l p(E ); l p F ) limn →∞ T n = A ∈ L(l p(E ); l p F )


43/94

(x j )∞ j =1 ∈ l p(E )

(y j )∞ j =1 := A((x j )∞ j =1 ) ∈ l p F .

ε > 0 n0 ∈N

|| T n (x j ) − y j || ≤ || (T n (x j ))∞ j =1 − (y j )∞ j =1 || C,p

= T n ((x j )∞ j =1 ) − A((x j )∞ j =1 )

C,p

< ε || (x j )∞ j =1 || p ,

n > n 0 limn →∞ T n (x j ) = y j , j ∈N

T (x j ) = y j j ∈N

(T (xn ))∞n =1 = ( yn )∞n =1 ∈ l p F , para toda ( xn )∞n =1 ∈ l p(E ),

T : l p(E ) → l p F

LCoh,p

p n ∈N

LnCoh,p n

p (LCoh,p , | | · | | Coh,p )

p

1 < p ≤ ∞ n ∈N T ∈ L(E 1,...,E n ; F ) T p

n ∈N

An : E 1 × · · · × E n → F : An (x1,...,x n ) = ψ1 (x1) · · · ψn (xn ) b ,


44/94

ψk ∈ E k , k = 1,...,n b ∈ F, p

m ∈N x(r )i ∈ E r ϕi ∈ F i = 1,...,m r = 1,...,n

m

i=1ϕi An x

(1)i ,...,x

(n )i

=m

i=1

ψ1 x(1)i · · · ψn x

(n )i ϕi(b)

≤ || ψ1| | · · · | | ψn ||m

i=1

x(1)i · · · x(n )i |ϕi(b)|

≤ || ψ1| | · · · | | ψn ||n

r =1

m

i=1

x(r )inp

1/np m

i=1

|ϕi(b)| p∗

1/p ∗

= ||ψ1| | · · · | | ψn |||| b||n

r =1

m

i=1

x(r )inp

1/np m

i=1

|ϕi(b)| p∗

|| b|| p∗1/p ∗

≤ || ψ1| | · · · | | ψn |||| b||n

r =1

m

i=1

x(r )inp

1/np

supw∈B F

m

i=1

|ϕi(w)| p∗

1/p ∗

= C n

r =1

m

i=1

x(r )inp

1/np

||(ϕi)mi=1 || w,p ∗ .

An p n

(LCoh,p , | | · | | Coh,p )

n ∈N T ∈ L(E 1,...,E n ; F ) p

|| T || ≤ || T || Coh,p .

m = 1

|ϕ(T (x1,...,x n )) | ≤ || T || Coh,p || (x1, 0, 0,...)|| np · · · | | (xn , 0, 0,...)|| np || (ϕ, 0, 0,...)|| w,p ∗ ,

|| T || ≤ || T || Coh,p

1 < p ≤ ∞ n ∈N LnCoh,p , | | · | | Coh,p n

LCoh,p (E 1,...,E n ; F )

| | · | | Coh,p n ∈N


45/94

E j F j = 1, ...n

Ar ∈ L(H r ; E r ) r = 1,...,n

T ∈ LCoh,p (E 1,...,E n ; F ) A ∈ L(F ; F 0) m ∈N x(r )i ∈ H r ϕi ∈ F 0

r = 1,...,n i = 1,...,m

m

i=1

ϕi((AT (A1,...,A n )) x(1)i ,...,x

(n )i

=m

i=1

(ϕiA) T A1 x(1)i ,...,A n x

(n )i

≤ || T || Coh,p (A1x(1)i )mi=1np

... (An x(n )i )

mi=1

np|| (ϕiA)mi=1 || w,p ∗

≤ || T || Coh,p || A1|| ... || An || (x(1)i )mi=1

np· · · (x(n )i )

mi=1

np|| A|||| (ϕi)mi=1 || w,p ∗

= ||A|||| T || Coh,p || A1|| ... || An || (x(1)i )mi=1np

· · · (x(n )i )mi=1np

|| (ϕi)mi=1 || w,p ∗

= C (x(1)i )mi=1

np· · · (x(n )i )

mi=1

np|| (ϕi)mi=1 || w,p ∗

AT (A1,...,A n ) ∈ LCoh,p (H 1,...,H n ; F 0)

|| AT (A1,...,A n )|| Coh,p ≤ || A|||| T || Coh,p || A1|| ... || An || .

|| idK n || Coh,p = 1 n ∈ N idK n

p

1 = || idK n || ≤ || idK n || Coh,p .

||ϕ|| w,p ∗ = ||ϕ|| p∗ K

m

i=1

ϕi idK n x(1)i ,...,x (n )i

=m

i=1

ϕi x(1)i · · · x

(n )i

≤m

i=1

||ϕi || x(1)i ... x(n )i

≤ m

i=1

||ϕi || p∗

1/p ∗ m

i=1

x(1)inp

1/np

· · · m

i=1

x(n )inp

1/np

= (x(1)i )mi=1

np· · · (x(n )i )mi=1

np|| (ϕi)mi=1 || w,p ∗ ,


46/94

m ∈ N x(r )i ,ϕi ∈ K r = 1,...,n i = 1,...,m

|| idK n || Coh,p ≤ 1 || idK n || Coh,p = 1

1 < p ≤ ∞ n ∈ N LnCoh,p , | | · | | Coh,p n

(T k)∞k=1 (LCoh,p (E 1,...,E n ; F ), | | · | | Coh,p )

|| · || ≤ || · || Coh,p (T k)∞k=1

L(E 1,...,E n ; F )

limk→∞

T k = T ∈ L(E 1,...,E n ; F ).

T ∈ LCoh,p (E 1,...,E n ; F ) T k

p T k k T k = ||T k || Coh,p ( T k)∞k=1 L(lnp (E 1),...,l np (E n ); l p F ) limk→∞ T k = A ∈L(lnp (E 1),...,l np (E n ); l p F )

(y j )∞ j =1 := A x(1) j ,...,x

(n ) j

∞

j =1∈ l p F ,

x(1) j

∞

j =1 , ..., x(n ) j

∞

j =1 ∈ lnp (E 1) × ... × lnp (E n )

ε > 0

k0 ∈N

T k x(1) j ,...,x

(n ) j − y j

≤ T k x(1) j ,...,x(n ) j

∞

j =1− (y j )∞ j =1

C,p

=

T k x

(1) j

∞

j =1, ..., x(n ) j

∞

j =1− A x(1) j

∞

j =1, ..., x(n ) j

∞

j =1 C,p

< ε x(1) j

∞

j =1 np· · · x

(n ) j

∞

j =1 np,

k > k 0 limk→∞ T k x(1) j ,...,x

(n ) j = y j j ∈ N

T x(1) j ,...,x(n ) j = y j j ∈N

T x(1) j ,...,x(n ) j

∞

j =1= ( y j )∞ j =1 ∈ l p F ,

x(1) j∞

j =1, ..., x(1) j

∞

j =1 ∈ lnp (E 1) × ... × lnp (E n )

T : lnp (E 1) × ... × lnp (E n ) → l p F T ∈ LCoh,p (E 1,...,E n ; F )


47/94

P Coh,p

p n ∈ N P nCoh,p n

p

P Coh,p

1 < p ≤ ∞ n ∈ N P nCoh,p , | | · | | Coh,p n


48/94

p

p

1 < p ≤ ∞ E i , F i = 1,...,n 1 p +

1 p∗ = 1 T ∈ L(E 1,...,E n ; F )


49/94

p T x(1) j1 , ..., x

(n ) jn j1 ,...,j n ∈N

∈ l p F ,

x(i) j

∞

j =1 ∈ l

p(E

i) i = 1,...,n

p E 1×·· ·× E n F L(E 1,...,E n ; F )

LmCoh,p (E 1,...,E n ; F )

T ∈ L(E 1,...,E n ; F ) 1 p + 1 p∗ = 1

i) T p

ii ) C > 0

∞

j1 ,...,j n =1

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn

≤ C ∞

j =1

x(1) j p

1/p

· · · ∞

j =1

x(n ) j p

1/p

(ϕ j1 ,...,j n ) j1 ,...,j n ∈N w,p ∗,

(ϕ j1 ,...,j n ) j1 ,...,j n ∈N ∈ lw p∗(F ) x(i) j

∞

j =1∈ l p(E i) i =

1,...,n

iii ) C > 0

m

j1 ,...,j n =1

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn

≤ C m

j =1

x(1) j p

1/p

· · · m

j =1

x(n ) j p

1/p

(ϕ j1 ,...,j n )m j1 ,...,j n =1 w,p ∗

,

m ∈ N , ϕ j1 ,...,j n ∈ F x(i) j ∈ E i i = 1, ..., n, j i =

1,...,m, j = 1,...,m

C ||T || mCoh,p LmCoh,p (E 1,...,E n ; F )

(i) ⇒ (ii ) T p

T : lw p∗(F ) × l p(E 1) × · · · × l p(E n ) −→ l1


50/94

(ϕ j1 ,...,j n ) j1 ,...,j n ∈N , x(1) j1

∞

j1 =1, ..., x(n ) jn

∞

jn =1 −→ ϕ j1 ,...,j n T x

(1) j1 , ..., x

(n ) jn j1 ,...,j n ∈N

(n + 1)

(xk)∞k=1 ∈ lw p∗(F ) × l p(E 1) × · · · × l p(E n )

xk → x ∈ lw p∗(F ) × l p(E 1) × · · · × l p(E n )

T (xk) → (z j1 ,...,j n ) j1 ,...,j n ∈N ∈ l1 ..

xk = (ϕk

j1 ,...,j n) j1 ,...,j n ∈

N , x(1)k,j 1

∞

j1 =1,..., x(n )

k,j n

∞

jn =1

x = (ϕ j1 ,...,j n ) j1 ,...,j n ∈N , x(1) j1

∞

j1 =1,..., x(n ) jn

∞

jn =1

,

(z j1 ,...,j n ) j1 ,...,j n ∈N = limk→∞ T (xk)

= limk→∞

ϕk j1 ,...,j n T x

(1)k,j 1 ,...,x

(n )k,j n j1 ,...,j n ∈N

.

T (x) = ( z j1 ,...,j n ) j1 ,...,j n ∈N .

T (x) = T (ϕ j1 ,...,j n ) j1 ,...,j n ∈N , x(1) j1

∞

j1 =1,..., x(n ) jn

∞

jn =1

= ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn j1 ,...,j n ∈N

,

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn = z j1 ,...,j n

j1,...,j n ∈N

limk→∞

ϕk j1 ,...,j n T x

(1)k,j 1 ,...,x

(n )k,j n = z j1 ,...,j n

x(i)k,j → x(i) j em E i e ϕ

k j1 ,...,j n → ϕ j1 ,...,j n em F ,


51/94

j ∈ N , i = 1,...,n j1,...,j n ∈ N T

limk→∞

ϕk j1 ,...,j n T x

(1)k,j 1 ,...,x

(n )k,j n = ϕ j1 ,...,j n T x

(1) j1 , ..., x

(n ) jn ,

j1,...,j n ∈N T

∞

j1 ,...,j n =1

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn

= ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn j1 ,...,j n ∈N 1

= T (ϕ j1

,...,j n ) j1

,...,j n ∈N , x(1)

j1

∞

j1 =1, ..., x(n )

jn

∞

jn =1 1

≤ T x(1) j∞

j =1 p· · · x(n ) j

∞

j =1 p(ϕ j1 ,...,j n ) j1 ,...,j n ∈N w,p ∗

.

(ii ) ⇒ (i) (ii ) ⇒ (iii )

(iii ) ⇒ (ii ) (ϕ j1 ,...,j n ) j1 ,...,j n ∈N ∈ lw p∗(F ) x(i) j

∞

j =1∈ l p(E i) i = 1,...,n

∞

j1 ,...,j n =1

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn

= supm

m

j1 ,...,j n =1

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn

≤ supm

C x(1) jm

j =1 p· · · x(n ) j

m

j =1 p(ϕ j1 ,...,j n )

m j1 ,...,j n =1 w,p ∗

= C x(1) j∞

j =1 p· · · x(n ) j

∞

j =1 p(ϕ j1 ,...,j n ) j1 ,...,j n ∈N w,p ∗

.

p p

p p

| | · | | mCoh,p ≤ || · || Coh,p .

T ∈ LCoh,p (E 1,...,E n ; F )

C µ BF


52/94

|ϕ(T (x1,...,x n )) | ≤ C || x1|| ... ||xn || B F |ψ(ϕ)| p∗dµ(ψ)1/p ∗

,

x j ∈ E j , ϕ ∈ F , j = 1,...,n m ∈N

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn ≤ C x

(1) j1 ... x

(n ) jn B F |ψ(ϕ j1 ,...,j n )| p∗dµ(ψ)

1/p ∗

,

ϕ j1 ,...,j n ∈ F x(i) j ∈ E i i = 1,...,n 1 ≤ j 1,...,j n ≤ m

m

j1 ,...,j n =1

ϕ j1 ,...,j n T x(1) j1 , ..., x

(n ) jn

≤ C m

j1 ,...,j n =1

x(1) j1 ... x(n ) jn B F |ψ(ϕ j1 ,...,j n )| p∗dµ(ψ)

1/p ∗

≤ C m

j1 ,...,j n =1

x(1) j1 ... x(n ) jn

p1/p m

j1 ,...,j n =1 B F |ψ(ϕ j1 ,...,j n )| p∗dµ(ψ)1/p ∗

= C m

j =1

x(1) j p

1/p

· · · m

j =1

x(n ) j p

1/p

B F m

j1 ,...,j n =1

|ψ(ϕ j1 ,...,j n )| p∗

dµ(ψ)1/p ∗

≤ C x(1) jm

j =1 p· · · x(n ) j

m

j =1 p supψ∈B

F

m

j1 ,...,j n =1|ψ(ϕ j1 ,...,j n )| p∗

1/p ∗

= C x(1) jm

j =1 p· · · x(n ) j

m

j =1 p(ϕ j1 ,...,j n )

m j1 ,...,j n =1 w,p ∗

,

T ∈ LmCoh,p (E 1,...,E n ; F )

LmCoh,p

p

n ∈ N LnmCoh,p n

p

(LmCoh,p , | | · | | mCoh,p )

LmCoh,p (E 1,...,E n ; F )


53/94

|| idK n || mCoh,p = 1

T p ||T || ≤ || T || mCoh,p m = 1

|ϕ(T (x1,...,x n )) | ≤ || T || mCoh,p || (x1, 0,...)|| p · · · | | (xn , 0,...)|| p|| (ϕ, 0,...)|| w,p ∗ ,

|| T || ≤ || T || mCoh,p

|| idK n || = 1 1 ≤ || idK n || mCoh,p

LCoh,p || idK n || Coh,p = 1

|| idKn

|| mCoh,p ≤ 1

n ∈N LnmCoh,p | | · | | mCoh,p n

Ai ∈ L(H i ; E i) i = 1,...,n T ∈ LmCoh,p (E 1,...,E n ; F )

A ∈ L(F ; G) m ∈ N ϕ j1 ,...,j n ∈ G x(i) j ∈ H i i = 1, ..., n, j i =

1,...,m, j = 1,...,m m

j1 ,...,j n =1

ϕ j1 ,...,j n AT (A1,...,A n ) x(1) j1 , ..., x

(n ) jn

=m

j1 ,...,j n =1

(ϕ j1 ,...,j n A) T A1 x(1) j1 ,...,A n x

(n ) jn

≤ || T || mCoh,p n

i=1

Ai x(i) j

m

j =1 p|| (ϕ j1 ,...,j n A)m j1 ,...,j n =1 || w,p ∗

≤ || A|||| T || mCoh,p || A1| | · · · | | An ||

n

i=1x

(i) j

m

j =1 p || (ϕ j1 ,...,j n )

m j1 ,...,j n =1 || w,p ∗ .

AT (A1,...,A n ) ∈ LmCoh,p (H 1,...,H n ; G)

|| AT (A1,...,A n )|| mCoh,p ≤ || A|||| T || mCoh,p || A1| | · · · | | An || .

p

p


54/94

n p

P nmCoh,p := P ∈ P n ; P̌ ∈ LnmCoh,p .

|| P || P mCoh,p = || P̌ || mCoh,p ,

LmCoh,p

P ∈ P (n E ; F ) k ≤ n a ∈ E d̂kP (a) ∈ P (kE ; F )

d̂kP (a)(x) = k!

(n − k)!P̆ (a, (n − k). . . ,a ,x , (k)...,x ) .

(n + 1) T n

T (a1,...,a n , b) = T aσ(1) ,...,a σ(n ) , b , σ {1,...,n } a1,...,a n ∈ E b ∈ G

P H n ∈ N E F P H (n E ; F ) := P (n E ; F ) ∩ P H


55/94

i) P H (n E ; F ) P (n E ; F ) P → ||P || H

ii ) P H (0E ; F ) = F E F

iii ) σ ≥ 1 n ∈ N k ≤ n a ∈ E E F P ∈ P H (n E ; F )

d̂kP (a) ∈ P H (kE ; F ) e1k!

d̂kP (a)H

≤ σn || P || H || a|| n − k .

(n E, G ; F ) (E, (n )· · ·,E ,G ; F )

J

n ∈N E 1,...,E n F J (E 1,...,E n ; F ) := L(E 1,...,E n ; F ) ∩ J L(E 1,...,E n ; F ) | | · | | J J

C ≥ 1 n ∈ N E F A ∈ J (n E, K ; F ) n

A1 ∈ J (n E ; F ) e ||A1|| J ≤ C || A|| J ,

A1 : E n

→ F

A1(x1,...,x n ) := A(x1,...,x n , 1)

M C P M

M σ = 2C

P mCoh,p

LmCoh,p

p P mCoh,p

p

LmCoh,p C = 1 P mCoh,p

p σ = 2


56/94

n ∈ N E F T ∈ LmCoh,p (n E, K ; F )

m

ϕ j1 ,...,j n ,j n +1 = ϕ j1 ,...,j n , se jn +1 = 1

0, se jn +1 = 2, ..., m ,e y jn +1 = 1, se jn +1 = 1

0, se jn +1 = 2, ..., m ,

j1,...,j n , j n +1 = 1,...,m ϕ j1 ,...,j n ∈ F x(i) j ∈ E i i =

1,...,n, j i = 1,...,m, j = 1,...,m

m

j1 ,...,j n =1

ϕ j1 ,...,j n T 1 x(1) j1 , ..., x

(n ) jn

=

m

j1 ,...,j n =1ϕ j1 ,...,j

n T x(1) j1 , ..., x

(n ) jn , 1

(3.8)=

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 T x(1) j1 , ..., x

(n ) jn , y jn +1

≤ || T || mCoh,p (y j )m j =1 p

n

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p ∗

= ||T || mCoh,pn

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n =1 || w,p ∗ ,

T 1 ∈ LmCoh,p (n E ; F ) ||T 1|| mCoh,p ≤ || T || mCoh,p LmCoh,p C = 1

σ = 2


57/94

p p

p

n

T ∈ L(E 1,...,E n ; F ) ak ∈ E k T a k ∈ L(E 1,...,E k− 1, E k+1 ,...,E n ; F )

ak k ar ∈ E r

r = 1,...,n r = k T a 1 ,...,a k − 1 ,a k +1 ,...,a n ∈ L(E k; F )

T a 1 ,...,a k − 1 ,a k +1 ,...,a n (x) = T (a1,...,a k− 1,x ,a k+1 ,...,a n ) .

P ∈ P (n E ; F ) k ≤ n a ∈ E P a k ∈ P (n − kE ; F )


58/94

P a k (x) := P̆ (ak , xn − k) = P̆ (a, (k)...,a,x, (n− k). . . ,x)

k = 1

P a

( U n , M n )N n =1 N U n

n M n n

N

I

N ∈ (N {1})∪{∞} ( U n , M n )N n =1 U 1 = M 1 = I I α1 α2 α3 α4

E, E 1, . . . , E n F n ∈{2, . . . , N } :

(CP 1) k ∈ {1, . . . , n } T ∈ Mn (E 1, . . . , E n ; F ) a j ∈ E j j ∈ {1, . . . , n } {k}

T a 1 ,...,a k − 1 ,a k +1 ,...,a n

∈ I (E k; F )

T a 1 ,...,a k − 1 ,a k +1 ,...,a n I ≤ α1 T M n a1 . . . ak− 1 ak+1 . . . an .

(CP 2) P ∈ U n (n E ; F ) a ∈ E P a n − 1 ∈ I (E ; F )

P a n − 1 I ≤ α2 P̌ M n an− 1 .

(CP 3) u ∈ I (E n ; F ) γ j ∈ E j

j = 1, . . . , n − 1

γ 1 · · · γ n − 1u ∈ Mn (E 1, . . . , E n ; F )

γ 1 · · · γ n − 1u M n ≤ α3 γ 1 · · · γ n − 1 u I .

(CP 4) u ∈ I (E ; F ) γ ∈ E γ n − 1u ∈ U n (n E ; F )

γ n − 1u U n ≤ α4 γ n − 1 u I .

(CP 5) P ∈ U n (n E ; F ) P̆ ∈ Mn (n E ; F )


59/94

I N ∈N∪{∞} ( U n , M n )N n =1 U 1 = M 1 = I

β 1 β 2 β 3 β 4

E, E 1,...,E n +1

F

n = 1, . . . , N − 1

(CH 1) T ∈ Mn +1 (E 1, . . . , E n +1 ; F ) a j ∈ E j j = 1, . . . , n + 1

T a j ∈ Mn (E 1, . . . , E j − 1, E j +1 , . . . , E n +1 ; F )

T a j M n ≤ β 1 T M n +1 a j .

(CH 2) P ∈ U n +1 (n +1 E ; F ) a ∈ E P a ∈ U n (n E ; F )

P a U n ≤ β 2 P̌ M n +1 a .

(CH 3) T ∈ Mn (E 1, . . . , E n ; F ) γ ∈ E n +1 γT ∈ Mn +1 (E 1, . . . , E n +1 ; F )

γT M n +1 ≤ β 3 T M n γ .

(CH 4) P ∈ U n (n E ; F ) γ ∈ E γP ∈ U n +1 (n +1 E ; F )

γP U n +1 ≤ β 4 P U n γ .

(CH 5) n = 1, . . . , N P ∈ U n (n E ; F ) P̆ ∈M n (n E ; F )

β i i = 1, ..., 4

( U n , M n )N n =1 β 1 = β 2 = β 3 = β 4 = 1 U 1 = M 1 = I

(P nCoh,p , LnCoh,p )∞n =1

n n p


60/94

p

i) p

(P nCoh,p , LnCoh,p )∞n =1 D p p

β i = 1 i = 1, ..., 4

(CH 5)

(CH 1) a1 ∈ E 1 T ∈ LCoh,p (E 1,...,E n +1 ; F )

m x( j )i ∈ E j , ϕi ∈ F , i = 1, ..., m , j = 1,...,n

m

i=1

ϕi T a 1 x(1)i ,...,x

(n )i

=m

i=1

ϕi T a1, x(1)i ,...,x

(n )i

≤ || T || Coh,p m

i=1

|| a1|| p x(1)i p

· · · x(n )i p

1/p

|| (ϕi)mi=1 || w,p ∗

= ||T || Coh,p || a1|| m

i=1

x(1)i p

· · · x(n )i p

1/p

|| (ϕi)mi=1 || w,p ∗ ,

T a 1 ∈ LCoh,p (E 2,...,E n +1 ; F ) ||T a 1 || Coh,p ≤ || T || Coh,p || a1||

T a j ∈ LCoh,p (E 1,...,E j − 1, E j +1 ,...,E n+1 ; F ), j = 2,...,n

|| T a j || Coh,p ≤ || T || Coh,p || a j ||

(CH 2) a ∈ E P ∈ P Coh,p (n +1 E ; F ) m ∈ N xi ∈


61/94

E, ϕ i ∈ F , i = 1,...,m

m

i=1

|ϕi(P a (xi)) | =m

i=1

|ϕi( P̌ (a, x i ,...,x i)) |

≤ || P̌ || Coh,p m

i=1

|| a || p|| x i || np1/p

|| (ϕi)mi=1 || w,p ∗

= || P̌ || Coh,p || a|| m

i=1

|| xi || np1/p

|| (ϕi)mi=1 || w,p ∗

P a ∈ P Coh,p (n E ; F ) ||P a || Coh,p ≤ || P̌ || Coh,p || a ||

(CH 3) T ∈ LCoh,p (E 1,...,E n ; F ) γ ∈ E n +1 m ∈ N

x( j )i ∈ E j , ϕi ∈ F , i = 1, ..., m , j = 1,...,n + 1

m

i=1

ϕi γT x(1)i ,...,x

(n )i , x

(n +1)i

=m

i=1

ϕi T x(1)i ,...,x

(n )i γ x

(n +1)i

=m

i=1

ϕi T x(1)i ,...,x

(n )i γ x

(n +1)i

≤ || T || Coh,p m

i=1

x(1)i p

· · · x(n − 1)i p

x(n )i γ x(n +1)i p 1/np

|| (ϕi)mi=1 || w,p ∗

≤ || T || Coh,p || γ || m

i=1

n +1

j =1

x( j )i p

1/p

|| (ϕi)mi=1 || w,p ∗

γT LCoh,p (E 1,...,E n , E n +1 ; F ) ||γT || Coh,p ≤ || T || Coh,p || γ ||

(CH 4) P ∈ P Coh,p (n E ; F ) γ ∈ E m

xi ∈ E, ϕi ∈ F , i = 1,...,m K = C

m

i=1

|ϕi(γP (xi)) | =m

i=1

|ϕi(γ (xi)P (x i)) |

=m

i=1

|ϕi(P ((γ (x i))1/n x i)) |

≤ || P || Coh,p m

i=1

|| (γ (xi))1/n x i || np1/p

|| (ϕi)mi=1 || w,p ∗

≤ || P || Coh,p || γ || m

i=1

|| x i || (n+1) p1/p

|| (ϕi)mi=1 || w,p ∗ ,


62/94

(γ (xi))1/n n γ (x i)

K = R

a i =

1, se γ (xi) ≥ 0

− 1, se γ (xi) < 0 ,

a iγ (xi) ≥ 0

m

i=1

|ϕi(γP (xi)) | =m

i=1

|γ (xi)||ϕi(P (xi)) |

=m

i=1

a iγ (xi)|ϕi(P (x i)) |

=m

i=1

|ϕi(P ((a iγ (x i))1/n x i)) |

≤ || P || Coh,p m

i=1

|| (a iγ (xi))1/n xi || np1/p

|| (ϕi)mi=1 || w,p ∗

= ||P || Coh,p m

i=1

|| | γ (xi)) |1/n xi || np1/p

|| (ϕi)mi=1 || w,p ∗

= ||P || Coh,p || γ ||

m

i=1|| xi ||

(n +1) p1/p

|| (ϕi)mi=1 || w,p ∗ .

γP P Coh,p (n+1 E ; F ) ||γP || Coh,p ≤ || P || Coh,p || γ ||

(P nCoh,p , LnCoh,p )∞n =1

D p p

(P nmCoh,p ,LnmCoh,p )∞n =1

n n

p

p

(P nmCoh,p , LnmCoh,p )∞n=1 D p p


63/94

β i = 1 i = 1, ..., 4

(CH 5)

(CH 1) an +1 ∈ E n +1 T ∈ LmCoh,p (E 1,...,E n +1 ; F )

T a n +1 ∈ LmCoh,p (E 1,...,E n ; F ) m ∈N

ϕ j1 ,...,j n ,j n +1 =ϕ j1 ,...,j n , se jn +1 = 1

0, se jn +1 = 2, ..., m ,

j1,...,j n , j n +1 = 1,...,m ϕ j1 ,...,j n ∈ F x(i) j ∈ E i i = 1,...,n, j i =

1,...,m, j = 1,...,m

m

j1 ,...,j n =1

ϕ j1 ,...,j n T a n +1 x(1) j1 , ..., x

(n ) jn

=m

j1 ,...,j n ,j n +1 =1

ϕ j1 ,...,j n ,j n +1 T x(1) j1 , ..., x

(n ) jn , an +1

≤ || T || mCoh,p || an +1 || p x(1) jm

j =1 p· · · x(n ) j

m

j =1 p|| (ϕ j1 ,...,j n ,j n +1 )m j1 ,...,j n ,j n +1 =1 || w,p ∗

= ||T || mCoh,p || an +1 || p x(1) jm

j =1 p

· · · x(n ) jm

j =1 p

|| (ϕ j1 ,...,j n )m j1 ,...,j n =1 || w,p ∗ ,

T a n +1 ∈ LmCoh,p (E 1,...,E n ; F ) e ||T a n +1 || mCoh,p ≤ || T || mCoh,p || an +1 || .

T a j ∈ LmCoh,p (E 1,...,E j − 1, E j +1 ,...,E n +1 ; F ), j = 1,...,n

|| T a j || mCoh,p ≤ || T || mCoh,p || a j ||

(CH 2) P ∈ P mCoh,p (n +1 E ; F ) a ∈ E (CH 5) P̆ ∈

LmCoh,p (n +1 E ; F ) (CH 1)

P̆ a ∈ LmCoh,p (n E ; F ) e || P̆ a || mCoh,p ≤ || P̆ || mCoh,p || a || .

(P a )∨ = P̆ a (CH 5)

P a ∈ U n (n E ; F ) e ||P a || mCoh,p ≤ || P || mCoh,p || a || .


64/94

(CH 3) γ ∈ E n +1 m ∈ N ϕ j1 ,...,j n ∈ F x(i) j ∈ E i

i = 1,...,n + 1 , j i = 1,...,m, j = 1,...,m

m

j1 ,...,j n +1 =1ϕ j1 ,...,j n +1 γT x(1) j1 , ..., x (n ) jn , x(n +1) jn +1

=m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 T x(1) j1 , ..., x

(n ) jn γ x

(n +1) jn +1 .

m 2

jn =1

m

j1 ,...,j n − 1 =1

ϕ̃ j1 ,...,j n T z (1)

j1 ,...,z (n )

jn ,

z (i) j i = x(i) j i , j i = 1, ..., m , i = 1,...,n − 1

z (n ) jn = x(n ) jn γ x

(n +1) j1 , j n = 1,...,m

z (n )m + jn = x(n ) jn γ x

(n+1) j2 , jn = 1,...,m

z (n )(m − 1)m + jn = x(n ) jn γ x

(n +1) jm , j n = 1,...,m

ϕ̃ j1 ,...,j n = ϕ j1 ,...,j n ,1, j n = 1,...,m

ϕ̃ j1 ,...,m + jn = ϕ j1 ,...,j n ,2, jn = 1,...,m

ϕ̃ j1 ,..., (m − 1)m + jn = ϕ j1 ,...,j n ,m , j n = 1, ..., m .


65/94

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 T x(1) j1 , ..., x

(n ) jn γ x

(n +1) jn +1

=m

jn +1

m

j1 ,...,j n =1

ϕ j1 ,...,j n +1 T x(1) j1 , ..., x

(n ) jn γ x

(n +1) jn +1

=m

j1 ,...,j n =1

ϕ j1 ,...,j n ,1 T x(1) j1 , ..., x

(n ) jn γ x

(n +1) j1 +

+m

j1 ,...,j n =1

ϕ j1 ,...,j n ,2 T x(1) j1 , ..., x

(n ) jn γ x

(n +1) j2 + · · ·

· · · +m

j1 ,...,j n =1

ϕ j1 ,...,j n ,m T x(1) j

1 , ..., x (n ) jn γ x

(n +1) jm

=m

j1 ,...,j n =1

ϕ̃ j1 ,...,j n T z (1)

j1 , ..., z (n )

jn +

+m

j1 ,...,j n =1

ϕ̃ j1 ,...,m + jn T z (1)

j1 ,...,z (n )m + jn + · · ·

· · · +m

j1 ,...,j n =1

ϕ̃ j1 ,..., (m − 1)m + jn T z (1)

j1 , ..., z (n )(m − 1)m + jn

=m 2

jn =1

m

j1 ,...,j n − 1 =1

ϕ̃ j1 ,...,j n T z (1) j1 , ..., z (n )

jn .

T ∈ LmCoh,p (E 1,...,E n ; F )

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 γT x(1) j1 , ..., x

(n ) jn , x

(n +1) jn +1

=m 2

jn =1

m

j1 ,...,j n − 1 =1

ϕ̃ j1 ,...,j n T z (1)

j1 ,...,z (n )

jn

=m,...,m,m 2

j1 ,...,j n − 1 ,j n =1

ϕ̃ j1 ,...,j n T z (1)

j1 ,...,z (n )

jn

≤ || T || mCoh,p z (n ) jnm 2

jn =1 p

n − 1

i=1

z (i) j im

j i =1 p|| (ϕ̃ j1 ,...,j n )

m,...,m,m 2 j1 ,...,j n − 1 ,j n =1 || w,p ∗

= ||T || mCoh,p x(n ) jn γ (x(n +1) jn +1 )

m

jn ,j n +1 =1 p

n − 1

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p ∗

(∗)

≤ || T || mCoh,p || γ ||

n +1

i=1 x(i)

j

m

j =1 p || (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p

∗

,


66/94

(∗)

x(n ) jn γ x(n +1) jn +1

m

jn ,j n +1 =1 p

=m

jn ,j n +1 =1

γ x(n +1) jn +1 p

x(n ) jn

p1/p

= γ x(n +1) jm

j =1 px(n ) j

m

j =1 p

γ γT ∈ LmCoh,p (E 1, . . . , E n +1 ; F ) γT mCoh,p ≤ T mCoh,p γ .

(CH 4) γ ∈ E

(x1,...,x n +1 ) ∈ E n +1 → 1n + 1

n +1

k=1

γ (xk) P̆ (x1, [k]· · ·, xn +1 ) ∈ F ,

[k]· · · k (n + 1)

γP

(γP )∨ m ∈ N

ϕ j1 ,...,j n ∈ F x(i) j ∈ E i i = 1,...,n + 1 , j i = 1,...,m, j = 1,...,m

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 (γP )∨ x(1) j1 , ..., x

(n +1) jn +1

= 1n + 1

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1

n +1

k=1

γ x(k) jk P̌ x(1) j1 ,

[k]· · ·, x (n +1) jn +1

≤ 1n + 1

m

j1 ,...,j n +1 =1

n +1

k=1

ϕ j1 ,...,j n +1 γ x(k) jk

P̌ x(1) j1 , [k]· · ·, x (n+1) jn +1

= 1n + 1

n +1

k=1

ϕ j1 ,...,j n +1 γ x(k) jk

P̌ x(1) j1 , [k]· · ·, x (n +1) jn +1

m

j1 ,...,j n +1 =1 1

≤ 1n + 1

n +1

k=1

ϕ j1 ,...,j n +1 γ x(k) jk

P̌ x(1) j1 , [k]· · ·, x (n +1) jn +1

m

j1 ,...,j n +1 =1 1

= 1n + 1

n +1

k=1

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 γ x(k) jk

P̌ x(1) j1 , [k]· · ·, x (n +1) jn +1

= 1n + 1

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(1) j1 x

(2) j2 , · · · , x

(n +1) jn +1 + · · ·

· · · +m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(n +1) jn +1 x(1) j1 , · · · , x(n ) jn .


67/94

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 (γP )∨ x(1) j1 , ..., x

(n +1) jn +1

= 1n + 1

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(1) j1 x

(2) j2 , · · · , x

(n +1) jn +1 + · · ·

· · · +m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(n +1) jn +1 x

(1) j1 , · · · , x

(n ) jn .

(CH 3)

m 2

j2 =1

m

j3 ,...,j n +1 =1

ϕ̃ j2 ,...,j n +1 P̌ z (2)

j2 ,...,z (n +1)

jn +1

ϕ̃ j2 ,...,j n +1 z (k)

jk k = 2,...,n + 1

(CH 3)

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(1) j1 x

(2) j2 , · · · , x

(n +1) jn +1

≤ || P̌ || mCoh,p || γ ||n +1

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p ∗ .

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 (γP )∨ x(1) j1 , ..., x

(n +1) jn +1

= 1

n + 1

m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(1) j1 x

(2) j2 , · · · , x

(n +1) jn +1 + · · ·

· · · +m

j1 ,...,j n +1 =1

ϕ j1 ,...,j n +1 P̌ γ x(n +1) jn +1 x

(1) j1 , · · · , x

(n ) jn

≤ 1n + 1

|| P̌ || mCoh,p || γ ||n +1

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p ∗ + · · ·

· · · + || P̌ || mCoh,p || γ ||n +1

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p ∗

= || P̌ || mCoh,p || γ ||n +1

i=1

x(i) jm

j =1 p|| (ϕ j1 ,...,j n +1 )m j1 ,...,j n +1 =1 || w,p ∗ .


68/94

γP ∈ P mCoh,p (n +1 E ; F )

|| γP || mCoh,p ≤ || P̌ || mCoh,p || γ || = ||P || mCoh,p || γ || .

(P nmCoh,p , LnmCoh,p )∞n =1

D p p

p LCoh,p ,

LmCoh,p u : E → F p

ψ : E × · · · × E → F, denido por ψ(x1,...,x n ) = ϕ(x1) . . .ϕ(xn − 1)u(xn ) ,

0 = ϕ ∈ E LmCoh,p


69/94

1 < p < ∞

n ∈ N

E 1,...,E n

F

T : E 1 × · · · × E n −→ F p


70/94

a = ( a1,...,a n ) ∈ E 1 × · · · × E n

T a1 + x(1)i ,...,a n + x

(n )i − T (a1,...,a n )

∞

i=1∈ l p F

x( j )i∞

i=1∈ l p (E j ) , j = 1,...,n.

1 < p < ∞ n ∈ N E F n P : E −→ F p a ∈ E

(P (a + xi) − P (a))∞i=1 ∈ l p F

(xi)∞i=1 ∈ l p (E )

E 1 × · · · ×

E n F, p

L (E 1,...,E n ; F ) L(a )Coh,p (E 1,...,E n ; F ) .

E 1 ×· · ·× E n F

p LevCoh,p (E 1,...,E n ; F ) .

p

E F a P (n E ; F ) P (a )Coh,p (n E ; F )

n P : E −→ F

p P evCoh,p (n E ; F ).

l p(E j ) j = 1,...,n l p(E )

LCoh,p

(E 1,...,E

n; F ) L(0)

Coh,p (E

1,...,E

n; F )

a ∈ E P ∈ P (n E ; F )

i) P p a ∈ E


71/94

ii ) P̌ p (a,...,a ) ∈ E n .

ii ) ⇒ i)

i) ⇒ ii ) P p a ∈ E

x( j )i∞

i=1∈ l p(E ) j = 1,...,n

x0 ∈ E i ∈N

n!2n P̌ a + x(1)i ,...,a + x(n )i − P (a,...,a )

=ε i = ± 1

ε1 · · · εn P x0 +n

k=1

εk a + x(k)i −

εi = ± 1

ε1 · · · εn P (x0 + ε1a + · · · + εn a)

=ε i = ± 1

ε1 · · · εn P x0 + n

k=1

εka + n

k=1

εkx(k)i − P (x0 + ε1a + · · · + εn a) ,

P̌ p a = 0

x0 = 0

a = 0 x0 = ( n + 1) a.

x0 +n

k=1

εka = x0 + ε1a + · · · + εn a = λa = 0 .

λa = 0

P (λa + x j ) − P (λa ) = P λ a + x jλ

− P (λa ) = λm P a + 1λ

x j − P (a)

P p λa a P

p (x0 + ( ε1a + · · · + εn a)) P̌

p (a,...,a ) .

L(a )Coh,p (E 1,...,E n ; F ) L(b)Coh,p (E 1,...,E n ; F )

p

a = ( a1,...,a n ) ∈ E 1 × · · · × E n T ∈ L(a )Coh,p (E 1,...,E n ; F )

i)

1 ≤ r < n

T a k 1 ,...,a k r ∈ L(0)Coh,p (E j1 ,...,E j s ; F )

{1,...,n } = { j1,...,j s}∪ {k1,...,k r }


72/94

{ j1,...,j s } ∩ {k1,...,k r } = ∅;

ii ) T ∈ L(b)Coh,p (E 1,...,E n ; F )

b ∈ {(λ1a1,...,λ n an ) ; λ i ∈K , i = 1,...,n } .

T p

(i) T a 1 ,...,a j − 1 ,a j +1 ,...,a n j = 1,...,n

T a 1 ,...,a j − 1 ,a j +1 ,...,a n x( j )i

= T a1 + 0 ,...,a j − 1 + 0 , a j + x( j )i , a j +1 + 0 ,...,a n + 0 − T (a1,...,a n ) .

T p a T a 1 ,...,a j − 1 ,a j +1 ,...,a n

p

T a 1 ,...,a n − 2

T a 1 ,...,a n − 2 x(n − 1)i , x

(n )i

∞

i=1

= T a1 + 0 , a2 + 0 ,...,a n − 2 + 0 , an − 1 + x(n − 1)i , an + x(n )i − T (a1,...,a n ) ∞i=1

− T a1, a2,...,a n − 1, x (n )i + T a1, a2,...,a n − 2, x(n − 1)i , an

∞

i=1

= T a1 + 0 , a2 + 0 ,...,a n − 2 + 0 , an − 1 + x(n − 1)i , an + x

(n )i − T (a1,...,a n )

∞

i=1

− T a 1 ,...,a n − 1 x(n )i + T a 1 ,...,a n − 2 ,a n x

(n − 1)i

∞

i=1,

T a 1 ,...,a n − 2

p

(ii ) b = ( λ1a1,...,λ n an ) λ j = 0 j

∞

i=1

ϕi T λ1a1 + x(1)i ,...,λ n an + x

(n )i − T (λ1a1,...,λ n an )

=∞

i=1

ϕi T λ1a1 + λ1λ1

x(1)i ,...,λ n an + λnλn

x(n )i − T (λ1a1,...,λ n an )

= |λ1 · · · λn |∞

i=1ϕi T a1 + 1λ1

x(1)i ,...,a n + 1λnx(n )i − T (a1,...,a n ) .


73/94

T p b

λ j = 0 j (i)

T p λ j = 0

j (i)

n = 3 λ1 = 0 , λ2 = 0

λ3 = 0

T (λ1a1 + xi , λ2a2 + yi , z i) − T (λ1a1, λ 2a2, 0)

= λ1λ2 T a1 + xiλ1

, a2 + yiλ2

, z i − T (a1, a2, 0)

= λ1λ2 T (a1, a2, z i) + T xiλ1 , a2, z i + T a1,

yiλ2 , z i + T

x iλ1 ,

yiλ2 , z i

= λ1λ2 T a 1 ,a 2 (z i) + T a 2xiλ1

, z i + T a 1yiλ2

, z i + T x iλ1

, yiλ2

, z i .

p

T ∈ LmCoh,p (E 1,...,E n ; F ) T ∈ LevCoh,p (E 1,...,E n ; F )

n = 2

T ∈ LmCoh,p (E 1, E 2; F ) (a, b) ∈ E 1 × E 2

(x i)∞i=1 ∈ l p(E 1)

(yi)∞i=1 ∈ l p(E 2) (ϕi)∞i=1 ∈ lw p∗(F )

∞

i=1

|ϕi (T (a + xi , b + yi) − T (a, b)) |

≤∞

i=1

|ϕi (T (a, y i)) | +∞

i=1

|ϕi (T (xi , b)) | +∞

i=1

|ϕi (T (xi , yi)) | .

(a, b) ∈ E 1 × E 2

ϕi,j =ϕ

i, se j = 1

0, se j ∈N − { 1}e (z j )∞ j =1 = ( a, 0, 0,...) ,


75/94

b) L(a )Coh,p (n E ; E ) = L(n E ; E ) a = ( a1,...,a n ) ∈ E n a i = 0 i ai = 0 i

a) ⇒ b) dim E = ∞ a = ( a1,...,a n ) ∈ E n a i = 0 i

a i = 0 i k ∈ {1,...,n } ai = 0

i = k ai = 0 i

i = k ϕi ∈ E ϕi(a i) = 1 T ∈ L(n E ; E )

T (x1,...,x n ) = ϕ1(x1)[k]· · ·ϕn (xn )xk .

T a 1 ,...,a k −1

,a k+1

,...,a n (x) = T (a1,...,a k− 1,x ,a k+1 ,...,a n ) = x x ∈

E T a 1 ,...,a k − 1 ,a k +1 ,...,a n p

T p a

b) ⇒ a) dim E < ∞ {e1,...,e r } {ϕ1,...,ϕ r }

E E ϕk(ei) = δ k,i x ∈ E

x = rk=1 ϕk(x)ek T ∈ L(n E ; E ) T (x1,...,x n ) = T

r

k1 =1

ϕk1 (x1)ek1 , ...,r

kn =1

ϕkn (xn )ekn

=r

k1 ,...,k n =1

ϕk1 (x1) · · · ϕkn (xn )T (e1,...,e n ) ,

T T

p

p

T ∈ L(0)Coh,p (E 1,...,E n ; F ) C > 0

∞

i=1

ϕi T x(1)i ,...,x

(n )i ≤ C

n

j =1

x( j )i∞

i=1 p|| (ϕi)∞i=1 || w,p ∗ ,

x( j )i∞

i=1 ∈ l p (E j ) , j = 1,...,n (ϕi)∞i=1 ∈ lw p∗(F ) C

L(0)Coh,p (E 1,...,E n ; F )


76/94

1 < p < ∞ a = ( a1,...,a n ) ∈ E 1×···× E n T ∈ LevCoh,p (E 1,...,E n ; F ) C = C (a1,...,a n )

∞

i=1

ϕi T a1 + x(1)i ,...,a n + x

(n )i − T (a1,...,a n ) ≤ C ,

x( j )i∞

i=1∈ l p(E j ) (ϕi)∞i=1 ∈ lw p∗(F ) x

( j )i

∞

i=1 p≤ 1 ||(ϕi)∞i=1 || w,p ∗ ≤ 1

j = 1,...,n

T ∈ LevCoh,p (E 1, E 2; F ) (a, b) ∈ E 1 × E 2

T a T b T p

∞

i=1

|ϕi (T (a + xi , b + yi) − T (a, b)) |

≤∞

i=1

|ϕi (T (a, y i)) | +∞

i=1

|ϕi (T (x i , b)) | +∞

i=1

|ϕi (T (x i , yi)) |

≤ C 1 (yi)∞i=1 p || (ϕi)∞i=1 || w,p ∗ + C 2 (x i)

∞i=1 p || (ϕi)

∞i=1 || w,p ∗

+ C 3 (xi)∞i=1 p (yi)∞i=1 p || (ϕi)

∞i=1 || w,p ∗

≤ C a,b ,

(xi)∞i=1 p ≤ 1 (yi)∞i=1 p ≤ 1 ||(ϕi)

∞i=1 || w,p ∗ ≤ 1

T ∈ L(E 1,...,E n ; F ) 1 p + 1 p∗ = 1

i) T p

ii ) a = ( a1,...,a n ) ∈ E 1 × · · · × E n C > 0

∞

i=1

ϕi T a1 + x(1)i ,...,a n + x

(n )i − T (a1,...,a n ) ≤

≤ C || a1|| + x(1)i∞

i=1 p

... || an || + x(n )i∞

i=1 p

||(ϕi)∞i=1 || w,p ∗ ,

x( j )i∞

i=1∈ l p(E j ), j = 1,...,n (ϕi)∞i=1 ∈ lw p∗(F )


77/94

iii ) a = ( a1,...,a n ) ∈ E 1 × · · · × E n C > 0

m

i=1 ϕi T a1 + x

(1)

i ,...,a n + x(n )

i − T (a1,...,a n ) ≤

≤ C || a1|| + x(1)im

i=1 p... || an || + x(n )i

m

i=1 p ||(ϕi)mi=1 || w,p ∗ ,

m ∈N , x( j )i ∈ E j , ϕi ∈ F , i = 1,... ,m , j = 1,...,n

C LevCoh,p (E 1,...,E n ; F )

| | · | | ev

(ii ) ⇒ (i) (ii ) ⇒ (iii )

(iii ) ⇒ (ii ) x( j )i∞

i=1∈ l p(E j ) , j = 1,...,n (ϕi)∞i=1 ∈ lw p∗(F )

∞

i=1

ϕi T a1 + x(1)i ,...,a n + x

(n )i − T (a1,...,a n )

= supm

m

i=1

ϕi T a1 + x(1)i ,...,a n + x

(n )i − T (a1,...,a n )

≤ supm

C || a1|| + x(1)im

i=1 p... || an || + x(n )i

m

i=1 p ||(ϕi)mi=1 || w,p ∗

= C || a1|| + x(1)i∞

i=1 p... || an || + x(n )i

∞

i=1 p ||(ϕi)∞i=1 || w,p ∗ .

(i) ⇒ (ii )

Gr = E r × l p (E r ) r = 1,...,n

(n + 1)

Φ (T ) : G1 × · · · × Gn × lw p∗(F ) −→ l1 (F )

(n +1) a1, x(1)i

∞

i=1, ..., an , x

(n )i

∞

i=1, (ϕi)∞i=1

ϕi T a1 + x(1)i ,...,a n + x

(n )i − T (a1,...,a n )

∞

i=1

Φ Φ (T )

F k, x (1)i∞

i =1,..., x ( m )i

∞

i =1,(ϕi )∞i =1

= {(b1,...,bn ) ∈ E 1 × · · · × E n tais que

Φ (T ) b1, x(1)i

∞

i=1, ..., bn , x

(n )i

∞

i=1, (ϕi)∞i=1

l1≤ k


78/94

E 1 ×· · ·× E n k x(r )i∞

i=1∈ B lp (E r ) , r = 1,...,n

(ϕi)∞i=1 ∈ B lwp∗ (F )

F k, x (1)i∞

i =1,..., x ( n )i

∞

i =1,(ϕi )∞i =1

=m∈N

F k, x (1)im

i =1,..., x ( n )i

m

i =1,(ϕi )mi =1

=m∈N

S − 1m ([0, k]) ,

S m : E 1 × · · · × E n → [0, ∞ ),

S m (

b1,...,bn) =

m

i=1 ϕi T b1 +

x(1)

i ,...,bn +

x(n )

i − T (

b1,...,bn)

.

F k := F k, x (1)i∞

i =1,..., x ( n )i

∞

i =1,(ϕi )∞i =1

,

x(1)i∞

i=1∈ Blp (E r ) r = 1,...,n

(ϕi)∞i=1 ∈ B lwp∗ (F ) E 1 × · · · × E n = k∈N F k k0 F k0 (b1,...,bn ) F k0

0 < ε < 1

Φ (T ) c1, x(1)i

∞

i=1, ..., cn , x

(n )i

∞

i=1, (ϕi)∞i=1

l1≤ k0 ,

|| cr − br || < ε x(r )i∞

i=1∈ Blp (E r ) , r = 1,...,n (ϕi)∞i=1 ∈ Blwp∗ (F )

(v1,...,vn )

vr , x(r )i

∞

i=1 < ε,

r = 1,...,n

vr < ε , x(r )i

∞

i=1 p< ε ,

|| (ϕi)∞i=1 || w,p∗ < ε

Φ (T )[((b1, (0)∞i=1 ) ,..., (bn , (0)∞i=1 ), (0)

∞i=1 )) +

+ v1, x(1)i

∞

i=1, ..., vn , x

(n )i

∞

i=1, (ϕi)∞i=1

l1

= Φ (T ) b1 + v1, x(1)i

∞

i=1, ..., bn + vn , x

(n )i

∞

i=1, (ϕi)∞i=1

l1≤ k0,


79/94

(br + vr ) − br = vr < ε ,

x(r )

j

∞

j =1 p < ε e ||(ϕi)∞

i=1 || w,p∗ < ε . Φ(T ) ε

((b1, (0)∞i=1 ) , ..., (bn , (0)∞i=1 , (0)

∞i=1 )) ∈ G1 × · · · × Gn × lw p∗(F ).

Φ(T )

∞

i=1

ϕi T b1 + x(1)i ,...,bn + x

(n )i − T (b1,...,bn )

= Φ (T ) b1, x(1)i

∞

i=1, ..., bn , x

(n )i

∞

i=1, (ϕi)∞i=1

l1

≤ Φ (T ) b1 + x(1)i∞

i=1 p· · · bn + x(n )i

∞

i=1 p ||(ϕi)∞i=1 || w,p ∗,

T = Φ (T ) .

| | · | | ev := Φ (·)

LevCoh,p (E 1,...,E n ; F ) .

P ∈ P evCoh,p (n E ; F ) 1 p + 1 p∗ = 1

i) P

p

ii ) C > 0 a ∈ E

∞

i=1

|ϕi (P (a + xi) − P (a)) | ≤ C || a || + (xi)∞i=1 pn

|| (ϕi)∞i=1 || w,p ∗ ,

(xi)∞i=1 ∈ l p(E ) (ϕi)∞i=1 ∈ lw p∗(F )

iii ) C > 0 a ∈ E

m

i=1|ϕi (P (a + xi) − P (a)) | ≤ C || a || + (xi)

mi=1 p

n

|| (ϕi)mi=1 || w,p ∗ ,


80/94

m ∈N , xi ∈ E, ϕi ∈ F , i = 1,...,m

C P evCoh,p (n E ; F ) | | · | | ev

n = 2 p∗∈ (1, ∞ )

Γ = (r, q ) ∈ [1, ∞ ) × (1, ∞ ) : 1r

= 1q

+ 1 p∗

.

C (a,b )r,q (E 1, E 2; F ) (a, b) ∈ E 1 × E 2 T ∈

L (E 1, E 2; F ) C ≥ 0

m

i=1

|ϕi (T (a + xi , b + yi) − T (a, b)) | r1/r

≤ C || a|| + (xi)mi=1 q || b|| + (yi)mi=1 q (ϕi)

mi=1 w,p ∗ ,

m ∈ N xi ∈ E 1 yi ∈ E 2 ϕi ∈ F i = 1,...,m q = np

T ∈ C (a,b)1,np (E 1, E 2; F ) T a ∈ C (0)1,np (E 2; F ) p < np

p < 2 p T a = 0 a ∈ E 1

T = 0 l p lnp

| | · | | ev

idK n ev ≥ idK n = 1 ,

idK n ev = 1

(LevCoh,p , · ev)


81/94

idK n ev ≤ 1

n = 2

m ∈ N a1, a2 ∈ K x(1)im

i=1 x(2)i

m

i=1 (ϕi)mi=1 x

(1)i , x

(2)i ,ϕi ∈ K

i = 1,...,m idK

p d p(idK ) = 1

m

i=1

ϕi idK 2 a1 + x(1)i , a2 + x

(2)i − idK 2 (a1, a2)

=m

i=1

ϕi a1x(2)i + a2x

(1)i + x

(1)i x

(2)i

≤ | a1|m

i=1

ϕi x(2)i + |a2|

m

i=1

ϕi x(1)i +

m

i=1

ϕi x(1)i x

(2)i

≤ | a1|m

i=1

ϕi x(2)i + |a2|

m

i=1

ϕi x(1)i +

+ m

i=1

ϕi x(1)i

m

i=1

ϕi x(2)i

= |a1|m

i=1

ϕi idK x(2)i + |a2|

m

i=1

ϕi idK x(1)i +

+ m

i=1

ϕi idK x(1)i

m

i=1

ϕi idK x(2)i

= |a1| +m

i=1

ϕi idK x(2)i |a2| +

m

i=1

ϕi idK x(1)i − | a1|| a2|

≤ |a1| +m

i=1

ϕi idK x(2)i |a2| +

m

i=1

ϕi idK x(1)i

≤ |a1| + x(2)im

i=1 p|| (ϕi)mi=1 || w,p ∗ |a2| + x

(1)i

m

i=1 p|| (ϕi)mi=1 || w,p ∗

≤ |a1| + x(2)im

i=1 p|a2| + x(1)i

m

i=1 p ||(ϕi)mi=1 || w,p ∗ ,

idK 2 ev ≤ 1

A j ∈ L(H j ; E j ) j = 1,...,n T ∈ LevCoh,p (E 1,...,E n ; F ) A ∈ L(F ; F 0)

m ∈ N (a1,...,a n ) ∈ H 1 × · · · × H n x(k)i ∈ E k ϕi ∈ F 0 k = 1,...,n


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i = 1,...,m

m

i=1ϕi (AT (A1,...,A n )) a1 + x

(1)i ,...,a n + x

(n )i − (AT (A1,...,A n ))( a1,...,a n )

=m

i=1

(ϕiA) T A1 a1 + x(1)i ,...,A n an + x

(n )i − T (A1(a1),...,A n (an ))

≤ || T || evn

r =1

|| Ar (a r )|| + (Ar x(r )i )mi=1

p || (ϕiA)mi=1 || w,p ∗

≤ || T || ev || A1|| ... || An ||n

r =1

|| a r || + (x(r )i )mi=1

p || A|||| (ϕi)mi=1 || w,p ∗

= || A|||| T || ev || A1|| ... || An ||n

r =1

|| a r || + (x(r )i )mi=1 p

||(ϕi)mi=1 || w,p ∗

= C || a1|| + (x(1)i )mi=1

p· · · || a1|| + (x(n )i )

mi=1

p ||(ϕi)mi=1 || w,p ∗

AT (A1,...,A n ) ∈ LevCoh,p (H 1,...,H n ; F 0)

|| AT (A1,...,A n )|| ev ≤ || A|||| T || ev || A1|| ... || An || .

(P evCoh,p , · ev )

idK n ev ≤ 1

m ∈ N a ∈ K (x i)mi=1 (ϕi)mi=1 xi ,ϕi ∈ K i = 1,...,m

idK n p

n ∈N idK n Coh,p = 1


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m

i=1

|ϕi(idK n (a + xi) − idK n (a)) | =∞

i=1

|ϕi (a + xi)n − an |

=∞

i=1

ϕi na n − 1xi +n2

an − 2x2i + · · · +n2

a2xn − 2i + naxn − 1i + x

ni

≤ n |a|n − 1∞

i=1

|ϕi(x i)| +n2

|a|n− 2∞

i=1

ϕi(x2i ) + · · · + n|a|∞

i=1

ϕi(xn − 1i ) +∞

i=1

|ϕi(xni )|

= n |a |n − 1∞

i=1

|ϕi(idK (xi)) | +n2

|a |n − 2∞

i=1

|ϕi(idK 2 (x i , x i)| + · · · +

+ n|a |∞

i=1

ϕi(idK n − 1 (xi ,...,x i

n − 1

)) +∞

i=1

ϕi(idK n (xi ,...,x i

n

)

≤ n |a|n − 1 || (xi)mi=1 || p || (ϕi)mi=1 || w,p ∗ +n2

|a |n − 2 || (xi)mi=1 || 2 p || (ϕi)mi=1 || w,p ∗ + · · · +

+ n|a | || (xi)mi=1 || n − 1 p ||(ϕi)mi=1 || w,p ∗ + ||(x i)mi=1 || n p || (ϕi)mi=1 || w,p ∗

≤ (|a | + || (xi)mi=1 || p)n || (ϕi)mi=1 || w,p ∗ ,

idK n ev ≤ 1

A1 ∈ L(E 0; E ) P ∈ P evCoh,p (n E ; F ) A2 ∈ L(F ; F 0) m ∈ N

a, x i ∈ E 0 ϕi ∈ F 0 i = 1,...,m m

i=1

|ϕi((A2P A1)(a + xi) − (A2P A1)(a)) |

=m

i=1

|(ϕiA2)(P (A1(a + xi)) − P (A1(a))) |

≤ || P || ev (|| A1(a)|| + || (A1x i)mi=1 || p)

n

|| (ϕiA2)mi=1 || w,p ∗

≤ || P || ev || A1|| n (|| a || + || (x i)mi=1 || p)n || A2|||| (ϕi)mi=1 || w,p ∗

= || A2|||| P || ev || A1|| n (|| a || + || (x i)mi=1 || p)n || (ϕi)mi=1 || w,p ∗

= C (|| a || + || (x i)mi=1 || p)n || (ϕi)mi=1 || w,p ∗

P evCoh,p

|| A2P A1|| ev ≤ || A2|||| P || ev || A1||n

.


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lw p (E )

1 < p ≤ ∞ 1 p + 1 p∗ = 1

lw p (E ) L(l p∗; E ) p = 1 l1(E ) L(c0; E ) (xi)∞i=1 ∈ lw p (E )

T ∈ L(l p∗; E )

T ((bi)∞i=1 ) =∞

i=1

bixi .

lw p (K ) l p(K )

K K ϕ ∈ K


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k ∈ N β k ∈ K ||β k || = 1 T I k(xk)αkβ k =

|T I k(xk)α k | (∗)

∞

k=1

T I k(xk)αkβ k = |T ((αkβ k xk)∞k=1 )| ≤ || T |||| (αk)∞k=1 || p ,

(∗) ≤ (|| T || + ε)|| (αk)∞k=1 || p .

(|| T I k || )∞k=1 ∈ l p∗ ε > 0 (T I k)∞k=1 ∈ l p∗(E )

|| (T I k)∞k=1 || p∗ ≤ || T || I (l p(E ))

l p∗(E ) I (T ) = ( T I k)∞k=1 T ∈ (l p(E ))

|| I || ≤ 1 IJ = idlp∗ (E ) JI = id(lp (E )) (l p(E ))

l p∗(E )

E ϕi ∈ E i = 1,...,m p ≥ 1

supψ∈B

E

m

i=1

|ψ(ϕi)| p1/p

= supy∈B E

m

i=1

|ϕi(y)| p1/p

.

1 < p < ∞ 1 = 1/p +1 /p ∗ p < q

(α k)∞k=1 ∈ l p

∗

(β k)∞k=1 ∈ lq

(λk)∞k=1 := ( αkβ k)

∞k=1 /∈ l1

1 > 1/q + 1 /p ∗ 0 < ε < 1− 1q −

1p∗

2

(α k)∞k=1 = 1

k( 1p∗ + ε)

∞

k=1∈ l p∗ e (β k)∞k=1 =

1

k(1q + ε)

∞

k=1∈ lq ,

(α kβ k)∞k=1 = 1

k(1q +

1p∗ +2 ε)

∞

k=1/∈ l1 .


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n ∈ N E 1,...,E n F K A : E 1 ×· · · × E n −→ F

(i) A

(ii ) A

(iii ) K

A(x1,...,x n ) ≤ K x1 · · · xn

(x1,...,x n ) ∈ E 1 × · · · × E n

(iv) A

(v) A

(vi) A


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90/94

E 1,...,E n F A : E 1 × · · · × E n −→ F n A

E 1,...,E n F (A j )∞ j =1 ⊂ L(E 1,...,E n ; F )

xi ∈ E i i = 1,...,n (A j (x1,...,x n ))∞ j =1

A(x1,...,x n ) := lim j →∞

A j (x1,...,x n ) ,

A ∈ L(E 1,...,E n ; F ) .

E F (P j )∞ j =1 ⊂ P (

n E ; F ) x ∈ E (P j x)∞ j =1

P (x) := lim j →∞

P j (x) ,

P ∈ P (n E ; F ) .


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p


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p p

p


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L p


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p

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