Open Sobre as extensões multilineares dos operadores absolutamente somantes

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❈❛♠♣✐♥❛ ●r❛♥❞❡

Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❉♦✉t♦r❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❙♦❜r❡ ❛s ❡①t❡♥sõ❡s ♠✉❧t✐❧✐♥❡❛r❡s ❞♦s

♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳

♣♦r

❉✐❛♥❛ ▼❛r❝❡❧❛ ❙❡rr❛♥♦ ❘♦❞rí❣✉❡③

❙♦❜r❡ ❛s ❡①t❡♥sõ❡s ♠✉❧t✐❧✐♥❡❛r❡s ❞♦s

♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳

♣♦r

❉✐❛♥❛ ▼❛r❝❡❧❛ ❙❡rr❛♥♦ ❘♦❞rí❣✉❡③

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ▼❛r✐♥❤♦ P❡❧❧❡❣r✐♥♦

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❆ss♦❝✐❛❞♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ ❯❋P❇✴❯❋❈●✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✳

❏♦ã♦ P❡ss♦❛ ✲ P❇ ❏❛♥❡✐r♦✴✷✵✶✹

†_{❊st❡ tr❛❜❛❧❤♦ ❝♦♥t♦✉ ❝♦♠ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❛♣❡s✳}

R696s Rodríguez, Diana Marcela Serrano.

Sobre as extensões multilineares dos operadores absolutamente somantes / Diana Marcela Serrano Rodríguez.- João Pessoa, 2014.

100f.

Orientador: Daniel Marinho Pellegrino Tese (Doutorado) - UFPB-UFCG

1. Matemática. 2. Operadores absolutamente somantes. 2.Operadores multilineares múltiplo somantes. 3. Operadores multilineares absolutamente somantes. 4. Teorema de Bohnenblust-Hille.

❘❡s✉♠♦

◆♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ✈❛♠♦s tr❛❜❛❧❤❛r ❝♦♠ ❞✉❛s ❣❡♥❡r❛❧✐③❛çõ❡s ❞♦s ❜❡♠ ❝♦♥❤❡❝✐❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳ ❆ ♣r✐♠❡✐r❛ ❡♥✈♦❧✈❡ ♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s ❡ ♥♦s ❢♦❝❛r❡♠♦s ♥✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ à ❞❡s✐❣✉❛❧❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡✳ ❊st❛ ❛✜r♠❛ q✉❡✱ ♣❛r❛

K₌R _♦✉C_{✱ ❡ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ m≥1✱ ❡①✐st❡♠ ❡s❝❛❧❛r❡s BK,m≥1 t❛✐s q✉❡}

N

X

i1,...,im=1

U(ei1, . . . , eim) m+12m

!m+1 2m

≤BK_,m sup

z1,...,zm∈DN

|U(z1, ..., zm)|

♣❛r❛ t♦❞❛ ❢♦r♠❛ m✲❧✐♥❡❛r U : KN _{× · · · ×K}N _{→ K ❡ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ N✱ ♦♥❞❡}

(ei)N_i=1 é ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ KN. ◆❡ss❛ ❧✐♥❤❛✱ ♥♦ss♦ ♦❜❥❡t✐✈♦ s❡rá ❛ ✐♥✈❡st✐❣❛çã♦ ❞❛s

♠❡❧❤♦r❡s ❝♦♥st❛♥t❡sBK_,m q✉❡ s❛t✐s❢❛③❡♠ ❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡✳

❆ s❡❣✉♥❞❛ ❣❡♥❡r❛❧✐③❛çã♦ ❡♥✈♦❧✈❡ ♦ ❡st✉❞♦ ❞♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ♥✉♠ ♣♦♥t♦❀ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ ✈❡rsã♦ ❛❜str❛t❛ ❞❡st❡s ♦♣❡r❛❞♦r❡s q✉❡ ❡♥❣❧♦❜❛ ✈ár✐❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❱❡r❡♠♦s q✉❡✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦s ❡s♣❛ç♦s ❞❡ s❡q✉ê♥❝✐❛s ❛❞❡q✉❛❞♦s✱ t❡r❡♠♦s ♦✉tr♦s t✐♣♦s ❞❡ ♦♣❡r❛❞♦r❡s ❝♦♠♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞❛ ♥♦ss❛ ✈❡rsã♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❖♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✱ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ❚❡♦r❡♠❛ ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② t✇♦ ❣❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ t❤❡ ✇❡❧❧✲❦♥♦✇♥ ❝♦♥❝❡♣t ♦❢ ❛❜s♦❧✉t❡❧② s✉♠♠✐♥❣ ♦♣❡r❛t♦rs✳ ❚❤❡ ✜rst ♦♥❡ ❝♦♥s✐sts ♦❢ t❤❡ ♠✉❧t✐♣❧❡ s✉♠♠✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♦♣❡r❛t♦rs ❛♥❞ ✐t ✐s ❢♦❝✉s❡❞ ♦♥ ❛ r❡s✉❧t ♦❢ ❝♦✐♥❝✐❞❡♥❝❡ t❤❛t ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❇♦❤♥❡♥❜❧✉st✲ ❍✐❧❧❡ ✐♥❡q✉❛❧✐t②✳ ❚❤✐s ✐♥❡q✉❛❧✐t② ❛ss❡rts t❤❛t✱ ❢♦rK₌R_♦rC_{❛♥❞ ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r}

m t❤❡r❡ ❡①✐sts ♣♦s✐t✐✈❡ s❝❛❧❛rs BK_,m ≥1 s✉❝❤ t❤❛t

N

X

i1,...,im=1

U(ei1, . . . , eim) m+12m

!m+1 2m

≤BK_,m sup

z1,...,zm∈DN

|U(z1, ..., zm)|

❢♦r ❡✈❡r②m✲❧✐♥❡❛r ♠❛♣♣✐♥❣U :KN_{×· · ·×K}N _{→K❛♥❞ ❡✈❡r② ♣♦s✐t✐✈❡ ✐♥t❡❣❡rN✱ ✇❤❡r❡}

(ei)N_i=1 ❞❡♥♦t❡s t❤❡ ❝❛♥♦♥✐❝❛❧ ❜❛s✐s ♦❢KN.■♥ t❤✐s ❧✐♥❡ ♦✉r ♠❛✐♥ ❣♦❛❧ ✐s t❤❡ ✐♥✈❡st✐❣❛t✐♦♥

♦❢ t❤❡ ❜❡st ❝♦♥st❛♥tsBK_,m s❛t✐s❢②✐♥❣ t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛❧✐t②✳

❚❤❡ s❡❝♦♥❞ ❣❡♥❡r❛❧✐③❛t✐♦♥ ✐♥✈♦❧✈❡s t❤❡ ❝♦♥❝❡♣t ♦❢ ❛❜s♦❧✉t❡❧② s✉♠♠✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♦♣❡r❛t♦rs ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t❀ ✇❡ ♣r❡s❡♥t ❛♥ ❛❜str❛❝t ✈❡rs✐♦♥ ♦❢ t❤❡s❡ ♦♣❡r❛t♦rs ✐♥✈♦❧✈✐♥❣ ♠❛♥② ♦❢ t❤❡✐r ♣r♦♣❡rt✐❡s✳ ❲❡ ♣r♦✈❡ t❤❛t✱ ❝♦♥s✐❞❡r✐♥❣ ❛♣♣r♦♣r✐❛t❡ s❡q✉❡♥❝❡ s♣❛❝❡s✱ ✇❡ ❤❛✈❡ ♦t❤❡r ❦✐♥❞ ♦❢ ♦♣❡r❛t♦rs ❛s ♣❛rt✐❝✉❧❛r ❝❛s❡s ♦❢ ♦✉r ✈❡rs✐♦♥✳

❑❡②✇♦r❞s✿ ❆❜s♦❧✉t❡❧② s✉♠♠✐♥❣ ♦♣❡r❛t♦rs✱ ▼✉❧t✐♣❧❡ s✉♠♠✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♦♣❡r❛t♦rs✱ ❆❜s♦❧✉t❡❧② s✉♠♠✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♦♣❡r❛t♦rs✱ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ✐♥❡q✉❛❧✐t②✳

❆❣r❛❞❡❝✐♠❡♥t♦s

❆♥t❡s ❞❡ ❛❣r❛❞❡❝❡r ❛ ❝❛❞❛ ✉♥❛ ❞❡ ❧❛s ♣❡rs♦♥❛s q✉❡ ❤✐❝✐❡r♦♥ ♣♦s✐❜❧❡ ❧❛ ❝✉❧♠✐♥❛❝✐ó♥ ❞❡ ❡st❡ s✉❡ñ♦✱ ♥❡❝❡s✐t♦ ❛❣r❛❞❡❝❡r❧❡ ❛ ➱❧✱ ♠✐ ❉✐♦s✱ q✉✐❡♥ ❢✉❡ ❡❧ q✉❡ ♣✉s♦ ❛ ❝❛❞❛ ✉♥❛ ❞❡ ❡s❛s ♣❡rs♦♥❛s ❡♥ ♠✐ ❝❛♠✐♥♦ ♣❛r❛ ♠♦str❛r♠❡ s✉ ✐♥♠❡♥s♦ ❛♠♦r✳ ●r❛❝✐❛s ♣♦r ♥✉♥❝❛ ❛❜❛♥❞♦♥❛r♠❡ ② ❜❡♥❞❡❝✐r♠❡ ❛ ❝❛❞❛ ❞í❛✳

❆ ♠✐ ❢❛♠✐❧✐❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ♠✐s ♣❛❞r❡s✱ ❏✉✈❡ ② ❈❤❡❧❛✳ P♦r s✉ ✐♥♠❡♥s♦ ❛♠♦r✱ ❡❧ ❛♣♦②♦ ❝♦♥st❛♥t❡ ② ❡❥❡♠♣❧♦ ❞❡ t❡♥❛❝✐❞❛❞✳ ●r❛❝✐❛s ♣♦r ❛②✉❞❛r♠❡ ❛ ♥♦ ❞❡s✐st✐r✳

❆ ♠✐ ❡s♣♦s♦✱ ❉❞❛♥✐❡❧ ◆úñ❡③ ❆❧❛r❝ó♥✳ P♦r ❧❛ ♣❛❝✐❡♥❝✐❛✱ ❡❧ ❛♠♦r✱ ❧❛ ❝♦♠♣❛ñí❛✳ P♦r ❧❛s ♥♦❝❤❡s ❡♥ ✈❡❧❛ ❝♦♥str✉②❡♥❞♦ s✉❡ñ♦s✱ t❛♥t♦ ❡♥ ❡❧ s✐❧❡♥❝✐♦ ❞❡ ❧♦s ❧✐❜r♦s✱ ❝♦♠♦ ❡♥ ❧❛s ❧❛r❣❛s ❝♦♥✈❡rs❛❝✐♦♥❡s ❡♥ ❧❛ ♦s❝✉r✐❞❛❞✳ ●r❛❝✐❛s ♣♦r ❡st❛r s✐❡♠♣r❡ ❝♦♥♠✐❣♦✳ ➼❊✉ t❡ ❛♠♦ ♠✉✐t♦ ♠❡✉ ❜❛❜②✦

❆❧ ♣r♦❢❡s♦r ❉❛♥✐❡❧ ▼✳ P❡❧❧❡❣r✐♥♦✳ ❆ é❧ s♦② ✐♥♠❡♥s❛♠❡♥t❡ ❣r❛t❛✱ ♥♦ só❧♦ ♣♦r s✉ ✐♠♣❡❝❛❜❧❡ ❧❛❜♦r ❝♦♠♦ ♦r✐❡♥t❛❞♦r✱ s✐♥♦ ♣♦r s✉ ❝❛❧✐❞❡③ ❞❡ ♣❡rs♦♥❛✳ ◆♦ t❡♥❣♦ ♥✐ ❝♦♠♦ ❧✐st❛r s✉s ✈✐rt✉❞❡s✱ s✐♠♣❧❡♠❡♥t❡ ➼❣r❛❝✐❛s ♣♦r t♦❞♦✦

❆ ❞♦s ❡①❝❡❧❡♥t❡s ♣r♦❢❡s♦r❡s ② ❣r❛♥❞❡s ♣❡rs♦♥❛s✱ ❊✈❡r❛❧❞♦ ❙♦✉t♦ ❞❡ ▼❡❞❡✐r♦s ② ❏✉❛♥ ❙❡♦❛♥❡ ❙❡♣ú❧✈❡❞❛✳

❆ ♠✐ ❤❡r♠❛♥✐t❛✱ ❏❤❛③❛✐r❛ ▼❛♥t✐❧❧❛ Pér❡③✳ ●r❛❝✐❛s ♣♦r ❛❝♦♠♣❛ñ❛r♠❡ ❡♥ ♠ás ✉♥❛ ❛✈❡♥t✉r❛✱ ♣♦r ❧❛s s♦♥r✐s❛s r♦❜❛❞❛s ② ❧♦s ♠♦♠❡♥t♦s ❝♦♠♣❛rt✐❞♦s✳ ◆❛❞✐❡ ♠❡❥♦r q✉❡ ❡❧❧❛ ♣❛r❛ ✈✐✈✐r ❡st❛ ❡①♣❡r✐❡♥❝✐❛✳

❆ ❡❧❧♦s✱ ♠✐s ❛♠✐❣♦s✱ ♠✐ ❢❛♠✐❧✐❛ ❇r❛s✐❧❡r❛✱ ❨❛♥❡✱ P❛♠♠❡❧❧❛✱ ◆❛❝✐❜✱ ●✉st❛✈♦✱ ❘❡❣✐♥❛❧❞♦✱ ❱❛❧❞❡❝✐r✱ ❑❛ré✱ ●✐❧s♦♥✱ ❏❛♥❛ ② ❘♦❞r✐❣♦✳ ●r❛❝✐❛s ♣♦r ❛❧❡❥❛r ❡s❡ ♠♦♥str✉♦ ❤♦rr✐❜❧❡ ❧❧❛♠❛❞♦ ❙♦❧❡❞❛❞✳

❆ t♦❞♦s ♠✐s ❛♠✐❣♦s ② ❢❛♠✐❧✐❛r❡s ❡♥ ❈♦❧♦♠❜✐❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ❧♦s q✉❡ ♠❡ ❣✉❛r❞❛r♦♥ ❡♥ s✉s ♦r❛❝✐♦♥❡s✳ ❙♦♥ ♠✉❝❤❛s ♣❡rs♦♥❛s ♣❛r❛ ♣♦♥❡r ❡♥ ✉♥❛s ❝✉❛♥t❛s ❧í♥❡❛s✳

❆ ❡s❡ ♣❡q✉❡ñ♦ ❣r✉♣♦ ❞❡ ❝♦❧♦♠❜✐❛♥♦s✱ ♠✉❡str❛ ❞❡ ❧❛ ❝♦r❞✐❛❧✐❞❛❞ ❞❡ ♥✉❡str❛ ❣❡♥t❡✱ ❆r♥♦❧❞♦ ❚❡❤❡rá♥✱ ▼❡r❝❛❧✉③ ❍❡r♥á♥❞❡③ ② ▲✉✐s ❆✳ ❆❧❜❛✳ ▼✉❝❤❛s ❣r❛❝✐❛s ♣♦r t♦❞♦ ❡❧ ❛♣♦②♦ ② ♣♦r ❧❛ ❛♠✐st❛❞ ❝♦♥str✉✐❞❛✳

❆ ❧♦s ♣r♦❢❡s♦r❡s ▼❛r② ▲✐❧✐❛♥ ▲♦✉r❡♥ç♦✱ ▲✉✐③❛ ❆♠á❧✐❛ ▼♦r❛❡s✱ ❱✐♥í❝✐✉s ❱✐❡✐r❛ ❋á✈❛r♦✱ ❏♦❡❞s♦♥ ❙✐❧✈❛ ❞♦s ❙❛♥t♦s ② ❈❧❡♦♥ ❞❛ ❙✐❧✈❛ ❇❛rr♦s♦ ♣♦r ❧❛s ❝♦♥tr✐❜✉❝✐♦♥❡s ❞❛❞❛s q✉❡ ✐♥❞✉❞❛❜❧❡♠❡♥t❡ ♠❡❥♦r❛r♦♥ ❡st❡ tr❛❜❛❥♦✳

❆❧ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛s ❞❡ ❧❛ ❯❋P❇ ♣♦r ❧❛ ❝♦♥✜❛♥③❛ ❞❡♣♦s✐t❛❞❛ ② ❛ ❧❛ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ♥í✈❡❧ ❙✉♣❡r✐♦r ✭❈❆P❊❙✮✱ ♣♦r ❡❧ ❛♣♦②♦ ✜♥❛♥❝✐❡r♦✳

❉❡❞✐❝❛tór✐❛

❆ ♠✐s ✈✐❡❥♦s✲

✲② ❛ tí ♠✐ ✈✐❡❥✐t♦ ❉❞✲

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ◆♦t❛çã♦ ❡ t❡r♠✐♥♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✶ ❆ ♦r✐❣❡♠✳ ❖♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳ ✻

✶✳✶ ❖♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✷ ❖ ❚❡♦r❡♠❛ ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ✶✻

✷✳✶ Pr✐♠❡✐r♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❖ ▲❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸ ❙♦❜r❡ ❛s ❝♦♥st❛♥t❡s ót✐♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹ ❈❛s♦ ❝♦♠♣❧❡①♦ ✲ π, e❡ γ s❡ ❡♥❝♦♥tr❛♠ ✲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷✳✺ ❱❛r✐❛çõ❡s ♥♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✸ ❯♠❛ ❢ór♠✉❧❛ ❢❡❝❤❛❞❛ ♣❛r❛ ❛s ❝♦♥st❛♥t❡s r❡❝✉rs✐✈❛s ❞♦ ❚❡♦r❡♠❛ ❞❡

❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡✳ ✺✶

✸✳✶ ▼❡❧❤♦r❛♥❞♦ ♦ r❡s✉❧t❛❞♦ ♣❛r❛ ❛s ❝♦♥st❛♥t❡s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✷ ❈♦♠♣❛r❛♥❞♦ ❛s ❝♦♥st❛♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✹ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ γ(s;s1,...,sm)✲s♦♠❛♥t❡s ✻✺ ✹✳✶ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✷ ❘❡s✉❧t❛❞♦s ❞♦ t✐♣♦ ❉✈♦r❡t③❦②✲❘♦❣❡rs ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹

■♥tr♦❞✉çã♦

❊♠ ✶✽✷✾✱ ♦ ♠❛t❡♠át✐❝♦ ❏✳ P✳ ●✳ ▲✳ ❉✐r✐❝❤❧❡t ❡st❛❜❡❧❡❝❡✉ ♦ r❡s✉❧t❛❞♦ ❝❧áss✐❝♦ ❞❡ ❆♥á❧✐s❡ q✉❡ ❛✜r♠❛ q✉❡✱ ❡♠ R_{✱ ✉♠❛ sér✐❡ ❝♦♥✈❡r❣❡ ❛❜s♦❧✉t❛♠❡♥t❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱}

❝♦♥✈❡r❣❡ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡✳ P❛r❛ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ❛ s✐t✉❛çã♦ é ✉♠ ♣♦✉❝♦ ❞✐❢❡r❡♥t❡✿ ♣♦r ❡①❡♠♣❧♦✱ ♥♦s ❡s♣❛ç♦s ℓp✱ ♣❛r❛ 1 < p < ∞✱ ❛ sér✐❡ Pe_nn é

✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡ ♠❛s ♥ã♦ é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡✳

P♦st❡r✐♦r♠❡♥t❡✱ ❇❛♥❛❝❤ ♣r♦♣ôs ❡♠ ❬✸✱ ♣✳ ✹✵❪ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ sér✐❡s ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s q✉❡ ♥ã♦ sã♦ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s ❡♠ q✉❛❧q✉❡r ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ♣r♦❜❧❡♠❛ ❝♦♥t❡♠♣❧❛❞♦ t❛♠❜é♠ ♥♦ ❙❝♦tt✐s❤ ❇♦♦❦ ✭❬✸✷✱ Pr♦❜❧❡♠❛ ✶✷✷❪✮ ❡ r❡s♦❧✈✐❞♦ ❛✜r♠❛t✐✈❛♠❡♥t❡✱ q✉❛s❡ ✈✐♥t❡ ❛♥♦s ❞❡♣♦✐s✱ ♣♦r ❆✳ ❉✈♦r❡t③❦② ❡ ❈✳❆✳ ❘♦❣❡rs ✭❬✷✵❪✮✳ ■♥t❡r❡ss❛❞♦ ♥❡st❡ r❡s✉❧t❛❞♦✱ ❆❧❡①❛♥❞❡r ●r♦t❤❡♥❞✐❡❝❦ r❡❛❧✐③♦✉ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞✐❢❡r❡♥t❡ ❞❡st❡ ❡♠ ❬✷✷❪ ❡ ❛♣r❡s❡♥t♦✉ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s q✉❡ ✧tr❛♥s❢♦r♠❛✧sér✐❡s ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s ❡♠ sér✐❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s ❡♠ ❬✷✸❪❀ t❛✐s ♦♣❡r❛❞♦r❡s sã♦ ♦s ❞❡♥♦♠✐♥❛❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳ ◆♦ ❡♥t❛♥t♦✱ ❢♦✐ ❛♣❡♥❛s ♥♦s ❛♥♦s ✻✵ q✉❡ ❡st❛ ❝❧❛ss❡ ❢♦✐ ❞❡✈✐❞❛♠❡♥t❡ ✐♥tr♦❞✉③✐❞❛ ❡ ❞✐✈✉❧❣❛❞❛ ❛tr❛✈és ❞❡ tr❛❜❛❧❤♦s ❞❡ ❞✐❢❡r❡♥t❡s ♠❛t❡♠át✐❝♦s ❝♦♠♦ P✐❡ts❝❤ ✭❬✹✻❪✮✱ ▲✐♥❞❡♥str❛✉ss ✭❬✷✻❪✮ ❡ P❡➟❝③②➠s❦✐ ✭❬✸✸❪✮✳

s❡♥t✐❞♦✱ ❛ ❡ssê♥❝✐❛ ❞❛ t❡♦r✐❛ ❧✐♥❡❛r✳ ◆♦ ♠❡s♠♦ ❛♥♦✱ ▼✳ ❈✳ ▼❛t♦s ✭❬✸✵❪✮ ❞❡✜♥✐✉ ♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ♥✉♠ ♣♦♥t♦ ❡✱ ❞❡s❞❡ ❡♥tã♦✱ ✈❛r✐♦s sã♦ ♦s ❛✉t♦r❡s q✉❡ tê♠ tr❛❜❛❧❤❛❞♦ s♦❜r❡ ❡st❛ ❧✐♥❤❛❀ ✈❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✹❪ ❡ ❬✶✷❪✳

❇❛s❡❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ♥♦ ❝♦♥t❡①t♦ ❞❛s ❡①t❡♥sõ❡s ❛♦ ❝❛s♦ ♠✉❧t✐❧✐♥❡❛r ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♥❛s t❡♦r✐❛s ❞♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ♥✉♠ ♣♦♥t♦ ❡ ❞♦s ♦♣❡r❛❞♦r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✱ ❛♠❜❛s ❞❡s❡♥✈♦❧✈✐❞❛s ♣♦r ▼❛t♦s✳

❙♦❜r❡ ♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✱ ❡st✉❞❛♠♦s ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛ ♣r♦✈❛❞♦ ❡♠ ❬✶✺❪ ❡ ❬✹✵❪✱ q✉❡ é ❡q✉✐✈❛❧❡♥t❡ à ❢❛♠♦s❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡✳ ❊st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♣✉❜❧✐❝❛❞❛ ❡♠ 1931 ♥❛ ♣r❡st✐❣✐♦s❛ r❡✈✐st❛

❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❛✜r♠❛ q✉❡✱ ♣❛r❛ K₌R _♦✉ C _{❡ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ m ≥ 1✱}

❡①✐st❡♠ ❡s❝❛❧❛r❡sBK,m≥1 t❛✐s q✉❡ N

X

i1,...,im=1

U(ei1, . . . , eim) m+12m

!m+1 2m

≤BK_,m sup

z1,...,zm∈DN

|U(z1, ..., zm)| ✭✶✮

♣❛r❛ t♦❞❛ ❢♦r♠❛ m✲❧✐♥❡❛r U : KN _{× · · · ×K}N _{→ K ❡ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ N✱ ♦♥❞❡}

(ei)N_i=1 é ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❞❡ KN✳ ❉❡s❞❡ ✶✾✸✶ ❛té ❛ ❞é❝❛❞❛ ♣❛ss❛❞❛✱ ❛♣❛r❡❝❡r❛♠ ✈ár✐❛s

❞❡♠♦♥str❛çõ❡s ❡ ❛♣❧✐❝❛çõ❡s ❞❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ ❞✐✈❡rs❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛ ❡✱ ♣♦r s✉❛ ✈❡③✱ ❢♦r❛♠ ♦❜t✐❞❛s ❢ór♠✉❧❛s ❝❛❞❛ ✈❡③ ♠❡♥♦r❡s ♣❛r❛ ❛s ❝♦♥st❛♥t❡s (BK_,m)_m

∈N. ❈♦♥t✉❞♦✱ ❡st❛s ❢ór♠✉❧❛s t✐♥❤❛♠ ❡♠ ❝♦♠✉♠ ♦ ❢❛t♦ ❞❡ ♣♦ss✉ír❡♠ ✉♠ ❝r❡s❝✐♠❡♥t♦ ❞♦ t✐♣♦ ❡①♣♦♥❡♥❝✐❛❧✳

◆♦s ú❧t✐♠♦s ❛♥♦s✱ ❛ ❜✉s❝❛ ♣❡❧❛s ❝♦♥st❛♥t❡s ♥❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡ t❡♠ tr❛③✐❞♦ ❛ ❝♦♥s✐❞❡r❛çã♦ ✉♠ ❢❛t♦ s✉r♣r❡❡♥❞❡♥t❡✿ ❛s ❝♦♥st❛♥t❡s ót✐♠❛s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ♣♦ss✉❡♠ ❝r❡s❝✐♠❡♥t♦✱ ♥♦ ♠í♥✐♠♦✱ s✉❜❡①♣♦♥❡♥❝✐❛❧✳ ❊♠ ❬✶✽❪✱ ❉✐♥✐③✱ ▼✉ñ♦③✕❋❡r♥á♥❞❡③✱ P❡❧❧❡❣r✐♥♦ ❡ ❙❡♦❛♥❡✕❙❡♣ú❧✈❡❞❛ tr♦✉①❡r❛♠ à ❧✉③ ❛ ❡✈✐❞ê♥❝✐❛ ❞❡st❡ ❢❛t♦ ♠❡❞✐❛♥t❡ ✉♠ ❡st✉❞♦ ❞❛s ❝♦♥st❛♥t❡s ❞❛❞❛s ♥♦ tr❛❜❛❧❤♦ ❬✹✵❪ ❞♦s ❞♦✐s ú❧t✐♠♦s ❛✉t♦r❡s ♠❡♥❝✐♦♥❛❞♦s✳ ❊st❛s ❝♦♥st❛♥t❡s✱ ❞❛❞❛s ❡♠ ❬✹✵❪✱ s❡rã♦ tr❛t❛❞❛s ♥❛ ❙❡çã♦ ✷✳✶✱ ❡ ❞❡♥♦t❛❞❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦ ♣♦r (CK_,m)_m

∈N✳ ❊❧❛s sã♦ ❛s ♠❡❧❤♦r❡s

❝♦♥st❛♥t❡s ❝♦♥❤❡❝✐❞❛s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ❡ ❢♦r❛♠ ❝♦♥❝❡❜✐❞❛s ♠❡❞✐❛♥t❡ ✉♠❛ ❛♥á❧✐s❡ ❡①❛✉st✐✈❛ ❞❛s ✐❞❡✐❛s ❞❡ ❉❡❢❛♥t ❡t ❛❧✳ ❞♦ ❛rt✐❣♦ ❬✶✺❪✳

❉❡♥♦t❛r❡♠♦s ♣♦r KK_,m ❛s ❝♦♥st❛♥t❡s ót✐♠❛s ✭♠❡♥♦r❡s✮ q✉❡ s❛t✐s❢❛③❡♠ ✭✶✮✳ ❯♠❛ ❞❛s ❝♦♥tr✐❜✉✐çõ❡s ♣r✐♥❝✐♣❛✐s ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ♠♦str❛ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛C t❛❧ q✉❡✿

KK_,m+1−KK_,m < C

m0.473

♣❛r❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥✜♥✐t❛ ❞❡m✳ ❆ ❢❡rr❛♠❡♥t❛ ❝❡♥tr❛❧ ♣❛r❛ ♣r♦✈❛r ❡st❛ ❡st✐♠❛t✐✈❛ ❡

♦✉tr♦s t❡♦r❡♠❛s r❡❧❛❝✐♦♥❛❞♦s q✉❡ tr❛t❛r❡♠♦s ♥❛ ❙❡çã♦ ✷✳✸✱ é ✉♠ r❡s✉❧t❛❞♦ ❞❡ ✐♥t❡r❡ss❡ ✐♥❞❡♣❡♥❞❡♥t❡✿ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♥st❛♥t❡s q✉❡ s❛t✐s❢❛③❡♠ ✭✶✮ t❛❧ q✉❡ ♦ ❧✐♠✐t❡ ❞❛ ❞✐❢❡r❡♥ç❛ ❞♦s t❡r♠♦s ❝♦♥s❡❝✉t✐✈♦s é ③❡r♦✳ ❊st❡ r❡s✉❧t❛❞♦✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ▲❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ✱ s❡rá ❞❛❞♦ ♥❛ ❙❡çã♦ ✷✳✷✳ P♦r ♠❡✐♦ ❞❡st❡✱ t❛♠❜é♠ ❝♦♥s❡❣✉✐r❡♠♦s ❞❡♠♦♥str❛r q✉❡ ❛s ❝♦♥st❛♥t❡s ót✐♠❛s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ♣♦ss✉❡♠ ❝r❡s❝✐♠❡♥t♦ s✉❜♣♦❧✐♥♦♠✐❛❧ ❞♦ t✐♣♦ p✲s✉❜✲❤❛r♠ô♥✐❝♦ ♣❛r❛ p _≈ 0.526322✱ ♥♦

❝❛s♦ ❞♦s r❡❛✐s✱ ❡p_≈0.304975♥♦ ❝❛s♦ ❞♦s ❝♦♠♣❧❡①♦s✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s

q✉❡

KR_,m<1.65 (m−1)0.526322+ 0.13

❡

KC_,m<1.41 (m−1)0.304975−0.04.

❖s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s tê♠ s❡✉s r❡s♣❡❝t✐✈♦s ❛♥á❧♦❣♦s s❡ ♠♦❞✐✜❝❛♠♦s ♥❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✮ ♦ ❡①♣♦❡♥t❡ 2m

m+1 ♣♦r

2mt

(m−1)t+2 ♣❛r❛ ♦ ♣❛râ♠❡tr♦ t ✈❛r✐❛♥❞♦ ❡♥tr❡ 1

❡ 2❀ t❛✐s ♠♦❞✐✜❝❛çõ❡s s❡rã♦ ❝❤❛♠❛❞❛s ❞❡ ✈❛r✐❛çõ❡s ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇♦❤♥❡♥❜❧✉st✲

❍✐❧❧❡✱ ❢❛③❡♥❞♦ ♣❛rt❡ ❞♦ ❈❛♣ít✉❧♦ ✷ ❡ ❡♥❝♦♥tr❛♥❞♦✲s❡ ❝♦♥t✐❞♦s ♥♦ tr❛❜❛❧❤♦✿

• ❬✸✼❪ ❚❤❡r❡ ❡①✐st ♠✉❧t✐❧✐♥❡❛r ❇♦❤♥❡♥❜❧✉st✕❍✐❧❧❡ ❝♦♥st❛♥ts (Cn)∞n=1 ✇✐t❤

limn→∞(Cn+1−Cn) = 0✳ ❉✳ ◆úñ❡③✲❆❧❛r❝ó♥✱ ❉✳ P❡❧❧❡❣r✐♥♦✱ ❏✳ ❇✳ ❙❡♦❛♥❡✲❙❡♣ú❧✈❡❞❛✱

❉✳ ▼✳ ❙❡rr❛♥♦✲❘♦❞rí❣✉❡③✳ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳✱ ✷✻✹✭✷✮✿ ✹✷✾✕✹✻✸✱ ✭✷✵✶✸✮✳

P♦r ♦✉tr❛ ♣❛rt❡✱ ♥♦ ❈❛♣ít✉❧♦ ✸ ♦❜t❡♠♦s ✉♠❛ ❢ór♠✉❧❛ ❢❡❝❤❛❞❛ q✉❡ s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♠✉❧t✐❧✐♥❡❛r ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡ ❡ q✉❡ s❡ ❛♣r♦①✐♠❛✱ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ❜❡♠✱ ❞❛s ❝♦♥st❛♥t❡s (CK_,m)_m

∈N ♠❡♥❝✐♦♥❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ❛s q✉❛✐s ❡stã♦ ❞❛❞❛s ♣♦r ✉♠❛ ❢ór♠✉❧❛ r❡❝✉rs✐✈❛ q✉❡ ❡♥✈♦❧✈❡ ❛ ❢✉♥çã♦ Γ✱ t♦r♥❛♥❞♦ ❝♦♠♣❧✐❝❛❞♦ ♦ ❝á❧❝✉❧♦

❞❡st❛s✳ ❆ss✐♠✱ ❛♣r❡s❡♥t❛♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ ❛ ♠❡♥♦r ❢ór♠✉❧❛ ❢❡❝❤❛❞❛ ♣❛r❛ ❝♦♥st❛♥t❡s q✉❡

s❛t✐s❢❛③❡♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡✳ ❊st❡ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛✲s❡ ♥❛ ♣✉❜❧✐❝❛çã♦✿

• ❬✺✹❪ ■♠♣r♦✈✐♥❣ t❤❡ ❝❧♦s❡❞ ❢♦r♠✉❧❛ ❢♦r s✉❜♣♦❧②♥♦♠✐❛❧ ❝♦♥st❛♥ts ✐♥ t❤❡ ♠✉❧t✐❧✐♥❡❛r ❇♦❤♥❡♥❜❧✉st✕❍✐❧❧❡ ✐♥❡q✉❛❧✐t✐❡s✳ ❉✳ ▼✳ ❙❡rr❛♥♦✲❘♦❞rí❣✉❡③✳ ▲✐♥✳ ❆❧❣✳ ❛♥❞ ✐ts ❆♣♣❧✳✱ ✹✸✽✭✼✮✿ ✸✶✷✹✕✸✶✸✽✱ ✭✷✵✶✸✮✳

❋✐♥❛❧♠❡♥t❡✱ ❧❡♠❜r❛♠♦s q✉❡✱ ❡♠ ❬✺✶❪✱ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❛❜str❛t❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ♣❛r❛ ❡s♣❛ç♦s ❞❡ s❡q✉ê♥❝✐❛s ♠✉✐t♦ ❣❡r❛✐s✱ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ❡ ❡①♣❧♦r❛❞❛✳ ❊ss❛ ❝❧❛ss❡ ❢♦✐ ❝❤❛♠❛❞❛ ❞❡ ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ λ✲s♦♠❛♥t❡s✳ ❆

❜✉s❝❛ ❞❡ ❛♠❜✐❡♥t❡s ❛❜str❛t♦s ♦♥❞❡ ✉♠❛ t❡♦r✐❛ ♠❛✐s ❣❡r❛❧ ❛✐♥❞❛ s❡❥❛ ✈á❧✐❞❛ t❡♠ s✐❞♦ ✐♥✈❡st✐❣❛❞❛ ❡♠ ❞✐❢❡r❡♥t❡s ❛rt✐❣♦s✱ ❛✐♥❞❛ ♣❛r❛ ♦♣❡r❛❞♦r❡s ♥ã♦✲♠✉❧t✐❧✐♥❡❛r❡s ✭❝✐t❛♠♦s ❬✸✶✱ ✹✷✱ ✹✸❪ ❡ ❛s r❡❢❡rê♥❝✐❛s ❛❧✐ ❝♦♥t✐❞❛s✮✳

◆♦ ❈❛♣ít✉❧♦ ✹✱ ✐♥tr♦❞✉③✐♠♦s ✉♠❛ ❛❜♦r❞❛❣❡♠ s❡♠❡❧❤❛♥t❡ ♣❛r❛ ♦ ❝❛s♦ ♠✉❧t✐❧✐♥❡❛r✳ ▼♦str❛♠♦s q✉❡ ✈ár✐♦s r❡s✉❧t❛❞♦s ♠✉❧t✐❧✐♥❡❛r❡s ❝♦♥❤❡❝✐❞♦s ✭❡ t❛♠❜é♠ ❛❧❣✉♥s ♦✉tr♦s ♥♦✈♦s r❡s✉❧t❛❞♦s✮ sã♦ ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s ❞❡ ♥♦ss❛ ❝❧❛ss❡ ❛❜str❛t❛✱ ❛ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ γ(s;s1,...,sm)✲s♦♠❛♥t❡s ♥✉♠ ♣♦♥t♦✳ ❈♦♠♦ é ❞❡ s❡ ❡s♣❡r❛r✱ ✈ár✐♦s ❞♦s r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ♥✉♠ ♣♦♥t♦ ❛♣❛r❡❝❡♠ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦s ♥♦ss♦s r❡s✉❧t❛❞♦s ♥♦ ❛♠❜✐❡♥t❡ ❛❜str❛t♦✳

❆❧é♠ ❞✐ss♦✱ ❛ ❝❧❛ss❡ ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ γ(s;s1,...,sm)✲s♦♠❛♥t❡s ❝♦♥té♠✱ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ♦✉tr❛s ❝❧❛ss❡s ❞❡ ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❡st✉❞❛❞❛s r❡❝❡♥t❡♠❡♥t❡✱ t❛✐s ❝♦♠♦ ♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s q✉❛s❡ s♦♠❛♥t❡s ❡ ❈♦❤❡♥ ❢♦rt❡♠❡♥t❡ s♦♠❛♥t❡s✳ ❊st❡s r❡s✉❧t❛❞♦s ❢❛③❡♠ ♣❛rt❡ ❞♦ ❛rt✐❣♦✿

• ❬✺✸❪ ❆❜s♦❧✉t❡❧② γ₋s✉♠♠✐♥❣ ♠✉❧t✐❧✐♥❡❛r ♦♣❡r❛t♦rs ❉✳ ▼✳ ❙❡rr❛♥♦✲❘♦❞rí❣✉❡③✳ ▲✐♥✳

❆❧❣✳ ❛♥❞ ✐ts ❆♣♣❧✳✱ ✹✸✾✭✶✷✮✿ ✹✶✶✵✕✹✶✶✽✱ ✭✷✵✶✸✮✳

◆♦t❛çã♦ ❡ t❡r♠✐♥♦❧♦❣✐❛

◆♦ ❞❡❝♦rr❡r ❞❡st❡ t❡①t♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s ❛♣❡♥❛s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s♦❜r❡ ♦ ❝♦r♣♦

K _{❞♦s ♥ú♠❡r♦s r❡❛✐s} R _{♦✉ ❞♦s ❝♦♠♣❧❡①♦s} C_{✳ ◆ã♦ ❢❛r❡♠♦s ❞✐st✐♥çã♦ ❡♥tr❡ ♦s t❡r♠♦s}

✧❛♣❧✐❝❛çã♦✧✱ ✧❢✉♥çã♦✧✱ ✧♠❛♣❛✧♦✉ ✧♦♣❡r❛❞♦r✧✳ ◆❛ ♠❛✐♦r ♣❛rt❡ ❞❡st❡ t❡①t♦✱ X✱ Y✱ E✱ F✱ G✱ H✱ Xi✱ Yi, ... ❞❡♥♦t❛rã♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳ ❆ ♥♦r♠❛ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ E s❡rá ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛❞❛ ♣♦r _k·k; q✉❛♥❞♦ ♠❛✐♦r ♣r❡❝✐sã♦ ❢♦r ♥❡❝❡ssár✐❛✱ ✉s❛r❡♠♦s

k·kE. ❖ sí♠❜♦❧♦ BE ❞❡♥♦t❛rá ❛ ❜♦❧❛ ✉♥✐tár✐❛ ❢❡❝❤❛❞❛ {x∈E;kxk ≤1} ❞❡ ✉♠ ❡s♣❛ç♦

❞❡ ❇❛♥❛❝❤ E.

❖ ❞✉❛❧ t♦♣♦❧ó❣✐❝♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ E s❡rá ❞❡♥♦t❛❞♦ ♣♦r E′_{✱ ❧❡♠❜r❡♠♦s}

q✉❡ E′ _=L(E,K_{) ❡✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ❞❡♥♦t❛r❡♠♦s ♣♦r L(E1, ..., Em;F) ♦ ❡s♣❛ç♦ ❞❡}

❇❛♥❛❝❤ ❞❡ t♦❞❛s ❛s ❛♣❧✐❝❛çõ❡sm✲❧✐♥❡❛r❡s ❞❡E1×...×Em ❡♠ F✱ ❝♦♠ ❛ ♥♦r♠❛ ✉s✉❛❧ ❞♦

sup✳ ◗✉❛♥❞♦ E1 = ...=Em✱ ♦ ❡s♣❛ç♦ s❡rá ❞❡♥♦t❛❞♦ ❞❛ ❢♦r♠❛ L(nE;F) ❡ s❡ F =K✱

❞❡♥♦tá✲❧♦✲❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ♣♦rL(E1, ..., Em).

❉✐r❡♠♦s q✉❡ T _{∈ L}(E;F) é ❞❡ ♣♦st♦ ✜♥✐t♦ q✉❛♥❞♦ ❛ ❞✐♠❡♥sã♦ ❞❡ T(F) ❢♦r

✜♥✐t❛✳ ❆ ✐♠❛❣❡♠ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r T : V _→ W ❡♥tr❡ ❞♦✐s ❡s♣❛ç♦s q✉❛✐sq✉❡r V✱

W_{❞❡ ✉♠ ✈❡t♦r v ∈}V _{s❡rá ❡①♣r❡ss❛ ♣♦r T v ♦✉ T(v)✳}

❖s ♥ú♠❡r♦s r❡❛✐s p, q s❡rã♦ t❛✐s q✉❡ 1 _≤ p, q < _∞✳ ❖ ❝♦♥❥✉❣❛❞♦ ❞❡ p s❡rá

❞❡♥♦t❛❞♦ ♣♦rp′_{✱ p}′ _{∈(1,∞)✳ ❊st❡ é t❛❧ q✉❡} 1

p +

1

p′ = 1✳ P❛r❛p= 1✱ p′ =∞✳

❈❤❛♠❛♠♦s ❛ ❛t❡♥çã♦ ♣❛r❛ ♦s s❡❣✉✐♥t❡s ❡s♣❛ç♦s q✉❡ s✉r❣✐rã♦ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✿ ❙❡ 1_≤p < _∞✱ lp(X) :=(xn)∞_n=1 ∈XN; P_nkxnkp <∞ . ❙❡X =K✱ ♦ ❡s♣❛ç♦ lp(X)s❡rá ❞❡♥♦t❛❞♦ s✐♠♣❧❡s♠❡♥t❡ ♣♦r lp.

l_∞(X)é ♦ ❡s♣❛ç♦ ❞❛s s❡q✉ê♥❝✐❛s ❧✐♠✐t❛❞❛s ❞❡ X✳ ◆♦✈❛♠❡♥t❡✱ s❡X =K❞❡♥♦t❛✲

r❡♠♦sl_∞(X) ♣♦rl_∞. lN

❈❛♣ít✉❧♦ ✶

❆ ♦r✐❣❡♠✳ ❖♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡

s♦♠❛♥t❡s✳

❏♦❤❛♥ ❉✐r✐❝❤❧❡t ✭1805 ₋ 1859✮✱ ♠❛t❡♠át✐❝♦ ❛❧❡♠ã♦✱ ❞❡♠♦♥str♦✉ q✉❡ ✉♠❛

s❡q✉ê♥❝✐❛ ❞❡ ❡s❝❛❧❛r❡s é ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠á✈❡❧ ♣r❡❝✐s❛♠❡♥t❡ q✉❛♥❞♦ é ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ s♦♠á✈❡❧ ❡✱ ♠❡❞✐❛♥t❡ ❛❥✉st❡s ❢❡✐t♦s ♥❡st❛ ❞❡♠♦♥str❛çã♦✱ ❡st❡♥❞❡✉✲s❡ ❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ q✉❛❧q✉❡r ❡s♣❛ç♦ ♥♦r♠❛❞♦ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❆♥♦s ❞❡♣♦✐s✱ ♣♦r ❙t❡❢❛♥ ❇❛♥❛❝❤ ✭1892 ₋ 1945✮✱ s✉r❣✐✉ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ✉♠

❡s♣❛ç♦ ♥♦r♠❛❞♦ é ❝♦♠♣❧❡t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞❛ s❡q✉ê♥❝✐❛ ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠á✈❡❧ é ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ s♦♠á✈❡❧✳

◆❡st❛ ❞✐r❡çã♦✱ ❛ s❡❣✉✐♥t❡ q✉❡stã♦ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ❡♠ 1932 ♣♦r ❇❛♥❛❝❤ ❡♠ s❡✉

❧✐✈r♦ ❚❤é♦r✐❡ ❞❡s ♦♣ér❛t✐♦♥s ❧✐♥é❛✐r❡s✿ ✧❊①✐st❡✱ ❡♠ q✉❛❧q✉❡r ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❞❡ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛✱ ✉♠❛ sér✐❡ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡ q✉❡ ♥ã♦ é ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡❄✧✳

❊st❛ ♣❡r❣✉♥t❛ ❢❛③ ♣❛rt❡ ❞♦ ❧✐✈r♦ ❚❤❡ ❙❝♦tt✐s❤ ❇♦♦❦ ✭❖ ▲✐✈r♦ ❊s❝♦❝ês✮ ❡ ❧❡✈♦✉ q✉✐♥③❡ ❛♥♦s ♣❛r❛ s❡r r❡s♣♦♥❞✐❞❛ ♠❡❞✐❛♥t❡ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✷✵❪ ♣♦r ❆✳ ❉✈♦r❡t③❦② ❡ ❈✳ ❆✳ ❘♦❣❡rs✿ ✧❙ér✐❡s ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s ❝♦✐♥❝✐❞❡♠ ❝♦♠ sér✐❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♥✈❡r❣❡♥t❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ t❡♠ ❞✐♠❡♥sã♦ ✜♥✐t❛✧✳

❊st❡ r❡s✉❧t❛❞♦ ❝❤❛♠♦✉ ❛ ❛t❡♥çã♦ ❞❡ ❆❧❡①❛♥❞❡r ●r♦t❤❡♥❞✐❡❝❦ ✭1928₋✮✱ q✉❡

❞❡♠♦♥str❛çã♦ ♣❛r❛ ❡ss❡ r❡s✉❧t❛❞♦✳ ❊❧❡ ❞❡✜♥✐✉ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ❝♦♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿ ✧❙❡ (xn)∞n=1 é ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ s♦♠á✈❡❧✱ ❡♥tã♦ (T xn)∞n=1 é

❛❜s♦❧✉t❛♠❡♥t❡ s♦♠á✈❡❧✱ ❞❡s❞❡ q✉❡ T s❡❥❛ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦ ❡♥tr❡

❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✧✳ ❊st❡s sã♦✱ ♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳ ▼❡❞✐❛♥t❡ ❞✐❢❡r❡♥t❡s r❡s✉❧t❛❞♦s ❡ s❡✉ tr❛❜❛❧❤♦ ❘és✉♠é ❞❡ ❧❛ t❤é♦r✐❡ ♠étr✐q✉❡ ❞❡s ♣r♦❞✉✐ts t❡♥s♦r✐❡❧s t♦♣♦❧♦❣✐q✉❡s ❛♣r❡s❡♥t❛❞♦ ❡♠ 1956 ✭✈❡❥❛ ❬✷✷❪✱ ❬✷✸❪✮✱ ●r♦t❤❡♥❞✐❡❝❦✱ ❡♠

❛❧❣✉♠ s❡♥t✐❞♦✱ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♦ ❛♥❝❡str❛❧ ❞❛ t❡♦r✐❛ ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳ ❊♥tr❡t❛♥t♦✱ ❛ ♥♦çã♦ ❞❡ ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ❡♠ s✉❛ ❢♦r♠✉❧❛çã♦ ♠♦❞❡r♥❛✱ é ❞❡✈✐❞❛ ❛ ❆✳ P✐❡ts❝❤ ❡♠ ❬✹✻❪✱ ❇✳ ▼✐t✐❛❣✐♥ ❡ ❆✳ P❡➟❝③②➠s❦✐ ❡♠ ❬✸✸❪✳ ❊♠ 1968✱ ✉♠ tr❛❜❛❧❤♦ ❝é❧❡❜r❡✱ ❬✷✻❪✱ ❞❡ ❆✳ P❡➟❝③②➠s❦✐ ❡ ▲✐♥❞❡str❛✉ss✱ t♦r♥♦✉ ❛

t❡♦r✐❛ ❞❡ ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s ♠❛✐s ♣♦♣✉❧❛r ❡ ♠♦str♦✉ ❛♣❧✐❝❛çõ❡s à t❡♦r✐❛ ❞♦s ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳

✶✳✶ ❖♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s

◆❡st❛ s❡çã♦✱ ✈❛♠♦s ❡s❜♦ç❛r ✉♠ ❜r❡✈❡ ♣❛♥♦r❛♠❛ ❞❛ t❡♦r✐❛ ❞♦s ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s❀ ❜❛s❡ ❞❛ t❡♦r✐❛ ❞♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❡ ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✱ ♦s q✉❛✐s sã♦ ♥♦ss♦ ❡✐①♦ ❞❡ ❡st✉❞♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✶ ❙❡❥❛♠1_≤p_{≤ ∞}❡X✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❯♠❛ s❡q✉ê♥❝✐❛(xn)∞_n=1

❡♠ X é ❢♦rt❡♠❡♥t❡ p✲s♦♠á✈❡❧ s❡ (_kxnk)∞_n=1 ∈ ℓp.

❉❡♥♦t❛♠♦s ♣♦r ℓp(X) ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ❢♦rt❡♠❡♥t❡ p✲s♦♠á✈❡✐s ❡♠ X. ❊♠ ℓp(X)✱1≤p≤ ∞,❞❡✜♥✐♠♦s

k(xn)∞n=1kp :=

  

(P∞_n=1kxnkp)

1

p ✱ s❡ _{1≤p <∞}

supn∈Nkxnk✱ s❡ p=∞.

❖ ❡s♣❛ç♦ ℓp(X)✱ ♠✉♥✐❞♦ ❝♦♠ ❡st❛ ♥♦r♠❛✱ é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

❊♠ ℓp(X), ❛s s❡q✉ê♥❝✐❛s (x1, x2, , . . . , xn,0,0, . . .) s❡rã♦ ✐❞❡♥t✐✜❝❛❞❛s ❝♦♠ ❛s

s❡q✉ê♥❝✐❛s ✜♥✐t❛s (x1, x2, , . . . , xn)✳ ❖ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r ❡st❛s s❡q✉ê♥❝✐❛s ❢♦r♠❛

✉♠ s✉❜❡s♣❛ç♦ ❞❡♥s♦ ❡♠ ℓp(X)✱ ♣❛r❛ 1≤ p <∞✳ ❉❡ ❢❛t♦✱ ❞❛❞♦s ε >0 ❡ x = (xn)∞_n=1

❡♠ ℓp(X)✱ ❡①✐st❡ N ∈N t❛❧ q✉❡ ∞

X

n=N+1

kxnkp

!1 p

< ε.

❆❣♦r❛✱ é só ❡s❝♦❧❤❡r ❛ s❡q✉ê♥❝✐❛ ✜♥✐t❛x′ _{= (x}

1, x2, , . . . , xN), ❡✱ ❞❛í✱

kx₋x′_kp = ∞

X

n=N+1

kxnkp

!1 p

< ε.

P❛r❛p=_∞❞❡♠♦♥str❛✲s❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

❉❡✜♥✐çã♦ ✶✳✶✳✷ ❙❡❥❛♠1_≤p_{≤ ∞}❡X✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳ ❯♠❛ s❡q✉ê♥❝✐❛(xn)∞_n=1

❡♠ X é ❢r❛❝❛♠❡♥t❡p✲s♦♠á✈❡❧ s❡ (ϕ(xn))∞_n=1 ∈ ℓp ♣❛r❛ t♦❞♦ ϕ∈X′.

❉❡♥♦t❛♠♦s ♣♦r ℓw

p (X) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s❡q✉ê♥❝✐❛s ❢r❛❝❛♠❡♥t❡p✲s♦♠á✈❡✐s✳

ℓw

p (X),k·k

w p

é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✱ ♣❛r❛ t♦❞♦ p_∈[1,_∞],♦♥❞❡

k(xn)∞_n=1kw_p := sup

ϕ∈B_X′

k(ϕ(xn))_nk_p, ✭✶✳✶✮

◆♦t❡ q✉❡✱ ♣❛r❛1_≤p < _∞✱ ℓp(X)⊂ ℓwp (X).❉❡ ❢❛t♦✱ s❡ (xn)∞_n=1 ∈ℓp(X),❡♥tã♦

k(xn)∞n=1k

w

p = sup

ϕ∈B_X′

∞

X

j=1

|ϕ(xj)|p

!1 p

≤ sup

ϕ∈B_X′

∞

X

j=1

kϕ_kp_kxjkp

!1 p

= ∞

X

j=1

kxjkp

!1 p

=_k(xn)∞_n=1k_p.

◗✉❛♥❞♦ p = _∞✱ ✈❛♠♦s t❡r s❡♠♣r❡ q✉❡ ℓw

∞(X) = ℓ∞(X) ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛s ♥♦r♠❛s

❝♦✐♥❝✐❞❡♠✳ ❉❡ ❢❛t♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❍❛❤♥✲❇❛♥❛❝❤✱ t❡♠♦s

k(xn)nk_∞= sup

n kxnk= supn ϕsup∈B_X′

|ϕ(xn)|

!

= sup

ϕ∈B_X′

sup

n |ϕ(xn)|

=_k(xn)kw_∞

◗✉❛♥❞♦1_≤p <_∞✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ℓp(X)❡ℓwp (X)❛❝♦♥t❡❝❡rá s♦♠❡♥t❡ q✉❛♥❞♦

♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ X ♣♦ss✉✐r ❞✐♠❡♥sã♦ ✜♥✐t❛✳ ❊st❡ r❡s✉❧t❛❞♦ é ♦ q✉❡ s❡ ❝♦♥❤❡❝❡ ❝♦♠♦ ❛

✈❡rsã♦ ❢r❛❝❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❉✈♦r❡t③❦②✲❘♦❣❡rs✱

❚❡♦r❡♠❛ ✶✳✶✳✸ ✭❉✈♦r❡t③❦②✲❘♦❣❡rs✱ ✈❡rsã♦ ❢r❛❝❛✮ ❙❡❥❛ 1 _≤p <_∞. ❚♦❞♦ ❡s♣❛ç♦

❞❡ ❇❛♥❛❝❤ ❝♦♠ ❞✐♠❡♥sã♦ ✐♥✜♥✐t❛ ❝♦♥té♠ ✉♠❛ s❡q✉ê♥❝✐❛ ❢r❛❝❛♠❡♥t❡ p✲s♦♠á✈❡❧✱ q✉❡

♥ã♦ é ❢♦rt❡♠❡♥t❡ p✲s♦♠á✈❡❧✳

➱ ❢á❝✐❧ ✈❡r q✉❡ q✉❛♥❞♦ t❡♠♦s ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦ u ❡♥tr❡ ❡s♣❛ç♦s ❞❡

❇❛♥❛❝❤ X ❡ Y✱ ❡st❡ ✈❛✐ ❧❡✈❛r s❡q✉ê♥❝✐❛s (xn)∞_n=1 ∈ ℓp(X) ❡♠ s❡q✉ê♥❝✐❛s

(uxn)∞_n=1 ∈ ℓp(Y) ❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ s❡ (xn)∞_n=1 ∈ ℓwp (X)✱ ❡♥tã♦✱ (uxn)∞_n=1 ∈ ℓwp (Y)✳

❊♠ ♠❡❧❤♦r❡s t❡r♠♦s✱ s❡ u : X _→ Y é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❧✐♠✐t❛❞♦ ❡♥tr❡ ❡s♣❛ç♦s ❞❡

❇❛♥❛❝❤✱ ❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛

(xn)∞n=1 →(uxn)∞n=1

s❡♠♣r❡ ✐♥❞✉③ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❧✐♠✐t❛❞♦ uˆs _{: ℓ}

p(X) → ℓp(Y)✱ ❝♦♠♦ t❛♠❜é♠ ✉♠

♦♣❡r❛❞♦r ❧✐♥❡❛r ❧✐♠✐t❛❞♦uˆw _:ℓw

p (X)→ℓwp (Y)❡✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱

kuˆs_k=_kuˆw_k=_ku_k.

❉❡ ❢❛t♦✱ s❡ uˆs_:ℓ

p(X)→ℓp(Y), t❡♠♦s

kuˆs_k= sup

k(xn)∞n=1kp≤1

∞

X

n=1

kuxnkp

!1 p

≤ sup

k(xn)∞n=1kp≤1 ku_k

∞

X

n=1

kxnkp

!1 p

=_ku_k

❡

ku_k= sup

kxk≤1k

ux_k= sup

k(yn)∞n=1=(x,0,0,...)kp≤1

kuˆs((yn)∞_n=1)k_p ≤ kuˆsk.

▲♦❣♦✱ kuˆs_{k = kuk. ❈♦♠ r❛❝✐♦❝í♥✐♦ s✐♠✐❧❛r✱ é ❢á❝✐❧ ♠♦str❛r q✉❡ kuˆ}w_{k = kuk.}

❆ss✐♠✱ ♣♦❞❡♠♦s ❢❛❧❛r ♥♦r♠❛❧♠❡♥t❡ q✉❡u✱ ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦✱ ❧❡✈❛ ✧❢♦rt❡ ❡♠

❢♦rt❡✧ ❡ ✧❢r❛❝♦ ❡♠ ❢r❛❝♦✧✳ ❈♦♠♦ ℓp(X) ⊂ ℓwp (X) ❡♥tã♦✱ t❡♠♦s t❛♠❜é♠ q✉❡ u✱ ✉♠

♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦✱ ❧❡✈❛ ✧❢♦rt❡ ❡♠ ❢r❛❝♦✧✳ ❊ ✧❢r❛❝♦ ❡♠ ❢♦rt❡✧❄ ❊st❛ q✉❡stã♦ ♥♦s ❧❡✈❛ à ❞❡✜♥✐çã♦ ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✹ ✭❖♣❡r❛❞♦r ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡✮ ❙❡❥❛♠ 1 _≤ p, q < _∞ ❡ u : X _−→ Y ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❝♦♥tí♥✉♦ ❡♥tr❡ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳ ❉✐③❡♠♦s q✉❡ u é ❛❜s♦❧✉t❛♠❡♥t❡ (p;q)✲s♦♠❛♥t❡ ✭♦✉ (p;q)✲s♦♠❛♥t❡✮ q✉❛♥❞♦ ♦ ♦♣❡r❛❞♦r ✐♥❞✉③✐❞♦

ˆ

u: ℓw

q (X) →ℓp(Y)

(xn)∞_n=1 →(uxn)∞_n=1

❡st✐✈❡r ❜❡♠ ❞❡✜♥✐❞♦ ❡ ❢♦r ❧✐♥❡❛r ❡ ❧✐♠✐t❛❞♦✳

❉❡♥♦t❛♠♦s ♣♦r Qp,q(X;Y) ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣♦r t♦❞♦s ♦s ♦♣❡r❛❞♦r❡s

(p;q)✲s♦♠❛♥t❡s ❞❡ X ❡♠ Y✳ ◗✉❛♥❞♦ p = q✱ ❡s❝r❡✈❡♠♦s Q_p(X;Y) ♥♦ ❧✉❣❛r ❞❡

Q

p,q(X;Y). ◗✉❛♥❞♦ p = 1, ❞✐③❡♠♦s s✐♠♣❧❡s♠❡♥t❡ q✉❡ ♦ ♦♣❡r❛❞♦r é ❛❜s♦❧✉t❛♠❡♥t❡

s♦♠❛♥t❡✳

●❡r❛❧♠❡♥t❡✱ ♣❛r❛ ❞❡t❡r♠✐♥❛r q✉❛♥❞♦ ✉♠ ♦♣❡r❛❞♦r é✱ ♦✉ ♥ã♦✱ ❛❜s♦❧✉t❛♠❡♥t❡

(p;q)✲s♦♠❛♥t❡✱ ✉t✐❧✐③❛✲s❡ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳ ❊st❡ ❝❛r❛❝t❡r✐③❛ ❡st❡s ♦♣❡r❛❞♦r❡s ❛tr❛✈és

❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✺ ❙❡❥❛u_{∈ L}(X;Y). ❙ã♦ ❡q✉✐✈❛❧❡♥t❡s✿

(i) u é (p;q)✲s♦♠❛♥t❡❀ (ii) ❊①✐st❡ K >0 t❛❧ q✉❡

n

X

k=1

ku(xk)kp

!1 p

≤K sup

ϕ∈B_X′

n

X

k=1

|ϕ(xk)|q

!1 q

, ✭✶✳✷✮

♣❛r❛ q✉❛✐sq✉❡r x1, ..., xn ❡♠ X ❡ n ♥❛t✉r❛❧❀

(iii) ❊①✐st❡ K >0 t❛❧ q✉❡ ∞

X

k=1

ku(xk)kp

!1 p

≤K sup

ϕ∈B_X′

∞

X

k=1

|ϕ(xk)|q

!1 q

,

s❡♠♣r❡ q✉❡ (xk)∞_k=1 ∈ℓwq (X).

❉❡♥♦t❛♠♦s ♣♦r πp,q(u) ♦ í♥✜♠♦ ❞♦s K t❛✐s q✉❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✷✮ ❝♦♥t✐♥✉❛

✈á❧✐❞❛✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s πp,q(u) = kuˆk.

Pr♦♣♦s✐çã♦ ✶✳✶✳✻ Qp,q(X, Y), πp,q(·)

é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡✱ ❛❧é♠ ❞✐ss♦✱ ku_{k ≤}πp,q(u)

➱ ✐♠♣♦rt❛♥t❡ ♣❡r❝❡❜❡r q✉❡ s❡ p < q✱ ❡♥tã♦✱ s♦♠❡♥t❡ ♦ ♦♣❡r❛❞♦r ♥✉❧♦ ♣♦❞❡ s❡r

(p;q)✲s♦♠❛♥t❡✳ ❉❡ ❢❛t♦✱ é ❝❧❛r♦ q✉❡ ♣♦❞❡♠♦s s✉♣♦r X ₆= _{0_}✳ ❈♦♠♦ p < q, s❡♠♣r❡

♣♦❞❡♠♦s ❡♥❝♦♥tr❛r (λk)∞k=1 ❡♠ ℓq −ℓp. ❊♥tã♦✱ ♣❛r❛ 0 6= x ∈ X✱ (λkx)k∞=1 ∈ ℓwq (X).

❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❛ u ₆= 0 ❛❜s♦❧✉t❛♠❡♥t❡ (p;q)✲s♦♠❛♥t❡✳ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦

✶✳✶✳✺✱ ❡①✐st❡ K >0t❛❧ q✉❡

n

X

k=1

ku(λkx)kp

!1 p

≤K sup

ϕ∈B_X′

n

X

k=1

|ϕ(λkx)|q

!1 q

♣❛r❛ t♦❞♦n ♥❛t✉r❛❧✳ ❆ss✐♠✱

ku(x)_k

n

X

k=1

|λk|p

!1 p

≤K sup

ϕ∈BX′

|ϕ(x)_|

n

X

k=1

|λk|q

!1 q

,

✐st♦ é✱

ku(x)_k

n

X

k=1

|λk|p

!1 p

≤K_kx_k n

X

k=1

|λk|q

!1 q

.

❚♦♠❛♥❞♦sup_kxk≤1, ♦❜t❡♠♦s

ku_k n

X

k=1

|λk|p

!1 p

≤K n

X

k=1

|λk|q

!1 q

,

❡ ❝♦♥❝❧✉í♠♦s q✉❡ (λk)∞_k=1 ∈ℓp✱ ♦ q✉❡ s✐❣♥✐✜❝❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳

P♦rt❛♥t♦✱ ♣❛r❛ ❡✈✐t❛r ♦ ❝❛s♦ tr✐✈✐❛❧✱ ✈❛♠♦s s❡♠♣r❡ s✉♣♦r p_≥q✳

P❛r❛ ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ♦ ❧✐✈r♦ ❬✶✼❪ é ✉♠❛ ❡①❝❡❧❡♥t❡ r❡❢❡rê♥❝✐❛ ♣❛r❛ ❝♦♥s✉❧t❛s✳

❈♦♠♦ ♠❡♥❝✐♦♥❛♠♦s ♥♦ ✐♥✐❝✐♦ ❞❡st❡ t❡①t♦✱ ♥♦ss♦ tr❛❜❛❧❤♦ ❜❛s❡✐❛✲s❡ ♥♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ ❡ ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✱ ❞✉❛s ❣❡♥❡r❛❧✐③❛çõ❡s ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳ ❉❡st❛ ❢♦r♠❛✱ ❝♦♥✈é♠ ❛♣r❡s❡♥t❛r♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❞❛s ♠❡s♠❛s✳

✶✳✷ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥✲

t❡s

❖ ❡♥❢♦q✉❡✱ t❛♥t♦ ♠✉❧t✐❧✐♥❡❛r ❝♦♠♦ ♣♦❧✐♥♦♠✐❛❧✱ ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ❢♦✐ ✐♥✐❝✐❛❞♦ ♣♦r P✐❡ts❝❤ ✭❬✹✼❪✮ ❡✱ ❞❡s❞❡ ❡♥tã♦✱ ❞✐❢❡r❡♥t❡s ❛✉t♦r❡s tê♠ tr❛❜❛❧❤❛❞♦ ❡♠ ❛♠❜♦s ♦s ❝♦♥t❡①t♦s ♠❡♥❝✐♦♥❛❞♦s✳ ❖❜s❡r✈❡ ♣❛r❛ ❡①❡♠♣❧♦s ❞♦ ❝❛s♦ ♣♦❧✐♥♦♠✐❛❧ ❬✽❪✱ ❬✶✵❪ ❡ ❬✷✶❪✳ P❛r❛ ♦ ♠✉❧t✐❧✐♥❡❛r✱ ❬✶✵❪✱ ❬✸✵❪ ❡ ❬✹✹❪✳

P❛r❛ ♥♦ss♦ ✐♥t❡r❡ss❡ ❞❡st❛❝❛♠♦s ♦ tr❛❜❛❧❤♦ ❬✸✵❪✱ ♦♥❞❡ ▼✳❈✳ ▼❛t♦s ❞❡✜♥❡ ❛ s❡✲ ❣✉✐♥t❡ ❝❧❛ss❡✱ ❡①♣❧♦r❛❞❛ ♣♦r ❞✐❢❡r❡♥t❡s ❛✉t♦r❡s ✭✈❡❥❛ ❬✷❪✱ ❬✹❪✱ ❬✹✽❪✮✳

❉❡✜♥✐çã♦ ✶✳✷✳✶ ✭❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✳✮ ❙❡

1

p ≤

1

q1 +...+

1

qm✱ ✉♠ ♦♣❡r❛❞♦r ♠✉❧t✐❧✐♥❡❛r T ∈ L(E1, ..., Em;F) é ❛❜s♦❧✉t❛♠❡♥t❡

(p;q1, ..., qm)✲s♦♠❛♥t❡ ♥♦ ♣♦♥t♦ a= (a1, ..., am)∈E1×...×Em q✉❛♥❞♦ T a1+x1j, ..., am+xmj

−T(a1, ..., am)

∞

j=1 ∈ℓp(F)

♣❛r❛ t♦❞♦ xk j

∞

j=1 ∈ ℓ

w

qk(Ek). ❊st❛ ❝❧❛ss❡ é ❞❡♥♦t❛❞❛ ♣♦r L

a

as,(p;q1,...,qm). ◗✉❛♥❞♦ a é ❛

♦r✐❣❡♠✱ ❞❡♥♦t❛♠♦✲❧❛ ♣♦r Las,(p;q1,...,qm)✱ ❡ s❡ T é ❛❜s♦❧✉t❛♠❡♥t❡ (p;q1, ..., qm)✲s♦♠❛♥t❡ ❡♠ t♦❞♦ ♣♦♥t♦✱ ❡s❝r❡✈❡♠♦sT _{∈ L}ev

as,(p;q1,...,qm).

❯♠ r❡s✉❧t❛❞♦ út✐❧ ♥❡st❛ t❡♦r✐❛ é ❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣♦r ❞❡s✐❣✉❛❧❞❛❞❡s✱ ❛ q✉❛❧ ♥♦s ❞á ✉♠❛ ♥♦✈❛ ❢❡rr❛♠❡♥t❛✱ ❛❧é♠ ❞❛ ❞❡✜♥✐çã♦✱ ♣❛r❛ ♣r♦✈❛r s❡ ✉♠ ♦♣❡r❛❞♦r

T _{∈ L}(E1, ..., Em;F) é ❛❜s♦❧✉t❛♠❡♥t❡ (p;q1, ..., qm)✲s♦♠❛♥t❡✳ P❛r❛ ✉♠❛ ❞❡♠♦♥str❛✲

çã♦ ❞❡st❡s r❡s✉❧t❛❞♦s✱ r❡❝♦♠❡♥❞❛♠♦s ❬✹❪ ❡ ❬✼✱ ❚❡♦r❡♠❛ 1.2, ii)❪✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✷ T _{∈ L}as,(p;q1,...,qm)(E1, ..., Em;F) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C >0 t❛❧ q✉❡

T x(1)_j , ..., x(_jm)∞

j=1 p ≤C m Y s=1

x(_js)∞

j=1 w qs ✭✶✳✸✮

♣❛r❛ t♦❞♦ x(_js)∞

j=1 ∈ ℓ

w

qs(Es), s = 1, ..., m. ❆❧é♠ ❞✐ss♦✱ ❛ ♠❡♥♦r ❞❛s ❝♦♥st❛♥t❡s

C q✉❡ s❛t✐s❢❛③❡♠ ✭✶✳✸✮✱ q✉❡ s❡rá ❞❡♥♦t❛❞❛ ♣♦r _kT_kas✱ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ♥♦ ❡s♣❛ç♦

Las,(p;q1,...,qm)(E1, ..., Em;F)✳

❚❡♦r❡♠❛ ✶✳✷✳✸ P❛r❛ T _{∈ L}(E1, ..., Em;F)✱ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s:

(i) T _{∈ L}ev

as,(p;q1,...,qm)(E1, ..., Em;F) ;

(ii) ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C >0 t❛❧ q✉❡

Tb1+x(1)j , ..., bm+x(jm)

−T (b1, ..., bm)

n j=1 p

≤C _kb1k+

x(1)_j n

j=1 w q1 !

· · · kbmk+

x(_jm)n

j=1 w qm ! ,

(iii) ❊①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C > 0 t❛❧ q✉❡

Tb1+x(1)j , ..., bm+x(jm)

−T (b1, ..., bm)

∞ j= p ✭✶✳✹✮

≤C _kb1k+

x(1)_j ∞

j=1 w q1 !

· · · kbmk+

x(_jm)∞

j=1 w qm ! ,

♣❛r❛ t♦❞♦(b1, ..., bm)∈E1× · · · ×Em ❡

x(_jr)∞

j=1 ∈ℓ

w

qr, r= 1, ..., m.

❆❞❡♠❛✐s✱ ❛ ♠❡♥♦r ❞❛s ❝♦♥st❛♥t❡s C q✉❡ s❛t✐s❢❛③❡♠ ✭✶✳✹✮ ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠

Lev

as(E1, ..., Em;F) ❛ q✉❛❧ s❡rá ❞❡♥♦t❛❞❛ ♣♦r kTk_ev✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s

Las,(p;q1,...,qm)(E1, ..., Em;F),k·kas

❡Lev

as,(p;q1,...,qm)(E1, ..., Em;F),k·kev

sã♦ ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤✳

❙✉❣❡r✐♠♦s ♦ tr❛❜❛❧❤♦ ❬✸✾✱ ❙✉❜s❡çã♦ 5.4✱❪ ♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s s♦❜r❡s ❡st❡s

♦♣❡r❛❞♦r❡s✳

✶✳✸ ❖♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s

❊♠ ✷✵✵✸✱ Pér❡③✲●❛r❝í❛ ✭❬✹✹❪✮ ❡ ▼✳❈✳ ▼❛t♦s ✭❬✷✾❪✮✱ ❞❡ ❢♦r♠❛ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ♥♦✈❛ ❛❜♦r❞❛❣❡♠ ♠✉❧t✐❧✐♥❡❛r ♣❛r❛ ♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✿ ♦s ♦♣❡r❛❞♦r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✱ ❡♠❜♦r❛ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ▼✳❈✳ ▼❛t♦s r❡✲ ♠♦♥t❡ ❛ 1993 ✭✈❡❥❛ ❬✷✽❪✮✳ ❊st❛ ♥♦✈❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ♣❛ss♦✉ ❛ s❡r ✐♥t❡♥s❛♠❡♥t❡

❡①♣❧♦r❛❞❛ ❡✱ ❞❡s❞❡ ❡♥tã♦✱ é ❝♦♥s✐❞❡r❛❞❛ ❝♦♠♦ ✉♠❛ ❞❛s ❡①t❡♥sõ❡s ♠❛✐s ✜é✐s ❛♦ ❡s♣ír✐t♦ ❧✐♥❡❛r✳

❉❡✜♥✐çã♦ ✶✳✸✳✶ ✭❖♣❡r❛❞♦r ♠ú❧t✐♣❧♦ s♦♠❛♥t❡✮ ❯♠ ♦♣❡r❛❞♦r ♠✉❧t✐❧✐♥❡❛r

T _{∈ L}(E1, ..., En;F) é ♠ú❧t✐♣❧♦ (q;p1, ..., pn)✲s♦♠❛♥t❡ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C ≥ 0

t❛❧ q✉❡

m

X

j1,...,jn=1

T x1_j1, ..., xn_jnq

!1/q

≤C n

Y

k=1

xk_jkm_j=1w

pk

✭✶✳✺✮

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ m_∈N _{❡ x}k jk

m

jk=1 ∈Ek✱ k = 1, ..., n.

❙❡ T _{∈ L}(E1, ..., En;F) é ♠ú❧t✐♣❧♦ (q;p1, ..., pn)✲s♦♠❛♥t❡✱ ❡s❝r❡✈❡♠♦s T _∈ Πn

q;p1,...,pn(E1, ..., En;F)✱ ♦♥❞❡ Π

n

q;p1,...,pn(E1, ..., En;F) ❞❡♥♦t❛ ❛ ❝❧❛ss❡ ❞❡ t❛✐s

♦♣❡r❛❞♦r❡s✳ ❖ í♥✜♠♦ ❞♦sC q✉❡ s❛t✐s❢❛③❡♠ (1.5)é r❡♣r❡s❡♥t❛❞♦ ♣♦rπq;p1,...,pn(T). ❙❡ p1 = ... = pn = p✱ ❞✐③❡♠♦s q✉❡ T é ♠ú❧t✐♣❧♦ (q;p)✲s♦♠❛♥t❡ ❡ ❡s❝r❡✈❡♠♦s T _∈Πn

q;p(E1, ..., En;F).◆♦ ❝❛s♦ ❞❡q=p✱ ❞✐③❡♠♦s q✉❡ ♦ ♦♣❡r❛❞♦r é ♠ú❧t✐♣❧♦p✲s♦♠❛♥t❡

❡ ❞❡♥♦t❛♠♦s T _∈ Πn

p(E1, ..., En;F). ❙❡ E1 = ... = En = E✱ ❡s❝r❡✈❡♠♦s s✐♠♣❧❡s♠❡♥t❡

Πn

q;p1,...,pn(

n_{E;F) ❡ s❡ F =K✱ ❡s❝r❡✈❡♠♦sΠ}n

q;p1,...,pn(E1, ..., En).

➱ ✐♠♣♦rt❛♥t❡ ♠❡♥❝✐♦♥❛r q✉❡✱ ❝♦♠♦ ♥♦ ❝❛s♦ ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ❛ t❡♦r✐❛ ❞♦s ♦♣❡r❛❞♦r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s só ❢❛③ s❡♥t✐❞♦ q✉❛♥❞♦q _≥pj✱ ♣❛r❛

t♦❞♦1_≤j _≤n❀ ♦✉ s❡❥❛✱ s❡q < pj ♣❛r❛ ❛❧❣✉♠1≤j ≤n❡T ∈Πnq;p1,...,pn(E1, ..., En;F)✱ ❡♥tã♦✱ T s❡rá ♦ ♦♣❡r❛❞♦r ♥✉❧♦✳

❆❧é♠ ❞✐ss♦✱ t❡♠✲s❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ t❛♠❜é♠ q✉❡πq;p1,...,pn ❞❡✜♥❡ ✉♠❛ ♥♦r♠❛ ❡♠

Πn

q;p1,...,pn(E1, ..., En;F)✳ ❖ ❡s♣❛ç♦ Π

n

q;p1,...,pn(E1, ..., En;F), π(q;p1,...,pn)(.)

é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❡♥tr❡ ❛s ♥♦r♠❛s

kT_{k ≤}π(q;p1,...,pn)(T).

❈♦♠♦ ✈✐♠♦s ♥♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s✱ ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ s❡q✉ê♥❝✐❛s t❛♥t♦ ✜♥✐t❛s ❝♦♠♦ ✐♥✜♥✐t❛s❀ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ♥♦s ♠♦str❛ q✉❡ ❛❝♦♥t❡❝❡ ♦ ♠❡s♠♦ ❝♦♠ ♦s ♦♣❡r❛❞♦r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✿

Pr♦♣♦s✐çã♦ ✶✳✸✳✷ ❙❡❥❛♠ 1_≤p1, ..., pn≤q <∞ ❡ T ∈ L(E1, ..., En;F)✳ ❆s s❡❣✉✐♥t❡s

❛✜r♠❛çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✭✐✮ T é ♠ú❧t✐♣❧♦ (q;p1, ..., pn)✲s♦♠❛♥t❡✳

✭✐✐✮ P❛r❛ ❝❛❞❛ ❡s❝♦❧❤❛ ❞❡ s❡q✉ê♥❝✐❛s xj_ij∞

ij=1 ∈

ℓw

pj(Ej), t❡♠♦s

T x1i1, ..., x

n in

∞

i1,...,in=1 ∈ℓq(

Nn_;F)✳

◆❡st❡ ❝❛s♦✱ ♦ ♦♣❡r❛❞♦r ♠✉❧t✐❧✐♥❡❛r ❛ss♦❝✐❛❞♦

∧

T :ℓw_p1(E1)× · · · ×ℓwpn(En)→ℓq(N

n_;F)

❞❛❞♦ ♣♦r

∧

T x1_i1∞_i

1=1, ..., x

n in

∞

in=1

= T x1_i1, ..., xn_in∞_i

1,...,in=1

é ❝♦♥tí♥✉♦ ❡

∧

T

=π(q;p1,...,pn)(T).

❱❡❥❛ q✉❡ ♣❛r❛ ❝❛❞❛T _∈Πn

q;p1,...,pn(E1, ..., En;F),q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ♦♣❡r❛❞♦r

♠✉❧t✐❧✐♥❡❛r ❝♦♥tí♥✉♦ ❛ss♦❝✐❛❞♦

∧

T :ℓw_p1(E1)× · · · ×ℓwpn(En)→ℓq(N

n_;F)

❞❛❞♦ ♣♦r

∧

T x1_i1∞_i

1=1, ..., x

n in

∞

in=1

= T x1_i1, ..., xn_in∞_i

1,...,in=1,

❡st❛♠♦s ✐♥❞✉③✐♥❞♦ ✉♠❛ ❛♣❧✐❝❛çã♦

∧

θ : Πn

q;p1,...,pn(E1, ..., En;F) → L

n _ℓw

p1(E1), ..., ℓ

w

pn(En) ;ℓq(N

n_;F),

T _→T∧

q✉❡ é ❧✐♥❡❛r ❡ ✐s♦♠étr✐❝❛✳

❯♠ ❞♦s tó♣✐❝♦s ♠❛✐s ✐♥t❡r❡ss❛♥t❡s ❞❡ ❡st✉❞♦ ♥❛ t❡♦r✐❛ ❞♦s ♦♣❡r❛❞♦r❡s ❛❜s♦❧✉t❛♠❡♥t❡ s♦♠❛♥t❡s é ♦ ❞♦s r❡s✉❧t❛❞♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ♥❡st❡ ❝❛s♦✱ q✉❛♥❞♦ ❛ ❝❧❛ss❡ ❞♦s ♦♣❡r❛❞♦r❡s ♠✉❧t✐❧✐♥❡❛r❡s ❝♦♥tí♥✉♦s ❡♠ ❝❡rt♦s ❡s♣❛ç♦s ❞❡ ❇❛♥❛❝❤ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ❝❧❛ss❡ ❞♦s ♦♣❡r❛❞♦r❡s ♠ú❧t✐♣❧♦ s♦♠❛♥t❡s✳

❈♦♠♦ ❡①❡♠♣❧♦✱ ❞❛r❡♠♦s ❞♦✐s t❡♦r❡♠❛s✳ ❖ ♣r✐♠❡✐r♦ é ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛✱ q✉❡ é ❛ ✈❡rsã♦ ♠✉❧t✐❧✐♥❡❛r ❞♦ ❢❛♠♦s♦ ❚❡♦r❡♠❛ ❞❡ ●r♦t❤❡♥❞✐❡❝❦✳ ❖ s❡❣✉♥❞♦ ❡♥✈♦❧✈❡ ♦ ❢❛♠♦s♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦t✐♣♦ ❞❡ ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

❚❡♦r❡♠❛ ✶✳✸✳✸ ✭Pér❡③✲●❛r❝í❛✮ ❙❡ 1_≤p_≤2✱ ❡♥tã♦

Πn_p(nℓ1;ℓ2) = L(nℓ1;ℓ2).

❚❡♦r❡♠❛ ✶✳✸✳✹ ✭Pér❡③✲●❛r❝í❛❀ ❙♦✉③❛✮ ❙❡ F é ✉♠ ❡s♣❛ç♦ ❝♦♠ ❝♦t✐♣♦ q✱ t♦❞♦

♦♣❡r❛❞♦r ♠✉❧t✐❧✐♥❡❛r T :E1×...×En→F é ♠ú❧t✐♣❧♦ (q; 1)✲s♦♠❛♥t❡✳ ❖✉ s❡❥❛✱

Πn_(q;1)(E1, ..., En;F) = L(E1, ..., En;F)

❖ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡ ❡♥❝♦♥tr❛r ✉♠❛ ♣r♦✈❛ ❞♦s t❡♦r❡♠❛s ❛❝✐♠❛ ❡♠ ❬✹✹✱ ❈♦r♦✲ ❧ár✐♦ 5.24✱❪ ❡ ❬✼✱ ❚❡♦r❡♠❛ 2.2✱❪ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ P❛r❛ ♠❛✐s r❡s✉❧t❛❞♦s ❞❡ ❝♦✐♥❝✐❞ê♥❝✐❛✱

r❡❝♦♠❡♥❞❛♠♦s ♦s tr❛❜❛❧❤♦s ❬✾❪✱ ❬✹✺❪ ❡ ❬✺✷❪✳

❈❛♣ít✉❧♦ ✷

❖ ❚❡♦r❡♠❛ ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡

❋r❡❞ér✐❝❦ ❇♦❤♥❡♥❜❧✉st ❡ ❊✐♥❛r ❍✐❧❧❡✱ ♥♦ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦ ❖♥ t❤❡ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❉✐r✐❝❤❧❡t s❡r✐❡s✱ ♣✉❜❧✐❝❛❞♦ ❡♠ 1931 ♥❛ ♣r❡st✐❣✐♦s❛ r❡✲

✈✐st❛ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ❝♦♥s❡❣✉❡♠ s♦❧✉❝✐♦♥❛r ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥✲ ❝✐❛ ❛❜s♦❧✉t❛ ❞❡ ❇♦❤r ❢♦r♠✉❧❛❞♦ ❡♠1913✳ ❊♠❜♦r❛ ❡ss❡ s❡❥❛ ♦ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❞❡ss❡

❛rt✐❣♦✱ ♥❡st❡ ❡❧❡s ❝♦♥s❡❣✉✐r❛♠ t❛♠❜é♠ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ❢❛♠♦s❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ▲✐tt❧❡✇♦♦❞4/3✱ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇♦❤♥❡♥❜❧✉st✲❍✐❧❧❡✱ ❛ q✉❛❧ s❡rá ♥♦ss♦

❢♦❝♦ ❞❡ ❡st✉❞♦ ♥❡st❡ ❝❛♣ít✉❧♦ ❡ ♥♦ ♣ró①✐♠♦✳

❆ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏✳ ❊✳ ▲✐tt❧❡✇♦♦❞ 4/3✱ ❞❡♠♦♥str❛❞❛ ❡♠ ❬✷✼❪✱ ❛✜r♠❛ q✉❡ ♣❛r❛

t♦❞❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r U : lN

∞×l∞N→K ❡ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ N✱ ❡①✐st❡ ✉♠❛

❝♦♥st❛♥t❡LK ≥1 q✉❡ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡

N

X

i,j=1

|U(ei, ej)|

4 3

!3 4

≤LKkUk

❇♦❤♥❡♥❜❧✉st ❡ ❍✐❧❧❡ ♣❡r❝❡❜❡r❛♠ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❡ r❡s✉❧t❛❞♦ ❡✱ é ❝❧❛r♦✱ ❞❛s té❝♥✐❝❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ❞❡♠♦♥strá✲❧♦✱ ❥á q✉❡ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡st❡ ❧❤❡s ♣❡r♠✐t✐r✐❛ ❛ss✐♠ s♦❧✉❝✐♦♥❛r ♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✱ ❛ s❛❜❡r✿ ◗✉❛❧ ❛ ❧❛r❣✉r❛ ♠á①✐♠❛ ❞❛ ❢❛✐①❛ ✈❡rt✐❝❛❧L ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦ ♥❛ q✉❛❧ ✉♠❛ sér✐❡ ❞❡ ❉✐r✐❝❤❧❡tX

n

an·n−s ❝♦♥✈❡r❣❡

✉♥✐❢♦r♠❡♠❡♥t❡ ♠❛s ♥ã♦ ❛❜s♦❧✉t❛♠❡♥t❡❄