Campos hipoelíticos no plano

Campos hipoelíticos no plano

Campos hipoelíticos no plano

Camilo Campana

Orientador: Prof. Dr. Adalberto Panobianco Bergamasco

Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação - ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências - Matemática . VERSÃO REVISADA

USP – São Carlos

Abril de 2013

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

C186c

Campana, Camilo

Campos hipoelíticos no plano / Camilo Campana; orientador Adalberto Panobianco Bergamasco. -- São Carlos, 2013.

155 p.

Dissertação (Mestrado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2013.

1. campos vetoriais complexos. 2. hipoeliticidade. 3. uniformização. 4.

❆❣r❛❞❡❝✐♠❡♥t♦s

➚ ❉❡✉s✳

❆♦ ♣r♦❣r❛♠❛ ❞❡ ♣ós ❣r❛❞✉❛çã♦ ❞♦ ❯❙P✲■❈▼❈ ✭■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛ ❡ ❈♦♠♣✉t❛çã♦✮✳ ❆♦s ♣r♦❢❡ss♦r❡s ❡ ❢✉♥❝✐♦♥ár✐♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝♦♥❢❡❝çã♦ ❞❡ss❛ ❞✐ss❡rt❛çã♦✳

❊♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❡ ❛♠✐❣♦✱ ❆❞❛❧❜❡rt♦ P❛♥♦❜✐❛♥❝♦ ❇❡r❣❛♠❛s❝♦✱ ♣❡❧❛ ❞✐s✲ ♣♦♥✐❜✐❧✐❞❛❞❡✱ ❣r❛♥❞❡ ❞❡❞✐❝❛çã♦ ❡ ♣❡❧♦s ❝♦♥❤❡❝✐♠❡♥t♦s tr❛♥s♠✐t✐❞♦s✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ ♠❡✉s ♣❛✐s ▲♦✉r❞❡s ❡ ❏✉❞ç♦♥✳ ▼❡✉s ❛✈ós✱ ❏♦✈❡❧✐♥❛ ❡ ❊st❡✈❛❧❞♦ ❡ ❛♦s ♠❡✉s ❜✐s❛✈ós ▼❛r✐❛ ❡ ❆♠ér✐❝♦✳ ❚❛♠❜é♠ ❛ ♠✐♥❤❛ ♣r✐♠❛✱ ❆♥❛ ❈❛r♦❧✐♥❛ ❈❛♠♣❛♥❛ q✉❡ ♠❡ ❛❥✉❞♦✉ ♥♦ ✐♥í❝✐♦ ❞❛ ❣r❛❞✉❛çã♦✳

❆♦s ♠❡✉s ❛♠✐❣♦s✳ ❊♠ ❡s♣❡❝✐❛❧ ♣❛r❛ ❆❧♦♠✐r✱ ❉✐♦♥❡✱ ❙t❡✈❡✱ ▲❛ís ❡ ❘♦❞r✐❣♦✳ ❚❛♠❜é♠ ❛♦ ❛♠✐❣♦ ❱✐♥í❝✐✉s✱ ♣❡❧❛ ❛❥✉❞❛ ♥♦ ✐♥í❝✐♦ ❞♦ ♠❡str❛❞♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯❋❊❙✱ ❆❧❛♥❝❛r❞❡❦ P❡r❡✐r❛ ❆r❛✉❥♦✱ ❆♥❛ ❈❧❛✉❞✐❛ ▲♦❝❛t❡❧❧✐✱ ❏♦sé ❆r♠í♥✐♦ ❋❡rr❡✐r❛✱ ❏ú❧✐❛ ❙❤❛❡t③❧❡ ❲r♦❜❡❧✱ ▼❛❣❞❛ ❙♦❛r❡s ❳❛✈✐❡r ❡ ❘✐❝❛r❞♦ ❙♦❛r❡s ▲❡✐t❡✳

✐✈

➚ t♦❞♦s ♦s ❝✐t❛❞♦s ❡ ❛s ♣❡ss♦❛s q✉❡ ♥ã♦ ♠❡♥❝✐♦♥❡✐✱ ♠❛s q✉❡ ✜③❡r❛♠ ♣❛rt❡ ❞❛ ♠✐♥❤❛ tr❛❥❡tór✐❛✱ ✜❝❛ r❡❣✐str❛❞♦ ♦ ♠❡✉ ❛❣r❛❞❡❝✐♠❡♥t♦✳

❘❡s✉♠♦

❙❡❥❛ L ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ♥ã♦ s✐♥❣✉❧❛r ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❛❜❡rt♦ ❞♦ ♣❧❛♥♦✳ ❚r❡✈❡s ♣r♦✈♦✉ q✉❡ s❡ Lé ❧♦❝❛❧♠❡♥t❡ r❡s♦❧ú✈❡❧ ❡♥tã♦Lé ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ P❛r❛ ❝❛♠✲ ♣♦s ♣❧❛♥❛r❡s ❤✐♣♦❡❧ít✐❝♦s✱ ✈❛❧❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❛❞✐❝✐♦♥❛❧✱ ❛ s❛❜❡r✱ t♦❞❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ✭r❡str✐t❛ ❛ ✉♠ ❛❜❡rt♦ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✮ é ✉♠❛ ❛♣❧✐❝❛çã♦ ✐♥❥❡t✐✈❛ ✭❡ ❛❜❡rt❛✮❀ ✐st♦✱ ♣♦r s✉❛ ✈❡③✱ ✐♠♣❧✐❝❛ q✉❡ t♦❞❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ ❤♦♠♦❣ê♥❡❛ Lu = 0 é ❧♦❝❛❧♠❡♥t❡ ❞❛

❆❜str❛❝t

▲❡t L ❜❡ ❛ ♥♦♥s✐♥❣✉❧❛r ❝♦♠♣❧❡① ✈❡❝t♦r ✜❡❧❞ ❞❡✜♥❡❞ ♦♥ ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ t❤❡ ♣❧❛♥❡✳ ❚r❡✈❡s ♣r♦✈❡❞ t❤❛t ✐❢ L ✐s ❧♦❝❛❧❧② s♦❧✈❛❜❧❡ t❤❡♥ L ✐s ❧♦❝❛❧❧② ✐♥t❡❣r❛❜❧❡✳ ❋♦r ❤②♣♦❡❧❧✐♣t✐❝ ♣❧❛♥❛r ✈❡❝t♦r ✜❡❧❞s ❛♥ ❛❞❞✐t✐♦♥❛❧ ♣r♦♣❡rt② ❤♦❧❞s✱ ♥❛♠❡❧②✱ ❡✈❡r② ✜rst ✐♥t❡❣r❛❧ ✭r❡str✐❝t❡❞ t♦ ❛ s✉✣❝✐❡♥t❧② s♠❛❧❧ ♦♣❡♥ s❡t✮ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ✭❛♥❞ ♦♣❡♥✮ ♠❛♣♣✐♥❣❀ t❤✐s✱ ♦♥ ✐ts t✉r♥✱ ✐♠♣❧✐❡s t❤❛t ❡❛❝❤ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤♦♠♦❣❡♥❡♦✉s ❡q✉❛t✐♦♥ Lu= 0 ✐s ❧♦❝❛❧❧② ♦❢ t❤❡ ❢♦r♠u=h◦Z✱

✇❤❡r❡h✐s ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞Z ✐s ❛ ✜rst ✐♥t❡❣r❛❧ ♦❢ t❤❡ ✈❡❝t♦r ✜❡❧❞✳ ❚❤❡ ❝❡♥tr❛❧ ♣r♦❜❧❡♠ ♦❢ ✐♥t❡r❡st ✐♥ t❤✐s ✇♦r❦ ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❣❧♦❜❛❧ q✉❡st✐♦♥✱ t❤❛t ✐s✱ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❣❧♦❜❛❧✱ ✐♥❥❡❝t✐✈❡ ✜rst ✐♥t❡❣r❛❧s ❛♥❞ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❣❧♦❜❛❧ s♦❧✉t✐♦♥s ❛s ❝♦♠♣♦s✐t✐♦♥s ♦❢ t❤❡ ✜rst ✐♥t❡❣r❛❧ ✇✐t❤ ❛ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ Pré✲r❡q✉✐s✐t♦s ✺

✶✳✶ ◆♦t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✷✳✶ ❆ t♦♣♦❧♦❣✐❛ ❞❡ C∞

c (Ω) ❡ ♦ ❡s♣❛ç♦ D′(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✷✳✷ ▼✉❞❛♥ç❛ ❞❡ ❱❛r✐á✈❡✐s ❡♠ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸ ❖♣❡r❛❞♦r❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ ❆♥á❧✐s❡ ❈♦♠♣❧❡①❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✹✳✶ ❖ ❖♣❡r❛❞♦r ❞❡ ❈❛✉❝❤②✲❘✐❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹✳✷ ❆❧❣✉♥s t❡♦r❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹✳✸ ❈♦♥✈❡r❣ê♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ❢✉♥çõ❡s ❤♦❧♦♠♦r❢❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✶✳✺ ❈❛♠♣♦s ❱❡t♦r✐❛✐s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✺✳✶ ❈❛♠♣♦s ❱❡t♦r✐❛✐s ❈♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✳✺✳✷ ❋♦r♠❛s ❉✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✺✳✸ ▼✉❞❛♥ç❛ ❞❡ ❱❛r✐á✈❡✐s ❡♠ ❈❛♠♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷ ❙✉♣❡r❢í❝✐❡s ❞❡ ❘✐❡♠❛♥♥ ✹✼

✷✳✶ ❉❡✜♥✐çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✷ ❯♥✐❢♦r♠✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✷✳✷✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ ❯♥✐❢♦r♠✐③❛çã♦ ♣❛r❛ ❙✉♣❡r❢í❝✐❡s ❞❡ ❘✐❡♠❛♥♥ ❙✐♠♣❧❡s✲ ♠❡♥t❡ ❈♦♥❡①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✷✳✷✳✷ ❯♥✐❢♦r♠✐③❛çã♦ ♣❛r❛ ❙✉♣❡r❢í❝✐❡s ❞❡ ❘✐❡♠❛♥♥ ♥ã♦ ❙✐♠♣❧❡s♠❡♥t❡ ❈♦✲ ♥❡①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✸ ❆❧❣✉♥s ❝♦♠❡♥tár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✸ ❘❡s♦❧✉❜✐❧✐❞❛❞❡ ✺✼

✸✳✶ ❘❡s♦❧✉❜✐❧✐❞❛❞❡ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✸✳✷ ❆ ❈♦♥❞✐çã♦ ✭P✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✸ ❘❡s♦❧✉❜✐❧✐❞❛❞❡ ♥♦ ❝❛s♦ ❤✐♣♦❡❧ít✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✸✳✹ ❆ ❈♦♥❞✐çã♦ ✭◗✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✺ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ▲♦❝❛❧ ♥♦ P❧❛♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻

✹ ❈❛s♦ ❊❧ít✐❝♦ ✼✾

✹✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✷ ❊❧✐t✐❝✐❞❛❞❡ ❡ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾

✹✳✷✳✶ ❘❡❞✉çã♦ ❞❡ ✉♠ ❈❛♠♣♦ ❊❧ít✐❝♦ ❛ ✉♠ ♠ú❧t✐♣❧♦ ❞♦ ❖♣❡r❛❞♦r ❞❡ ❈❛✉❝❤②✲❘✐❡♠❛♥♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹✳✸ ❊str✉t✉r❛ ❉✐❢❡r❡♥❝✐á✈❡❧ ❈♦♠♣❧❡①❛ ✐♥❞✉③✐❞❛ ♣♦r ✉♠ ❈❛♠♣♦ ❊❧ít✐❝♦ ✳ ✳ ✳ ✳ ✳ ✽✾ ✹✳✹ ❙♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷

✺ ❈❛s♦ ❍✐♣♦❡❧ít✐❝♦ ✾✼

✺✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✺✳✷ ❍✐♣♦❡❧✐t✐❝✐❞❛❞❡ ❡ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✺✳✸ ❚❡♦r❡♠❛ ❞❡ ❆♣r♦①✐♠❛çã♦ ❡ ❊str✉t✉r❛ ❞❛s ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸ ✺✳✸✳✶ ❚❡♦r❡♠❛ ❞❡ ❆♣r♦①✐♠❛çã♦ ❞❡ ❇❛♦✉❡♥❞✐✲❚r❡✈❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸ ✺✳✸✳✷ ❊str✉t✉r❛ ▲♦❝❛❧ ❞❛s ❙♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✵ ✺✳✹ ❊str✉t✉r❛ ❉✐❢❡r❡♥❝✐á✈❡❧ ❈♦♠♣❧❡①❛ ✐♥❞✉③✐❞❛ ♣♦r ✉♠ ❈❛♠♣♦ ❍✐♣♦❡❧ít✐❝♦ ✳ ✳ ✶✸✻ ✺✳✺ ❙♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✾

❙❯▼➪❘■❖ ①✐

■♥tr♦❞✉çã♦

◆❛ ❞é❝❛❞❛ ❞❡1960 ♦ ❡st✉❞♦ ❞❡ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s

❢♦✐ ❛❧❛✈❛♥❝❛❞♦ ♣❡❧♦s tr❛❜❛❧❤♦s ♣✉❜❧✐❝❛❞♦s ♣♦r ◆✐r❡♥❜❡r❣ ❡ ❚r❡✈❡s✳ ❊♠ ✉♠ ❞❡ss❡s tr❛❜❛❧❤♦s ❢♦✐ ❢♦r♠✉❧❛❞❛ ❛ ❝♦♥❞✐çã♦ ✭P✮✱ ❛ q✉❛❧ ✉♠ ❜♦♠ t❡♠♣♦ ❞❡♣♦✐s s❡ ♠♦str♦✉ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ❞❡ t✐♣♦ ♣r✐♥❝✐♣❛❧ ✭♣♦r ❡①❡♠♣❧♦✱ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❝♦♠♣❧❡①♦s ♥ã♦ s✐♥❣✉❧❛r❡s✮✳ ❆ss✐♠✱ s✉r❣✐r❛♠ ♥♦✈♦s ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❝♦♠♣❧❡①♦s❀ ✉♠ ❞❡❧❡s ❝♦♥s✐st❡ ❡♠ ❞❡❝✐❞✐r s❡ ✉♠ ❝❛♠♣♦ é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱✐✳❡✳✱ s❡ ♣❛r❛ ❝❛❞❛ ♣♦♥t♦ p❞♦ ❞♦♠í♥✐♦ ❞❡ ✉♠ ❝❛♠♣♦ L✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ u ❞❡ L ✭✐✳❡✳✱ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❞❡✜♥✐❞❛ ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p✱ t❛❧ q✉❡ Lu = 0 ♥❡st❛ ✈✐③✐♥❤❛♥ç❛✮✱ q✉❡ s❡❥❛ C∞ _{❡ s❛t✐s❢❛ç❛ du 6= 0✳ ❆ ❢✉♥çã♦ u é ❝❤❛♠❛❞❛} ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❞♦ ❝❛♠♣♦ L✳ ❈♦♠♦ ♦ ♣ró♣r✐♦ ♥♦♠❡ ❞✐③✱ ❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❧♦❝❛❧ é ✉♠ ❝♦♥❝❡✐t♦ ❧♦❝❛❧✳ ◆♦ ❡♥t❛♥t♦✱ s✉r❣❡ ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✱ ❞❡ ❝❛rát❡r ❣❧♦❜❛❧✱ s♦❜r❡ ❡st❡ ♣r♦❜❧❡♠❛✿ s❡ L é ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ♥ã♦ s✐♥❣✉❧❛r ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❛❜❡rt♦ U ❞❡ Rn✱

❡①✐st❡ ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❣❧♦❜❛❧ Z✱ ✐✳❡✳✱ ❡①✐st❡ Z ❢✉♥çã♦ C∞ _{❞❡✜♥✐❞❛ ❡♠ t♦❞♦ ♦ ❛❜❡rt♦} U t❛❧ q✉❡

LZ = 0✱ dZ 6= 0 ❡♠ U❄

❊♠ ❬✸❪ ❡st❛ ♣❡r❣✉♥t❛ ❢♦✐ r❡s♣♦♥❞✐❞❛ ❡♠ ✉♠❛ ❝❧❛ss❡ ❞❡ ❝❛♠♣♦s✱ ♦s ❝❛♠♣♦s ❤✐♣♦❡❧ít✐❝♦s ❞❡✜♥✐❞♦s ❡♠ ❛❜❡rt♦s s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①♦s ❞♦ ♣❧❛♥♦✱ ❝♦♠ ♦ ❛❞✐❝✐♦♥❛❧ ❞❡ q✉❡ ❛ s♦❧✉çã♦ ♦❜t✐❞❛ é ✐♥❥❡t♦r❛✳

✷

✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ♥ã♦ s✐♥❣✉❧❛r ❤✐♣♦❡❧ít✐❝♦ ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❛❜❡rt♦ ❝♦♥❡①♦ Ω _⊂ R2✱ ♦ s❡✲

❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

(_∗) LZ = 0✱ dZ ₆= 0 ❡Z ✐♥❥❡t✐✈❛ ❡♠Ω

t❡♠ s♦❧✉çã♦ ❣❧♦❜❛❧ q✉❛♥❞♦ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❛❜❡rt♦ Ω é tr✐✈✐❛❧✱ ✐✳❡✳✱ Ω s✐♠♣❧❡s✲

♠❡♥t❡ ❝♦♥❡①♦✱ ❡ q✉❛♥❞♦ ♦ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ Ω é Z✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦_{π1(Ω) ={0}} ♦✉

q✉❛♥❞♦ π1(Ω) =Z✳

❯♠❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ♦ ♠ét♦❞♦ ✉s❛❞♦ ♣❛r❛ ♦❜t❡r ❛ s♦❧✉çã♦ é ✐♥s♣✐r❛❞♦ ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❝❛♠♣♦ Lé ❡❧ít✐❝♦✳ ❯♠ ❢❛t♦ ✭q✉❡ ❞✐s❝✉t✐r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✹✮ s♦❜r❡ ❝❛♠♣♦s ❡❧ít✐❝♦s é q✉❡ ❧♦❝❛❧♠❡♥t❡ ❡①✐st❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❛ q✉❛❧ ❣r♦ss♦ ♠♦❞♦ é ✉♠❛ s♦❧✉çã♦ ❞❡ L❡ tr❛♥s❢♦r♠❛ L ❡♠ ✉♠ ♠ú❧t✐♣❧♦ ❞♦ ♦♣❡r❛❞♦r ❞❡ ❈❛✉❝❤②✲❘✐❡♠❛♥♥✳ ❚♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ L ♥❡ss❛s ♥♦✈❛s ❝♦♦r❞❡♥❛❞❛s sã♦ ❢✉♥çõ❡s ❤♦❧♦♠♦r❢❛s✳ ❆ss✐♠✱ L✐♥❞✉③ ✉♠❛ ❡str✉t✉r❛ ❞✐❢❡r❡♥✲ ❝✐❛❧ ❝♦♠♣❧❡①❛ ❡♠ Ω✭❝❛♣ít✉❧♦ ✷✮ q✉❡ t♦r♥❛Ω✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ ❘✐❡♠❛♥♥✳ ◆♦ ❝❛s♦ ❡♠ q✉❡

π1(Ω) ={0} ♦✉π1(Ω) =Z✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r t❡♦r❡♠❛s ❞❡ ✉♥✐❢♦r♠✐③❛çã♦ ❞❡ s✉♣❡r❢í❝✐❡s ❞❡

❘✐❡♠❛♥♥ ♣❛r❛ ✏❝♦❧❛r✑ ❛s s♦❧✉çõ❡s ❧♦❝❛✐s ❞❡ L✱ ♦❜t❡♥❞♦ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ q✉❡ ❤❡r❞❛ ❛s ❜♦❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s s♦❧✉çõ❡s ❧♦❝❛✐s✳

P♦rt❛♥t♦✱ ❛♥t❡s ❞❡ tr❛t❛r ❞♦ ❝❛s♦ ❤✐♣♦❡❧ít✐❝♦✱ ✈❛♠♦s tr❛t❛r ❞♦ ❝❛s♦ ❡❧ít✐❝♦✳ ❈♦♠♦ ❡❧✐t✐❝✐❞❛❞❡ é ✉♠❛ ❤✐♣ót❡s❡ ♠❛✐s ❢♦rt❡ s♦❜r❡ ✉♠ ❝❛♠♣♦✱ é ♥❛t✉r❛❧ ♣❡♥s❛r q✉❡ t❡♥❤❛♠♦s ✉♠❛ t❡s❡ ♠❛✐s ❢♦rt❡ t❛♠❜é♠✱ ✐✳❡✳✱ ♥♦ ❝❛♣ít✉❧♦ ✹ q✉❡ tr❛t❛ ❞♦ ❝❛s♦ ❡❧ít✐❝♦✱ q✉❛♥❞♦L é ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❡❧ít✐❝♦ ❞❡✜♥✐❞♦ ❡♠ ✉♠ ❛❜❡rt♦ ❝♦♥❡①♦ Ω _⊂ R2✱ ✈❛♠♦s ♠♦str❛r

q✉❡ ♦ ♣r♦❜❧❡♠❛

(_∗′_{) LZ = 0✱ grad(ℜ{Z})✱ grad(ℑ{Z}) ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡Z ✐♥❥❡t✐✈❛ ❡♠Ω}

t❡♠ s♦❧✉çã♦ ❣❧♦❜❛❧ ♥♦ ❝❛s♦ ❡♠ q✉❡ π1(Ω) = {0} ♦✉ π1(Ω) = Z✳ ❚❛♠❜é♠ s❡rá ❢❡✐t♦ ✉♠

■♥tr♦❞✉çã♦ ✸

❈❛♣ít✉❧♦

1

Pré✲r❡q✉✐s✐t♦s

✶✳✶ ◆♦t❛çã♦

❙❡❥❛Z₊ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s _{{0,1,2,3, ...}}❡ s❡❥❛ Zn

+ ♦ ❝♦♥✲

❥✉♥t♦ ❞❛s n✲✉♣❧❛s ❞❡ Z₊✳ ❯♠ ❡❧❡♠❡♥t♦ ❞❡ Zn

+ é ❝❤❛♠❛❞♦ ❞❡ ♠✉❧t✐✲í♥❞✐❝❡ ❡ ❞❡♥♦t❛❞♦✱ ❡♠

❣❡r❛❧✱ ♣❡❧❛s ❧❡tr❛s ❣r❡❣❛s ♠✐♥ús❝✉❧❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ α= (α1, ..., αn)✱ β = (β1, ..., βn)✱

❡t❝✳ ❊♠ Zn

+ ❞❡✜♥✐♠♦s ❛ s♦♠❛ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ❞❡ Z+✿

α+β = (α1+β1, ..., αn+βn)❡ kα= (kα1, ...kαn)✳

❉❡♥♦t❛♠♦s ♣♦r |α| ❛ s♦♠❛ ❞❛s ❝♦♦r❞❡♥❛❞❛s ❞❡α✿

|α_|=

n

X

j=1

αj✱ ♣❛r❛ α∈Zn+

❡ ♣♦r α! ♦ ♣r♦❞✉t♦ ❞♦s ❢❛t♦r✐❛✐s ❞❛s ❝♦♦r❞❡♥❛❞❛s✿

α! =

n

Y

j=1

αj!✱ ♣❛r❛ α∈Zn+✳

❚❛♠❜é♠ ❞❡✜♥✐♠♦s ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ✐st♦ é✱ s❡ α✱ β ∈Zn

+✱

✻ Pré✲r❡q✉✐s✐t♦s

❉❡✜♥✐♠♦s

α β

=

n

Y

j=1

αj

βj

=

n

Y

j=1

αj!

(αj −βj)!βj

= α! (α−β)!β!✱

s❡ α_≥β ❡ α_β= 0 s❡α < β✳

❉❛❞♦x= (x1, ..., xn)∈Rn✱ ❞❡♥♦t❛♠♦s

xα ₌ n

Y

j=1

xjαj✱ ♣❛r❛ α∈Zn+✳

❙❡ 1≤j ≤n✱ ❞❡✜♥✐♠♦s

Dxj .

= 1

i ∂ ∂xj✱

q✉❡ ♣♦❞❡ s❡r ❞❡♥♦t❛❞♦ ♣♦r Dj✱ ♦♥❞❡

∂

∂xj ✐♥❞✐❝❛ ❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❡♠ r❡❧❛çã♦ ❛

xj ❡♠

Rn✱ q✉❡ ♣♦❞❡ s❡r ❞❡♥♦t❛❞❛ ♣♦r_∂j✳

P❛r❛ α_∈Zn✱ ❞❡✜♥✐♠♦s

Dxα = n

Y

j=1

Dxj

αj✳

❖❜s✳ ✶✳✶✳✶✳ ◗✉❛♥❞♦ α= 0✱ ❡♥tã♦ ∂α_{f =f✳}

❙❡❥❛ Ω _⊂ Rn ✉♠ ❛❜❡rt♦ ❡ s❡❥❛ _{k ∈} Z₊✳ ❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❞❡ ❝❧❛ss❡ _Ck ❡♠ _Ω✱

✐♥❞✐❝❛❞♦ ♣♦r Ck_{(Ω)✱ ❝♦♥s✐st❡ ❞❡ t♦❞❛s ❛s f✱ ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ❡♠ Ω ❛ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s}

❝♦♠ ❞❡r✐✈❛❞❛s ❝♦♥tí♥✉❛s ❛té ♦r❞❡♠ k✱ ✐st♦ é✱ f _∈ Ck_{(Ω) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ∂}α_{f é ❝♦♥✲}

tí♥✉❛ ♣❛r❛ t♦❞♦ α ∈ Zn

+ ❝♦♠ |α| ≤ k✳ ❆ss✐♠ C0(Ω) ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❛

✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❡♠ Ωq✉❡ sã♦ ❝♦♥tí♥✉❛s✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ C∞_{(Ω)✱ ❝♦♥s✐st❡ ❞❛s ❢✉♥çõ❡s} ❞❡✜♥✐❞❛s ❡♠ Ω ❛ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❝♦♠ ❞❡r✐✈❛❞❛s ❝♦♥tí♥✉❛s ❞❡ t♦❞❛s ❛s ♦r❞❡♥s✳ ▼❛✐s

❡①♣❧✐❝✐t❛♠❡♥t❡✱

C∞_{(Ω) =} \

k∈Z+

✶✳✶ ◆♦t❛çã♦ ✼

❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❖ s✉♣♦rt❡ ❞❡ ✉♠❛ ❢✉♥çã♦ f _∈C0_{(Ω)✱ ✐♥❞✐❝❛❞♦ ♣♦r supp(f)✱ é ♦ ❢❡❝❤♦}

❡♠ Ω ❞♦ ❝♦♥❥✉♥t♦ {x∈Ω ; f(x)6= 0}✳

❖❜s✳ ✶✳✶✳✸✳ ❖ s✉♣♦rt❡ ❞❡ f é ♦ ❝♦♠♣❧❡♠❡♥t❛r ❡♠ Ω ❞♦ ♠❛✐♦r ❛❜❡rt♦ ♦♥❞❡ f é ♥✉❧❛✳

■♥❞✐❝❛♠♦s ♣♦r Ck

c(Ω)✱ ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ Ck(Ω) ❞❛s ❢✉♥çõ❡s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❡

♣♦r C∞

c (Ω) ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ C∞(Ω) ❞❛s ❢✉♥çõ❡s ❞❡ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳

C∞

c (Ω) =

\

k∈Z+

C_ck(Ω)✳

❖❜s✳ ✶✳✶✳✹✳ Ck_{(Ω)✱ C}∞_{(Ω)✱ C}k

c(Ω) ❡ Cc∞(Ω) sã♦ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s s♦❜r❡ C✱ ❝♦♠ ❛s

♦♣❡r❛çõ❡s ✉s✉❛✐s ❞❡ ❢✉♥çõ❡s✳

❖ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ❡♠Ω✱ ✐♥❞✐❝❛❞♦ ♣♦rL1

loc(Ω)✱ é ♦ ❝♦♥❥✉♥t♦

❞❛s ❢✉♥çõ❡s f ▲❡❜❡s❣✉❡ ♠❡♥s✉rá✈❡✐s ❡♠Ω t❛✐s q✉❡

Z

K|

f(x)_|dx <_∞✱

♣❛r❛ t♦❞♦ K ⊂Ω ❝♦♠♣❛❝t♦✳

❖❜s✳ ✶✳✶✳✺✳ ❖ ❝♦♥❥✉♥t♦ L1

loc(Ω) é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ C✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥✲

❝✐❛ (fj)j∈N ❝♦♥✈❡r❣❡ ❡♠ L1_loc(Ω)✱ q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f ∈L1_loc(Ω)✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦

K _⊂Ω ❝♦♠♣❛❝t♦

Z

K|

fj(x)−f(x)|dx−→0✱ q✉❛♥❞♦ j −→ ∞✳

❖ ❡s♣❛ç♦ L1

loc(Ω) é ❝♦♠♣❧❡t♦✱ ✐✳❡✳✱ s❡ (fj)j∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠ L1_loc(Ω)✱

❡♥tã♦ ❡①✐st❡ f ∈L1

loc(Ω) t❛❧ q✉❡ fj −→f ❡♠ L1loc(Ω)✳

❈♦♠ ❛ ♥♦t❛çã♦ ♠✉❧t✐✲í♥❞✐❝❡✱ ❛ ❢♦r♠✉❧❛ ❞❡ ❚❛②❧♦r ❝♦♠ r❡st♦ ✐♥t❡❣r❛❧ ✜❝❛ ❞❛ ❢♦r♠❛✿

f(x+y) = X

|α|<k

∂αf(x)y

α

α! +k

Z 1

0

(1−t)k−1 X

|α|=k

∂αf(x+ty)y

α

α!dt

✽ Pré✲r❡q✉✐s✐t♦s

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s

❙❡❥❛ ❊ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ C✳ ❯♠❛ t♦♣♦❧♦❣✐❛ T ❡♠ ❊ é ❞✐t❛ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛

❡str✉t✉r❛ ✈❡t♦r✐❛❧ s❡ ❛s ♦♣❡r❛çõ❡s

❊×❊ −→ ❊ (x, y) _7−→ x+y ❡

C_×❊ _−→ ❊

(λ, x) _7−→ λx

sã♦ ❝♦♥tí♥✉❛s✳ ❖ ❡s♣❛ç♦ ❊ ♣r♦✈✐❞♦ ❞❡ ✉♠❛ t♦♣♦❧♦❣✐❛ q✉❡ ♣r❡s❡r✈❛ ❛ ❡str✉t✉r❛ ✈❡t♦r✐❛❧ é ❝❤❛♠❛❞♦ ❞❡ ❊s♣❛ç♦ ❱❡t♦r✐❛❧ ❚♦♣♦❧ó❣✐❝♦✱ ❡ é ✐♥❞✐❝❛❞♦ ♣❡❧❛ ❛❜r❡✈✐❛çã♦ ❊❱❚✳

❙❡❣✉❡ q✉❡ s❡ ❊ é ✉♠ ❊❱❚✱ ♣❛r❛ ❝❛❞❛ a_∈❊ ❡ λ₆= 0 ✜①❛❞♦s✱ ❛s ❛♣❧✐❝❛çõ❡s

x7→a+x e x 7→λx ✭✶✳✷✳✶✮

sã♦ ❤♦♠❡♦♠♦r✜s♠♦s✳ ❙❡❥❛ ❊ ✉♠ ❊❱❚✳ P❛r❛ ❡s♣❡❝✐✜❝❛r s✉❛ t♦♣♦❧♦❣✐❛✱ ❜❛st❛ ❞❛r ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ 0 ✭❡❧❡♠❡♥t♦ ♥✉❧♦ ❞♦ ❡s♣❛ç♦ ❊✮✳ ❊st❛ é ✉♠❛ ❢❛♠í❧✐❛B ❞❡ ✈✐③✐♥❤❛♥ç❛s

❞❡ 0 t❛❧ q✉❡ t♦❞❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ 0 ❝♦♥té♠ ❛❧❣✉♠ ❡❧❡♠❡♥t♦ ❞❡ B✳ ❆ss✐♠ ✉♠ ❝♦♥❥✉♥t♦

G⊂❊ é ❛❜❡rt♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

♣❛r❛ ❝❛❞❛ x∈G✱ ∃U ∈B t❛❧ q✉❡ _{x+U ⊂}G✳

❯♠ ❛❜❡rt♦ U ❞❡ ❊ é✿

• ❈♦♥✈❡①♦✿ ❙❡ x✱ y∈U ❡ 0≤t≤1 =⇒xt+ (1−t)y∈U✳

• ❊q✉✐❧✐❜r❛❞♦✿ ❙❡ x∈U✱|c|<1❝♦♠ c∈C_=⇒cx∈U✳

• ❆❜s♦r✈❡♥t❡✿ ∀x_∈❊✱ _∃t=tx >0 t❛❧ q✉❡ x∈tU ={ty ; y∈U}✳

❯♠ ❊❱❚ é ❧♦❝❛❧♠❡♥t❡ ❝♦♥✈❡①♦ s❡ é ❍❛✉s❞♦r✛ ❡ ❡①✐st❡ ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞❡ 0

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✾

❊①✐st❡ ✉♠ ♠♦❞♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❞❡ ❝♦♥str✉✐r ✉♠ ❊▲❈✳ ❱❛♠♦s ❢❛❧❛r ✉♠ ♣♦✉❝♦ ❞✐ss♦ ❛❣♦r❛✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❯♠❛ s❡♠✐♥♦r♠❛ ❡♠ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❊ é ✉♠❛ ❢✉♥çã♦

p:❊−→R

t❛❧ q✉❡✿

✶✳ p(cx) =|c|p(x)✱ c∈C✱ _x∈❊❀

✷✳ p(x+y)≤p(x) +p(y)✱ x✱ y∈❊✳

❖❜s✳ ✶✳✷✳✷✳ ❙❡❣✉❡ ❞❡ 1 q✉❡ p(0) = 0✳ ❉❡ 2 t❡♠♦s

0 =p(x₋x)_≤p(x) +p(x) = 2p(x)✱ _∀x_∈❊✳

❖✉ s❡❥❛✱ p(x)_≥0✱ _∀x_∈❊✳

❯♠❛ ❢❛♠í❧✐❛P ❞❡ s❡♠✐♥♦r♠❛s ❡♠ ❊ é ❞✐t❛ s❡♣❛r❛♥t❡ s❡ ♣❛r❛ t♦❞♦ _x∈❊✱ ❝♦♠_{x6= 0}✱

❡①✐st❡ p ∈ P t❛❧ q✉❡ _{p(x) 6= 0}✳ ❉❛❞❛ ✉♠❛ t❛❧ ❢❛♠í❧✐❛ P✱ s❡❥❛ B ❛ ❝♦❧❡çã♦ ❞❡ t♦❞♦s ♦s

❝♦♥❥✉♥t♦s ❞❛ ❢♦r♠❛

{x_∈❊ ; p(x)< ǫ_}✱ ❝♦♠ ǫ >0✱p_∈P

❡ ❞❛s ✐♥t❡rs❡çõ❡s ✜♥✐t❛s ❞❡ t❛✐s ❝♦♥❥✉♥t♦s✳ ❊♥tã♦ ♣r♦✈❛✲s❡ q✉❡Bé ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s

❞❡ 0 ❡♠ ✉♠❛ t♦♣♦❧♦❣✐❛ q✉❡ t♦r♥❛ ❊ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t♦♣♦❧ó❣✐❝♦ ❧♦❝❛❧♠❡♥t❡ ❝♦♥✈❡①♦✳

❊st❛ t♦♣♦❧♦❣✐❛ é ❝❤❛♠❛❞❛✿ ❛ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣❡❧❛s s❡♠✐♥♦r♠❛s ❡♠ P✳

❖❜s✳ ✶✳✷✳✸✳ ❙❡❥❛ ❊ ✉♠ ❊▲❈✳ ❙❡ U é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ 0 q✉❡ é ❝♦♥✈❡①❛✱ ❡q✉✐❧✐❜r❛❞❛ ❡

❛❜s♦r✈❡♥t❡✱ ❡♥tã♦ ♣r♦✈❛✲s❡

pU(x) = inf{t >0 ; x∈tU}✱

✶✵ Pré✲r❡q✉✐s✐t♦s

❙❡❥❛ ❛❣♦r❛ P ✉♠❛ ❢❛♠í❧✐❛ ❡♥✉♠❡rá✈❡❧ ❞❡ s❡♠✐♥♦r♠❛s✱ ❞✐❣❛♠♦s P _{= {p1, p2,· · · }}✳

P❛r❛ q✉❡ s❡❥❛ s❡♣❛r❛♥t❡ é ♣r❡❝✐s♦ q✉❡ ✈❛❧❤❛✿

x_∈❊, pk(x) = 0, ∀k ∈N=⇒x= 0. ✭✶✳✷✳✷✮

❆ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣❡❧❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐♥♦r♠❛s é ♠❡tr✐③❛✈❡❧✳ P♦r ❡①❡♠♣❧♦✿

d(x, y)=.

∞

X

k=1

1 2k

pk(x−y)

1 +pk(x−y)

, x, y∈❊ ✭✶✳✷✳✸✮

é ✉♠❛ ♠étr✐❝❛ q✉❡ ❞❡✜♥❡ ❛ ♠❡s♠❛ t♦♣♦❧♦❣✐❛ q✉❡ ❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐♥♦r♠❛sP ❡ é ✐♥✈❛r✐❛♥t❡

♣♦r tr❛♥s❧❛çõ❡s✳

❖❜s✳ ✶✳✷✳✹✳ ◆♦t❡ q✉❡ ✭✶✳✷✳✷✮ é ❝r✉❝✐❛❧ ♣❛r❛ ♠♦str❛r q✉❡ d é ✉♠❛ ♠étr✐❝❛✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♣♦❞❡✲s❡ ♣r♦✈❛r q✉❡ ❛ t♦♣♦❧♦❣✐❛ ❞❡ ✉♠ ❊▲❈ ❝♦♠ ❜❛s❡ ❡♥✉♠❡rá✈❡❧ é ♠❡tr✐③á✈❡❧ ❡ t❡♠ ✉♠❛ ♠étr✐❝❛ ✐♥✈❛r✐❛♥t❡ ♣♦r tr❛♥s❧❛çõ❡s q✉❡ ♣♦❞❡ s❡r ❣❡r❛❞❛ ♣♦r ✉♠❛ ❢❛♠í❧✐❛ ❡♥✉♠❡rá✈❡❧ ❞❡ s❡♠✐♥♦r♠❛s✳

P❛r❛ ♥ós ✉♠ ❊s♣❛ç♦ ❞❡ ❋ré❝❤❡t é ✉♠ ❊▲❈ ❝✉❥❛ t♦♣♦❧♦❣✐❛ é ❞❡✜♥✐❞❛ ♣♦r ✉♠❛ ❢❛♠í❧✐❛ ❡♥✉♠❡rá✈❡❧ s❡♣❛r❛♥t❡ ❞❡ s❡♠✐♥♦r♠❛s ❛s q✉❛✐s ❞❡✜♥❡♠ ❛ ♠étr✐❝❛ ❝♦♠♦ ❡♠ ✭✶✳✷✳✸✮ ❡ ♦ ❡s✲ ♣❛ç♦ ❝♦♠ ❡st❛ ♠étr✐❝❛ é ❝♦♠♣❧❡t♦✳

❊♠ t❡r♠♦s ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❡♥✉♠❡rá✈❡❧ ❞❡ s❡♠✐♥♦r♠❛s✱ q✉❡ ❣❡r❛♠ ❛ t♦♣♦❧♦❣✐❛✱ ❞✐❣❛♠♦s

{p1, p2, p3,· · · }✱ ♣♦❞❡♠♦s ❡♥✉♥❝✐❛r ✉♠ r❡s✉❧t❛❞♦ ❝♦♠♦ s❡❣✉❡✳ ❙❡❥❛ (xk) ✉♠❛ s❡q✉ê♥❝✐❛

❡♠ ❊✳ ❙❡ pm(xj − xl) −→ 0✱ q✉❛♥❞♦ j✱l −→ ∞✱ ∀m ∈ N✱ ❡♥tã♦ ∃x ∈ ❊ t❛❧ q✉❡

pm(xj −x)−→0✱ q✉❛♥❞♦ j −→ ∞✱∀m ∈N✳

❊①❡♠♣❧♦ ✶✳✷✳✺✳ ❙❡❥❛ Ω⊂Rn ✉♠ ❛❜❡rt♦✳ ❙❡❥❛ _(Ωj)j∈N t❛❧ q✉❡ _{Ωj ⊂ Ω} é ❛❜❡rt♦ ❝♦♠ _Ωj

❝♦♠♣❛❝t♦✱ Ωj ⊂Ωj+1✱ ♣❛r❛ t♦❞♦ j ∈N ❡

[

j∈N

Ωj = Ω✳

❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ C∞_{(Ω) ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣❡❧❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐♥♦r♠❛s}

pk,j(ϕ) =

X

|α|≤k

sup

Ωj

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✶✶

é ✉♠ ❡s♣❛ç♦ ❞❡ ❋ré❝❤❡t✳

❖❜s✳ ✶✳✷✳✻✳ ❆ s❡q✉ê♥❝✐❛ (Ωj)j∈N é ❝❤❛♠❛❞❛ ❞❡ s❡q✉ê♥❝✐❛ ❞❡ ❛❜❡rt♦s ❡s❣♦t❛♥t❡s ❞❡ Ω✳

❊①❡♠♣❧♦ ✶✳✷✳✼✳ ❙❡❥❛ K ⊂Ω ❝♦♠♣❛❝t♦✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧

C∞

c (K) = {ϕ∈C∞(Ω) ; supp(ϕ)⊂K}

❝♦♠ ❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐♥♦r♠❛s

Pk(ϕ) =

X

|α|≤k

sup

K |

∂αϕ|✱ k ∈Z₊✱

é ✉♠ ❡s♣❛ç♦ ❞❡ ❋ré❝❤❡t✳

❖❜s✳ ✶✳✷✳✽✳ ❊s♣❛ç♦s ❞❡ ❋ré❝❤❡t sã♦ ✐♠♣♦rt❛♥t❡s ♣♦rq✉❡ ❝❡rt❛s ❝♦♥s❡q✉ê♥❝✐❛s ❞❡ ❚❡♦r❡♠❛ ❞❡ ❇❛✐r❡ sã♦ ✈á❧✐❞❛s ♥❡❧❡s✳ P♦r ❡①❡♠♣❧♦✿ ❚❡♦r❡♠❛ ❞❛ ❛♣❧✐❝❛çã♦ ❛❜❡rt❛ ❞❛❞♦ ♣❡❧♦ t❡♦r❡♠❛ ❛❜❛✐①♦✳

❚❡♦r❡♠❛ ✶✳✷✳✾✳ ❙❡❥❛♠ ❊✱ ❋ ❞♦✐s ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t✳ ❙❡ T : ❊ −→❋ é ✉♠❛ ❛♣❧✐❝❛çã♦

❧✐♥❡❛r ❝♦♥tí♥✉❛ ❡ s♦❜r❡❥❡t✐✈❛✱ ❡♥tã♦ T é ❛❜❡rt❛✳

❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ T :❊−→❋ é ✉♠❛ ❜✐❥❡çã♦✱ ❡♥tã♦T−1 _{:❋−→❊ é ❝♦♥tí♥✉❛✳}

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✵✳ ❙❡❥❛ ❊ ✉♠ ❊▲❈ ❡ s❡❥❛ P ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐♥♦r♠❛s q✉❡ ❣❡r❛

❛ t♦♣♦❧♦❣✐❛ ❞❡ ❊✳ ❊♥tã♦ ❛ s❡♠✐♥♦r♠❛ q é ❝♦♥tí♥✉❛ ❡♠ ❊ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ {p1, p2, ..., pN} ⊂P ❡ ✉♠❛ ❝♦♥st❛♥t❡ C t❛❧ q✉❡

q(x)≤Cmax{p1(x), ..., pN(x)}, x ∈❊. ✭✶✳✷✳✹✮

❉❡♠♦♥str❛çã♦✿

❙✉♣♦♥❤❛ q ❝♦♥tí♥✉❛✳ ❊♥tã♦ q−1_{(−1,1)é ✉♠ ❛❜❡rt♦ ❞❡ ❊ q✉❡ ❝♦♥té♠ 0✳ ▲♦❣♦ ❡①✐st❡ ✉♠}

❛❜❡rt♦

A=

N

\

j=1

{y_∈❊ ; pj(y)< ǫj, pj ∈P, ǫj >0}

t❛❧ q✉❡ A ⊂ q−1_{(−1,1)✳ ❙❡❥❛ x ∈ ❊✳ ❙❡ p}

j(x) > 0 ♣❛r❛ ❛❧❣✉♠ j ∈ {1,2, ..., N}✱ ❡♥tã♦

✶✷ Pré✲r❡q✉✐s✐t♦s

t= 1 2

min{ǫ1, ..., ǫN}

max_{p1(x), ..., pN(x)}

t❡♠♦s tx _∈A_⊂q−1_{(−1,1)✳ ❆ss✐♠ q(tx)<1✳ ▲♦❣♦}

q(x)< 1 t = 2

max_{p1(x), ..., pN(x)}

min{ǫ1, ..., ǫN}

=Cmax_{p1(x), ..., pN(x)}✱

❝♦♠ C = 2/min{ǫ1, ..., ǫN}✳ ❆❣♦r❛✱ s❡ pj(x) = 0✱ ♣❛r❛ t♦❞♦ j ∈ {1, ..., N}✱ ❡♥tã♦

pj(tx) = 0✱ ♣❛r❛ t♦❞♦ j ∈ {1, ..., N}✱ ♣❛r❛ t♦❞♦ t > 0✱ ♦✉ s❡❥❛✱ tx ∈ A✱ ♣❛r❛ t♦❞♦

t > 0✳ ▲♦❣♦ q(tx)<1✱ ♣❛r❛ t♦❞♦t > 0✳ ❉❛í t❡♠♦s0≤q(x)< 1_t✱ ♣❛r❛ t♦❞♦ t >0✳ ❙❡❣✉❡

q✉❡ q(x) = 0 ❡ ❛ss✐♠ ✭✶✳✷✳✹✮ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡P ❞❡ ♠♦❞♦ q✉❡ ✭✶✳✷✳✹✮

s❡❥❛ s❛t✐s❢❡✐t❛✳ ❊♥tã♦ q é ❝♦♥tí♥✉❛ ♥♦ 0✳ ❈♦♠♦ q é ✉♠❛ s❡♠✐♥♦r♠❛ ❡♠ ❊✱ t❡♠♦s q✉❡ q s❛t✐s❢❛③

|q(x)−q(y)| ≤q(x−y)✱ x, y ∈❊✳

❉❛í✱ s❡❣✉❡ q✉❡ q é ❝♦♥tí♥✉❛ ❡♠ ❊✳

❖❜s✳ ✶✳✷✳✶✶✳ ❯♠ ❝♦r♦❧ár✐♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✱ é q✉❡ ❛ t♦♣♦❧♦❣✐❛ ❞❡ ❊ é ❣❡r❛❞❛ ♣❡❧❛ ❢❛♠í❧✐❛ ❞❡ t♦❞❛s ❛s s❡♠✐♥♦r♠❛s ❝♦♥tí♥✉❛s ❡♠ ❊✳

❈♦r♦❧ár✐♦ ✶✳✷✳✶✷✳ ❙❡❥❛ ❊ ❝♦♠♦ ♥❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛✳ ❊♥tã♦ ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r u ❡♠ ❊ é ❝♦♥tí♥✉♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ {p1, ..., pN} ⊂ P ❡ ✉♠❛

❝♦♥st❛♥t❡ C t❛✐s q✉❡

|hu, xi| ≤Cmax{p1(x), ..., pN(x)}✱ x∈❊✳

❉❡♠♦♥str❛çã♦✿

❙❡❥❛ u✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡♠ ❊✳ ❊♥tã♦ ❊∋x7→ |hu, xi| é ✉♠❛ s❡♠✐♥♦r♠❛ ❡♠ ❊✳ ❆ss✐♠

♦ r❡s✉❧t❛❞♦ s❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ♣r♦♣♦s✐çã♦ ✶✳✷✳✶✵✳

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✶✸

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✸✳ ❙❡❥❛♠ ❊ ❡ ❋ ❞♦✐s ❊▲❈ ❡ s❡❥❛♠P ❡Q❞✉❛s ❢❛♠í❧✐❛s ❞❡ s❡♠✐♥♦r♠❛s

q✉❡ ❣❡r❛♠ ❛s t♦♣♦❧♦❣✐❛s ❞❡ ❊ ❡ ❋ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦ ❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛rT :E −→F é ❝♦♥tí♥✉❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ t♦❞❛ q_∈Q✱ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦_{{p1, p2, ..., pN} ⊂} P ❡ ✉♠❛ ❝♦♥st❛♥t❡ _{C =C(q)} t❛❧ q✉❡

q(T(x))_≤Cmax_{p1(x), ..., pN(x)}✱ x∈❊✳

❉❡♠♦♥str❛çã♦✿ ❖ ❛r❣✉♠❡♥t♦ ❞❡st❛ ❞❡♠♦str❛çã♦ é ❛♥❛❧♦❣♦ ❛♦ ❛r❣✉♠❡♥t♦ ✉s❛❞♦ ♥❛ ❞❡♠♦♥tr❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ✶✳✷✳✶✵✳

❖❜s✳ ✶✳✷✳✶✹✳ ❊①✐st❡ ✉♠❛ t❡❝♥✐❝❛ ❡♠ ❛♥á❧✐s❡ ♣❛r❛ ♦❜t❡r ❞❡s✐❣✉❛❧❞❛❞❡s q✉❡ é ❛ ❛♣❧✐❝❛çã♦ ❞♦ t❡♦r❡♠❛ ✶✳✷✳✾ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ♣r♦♣♦s✐çã♦ ✶✳✷✳✶✸ ❛ ❡s♣❛ç♦s ❞❡ ❋ré❝❤❡t ❛❞❡q✉❛❞♦s✳ ❊ss❛ t❡❝♥✐❝❛ s❡rá ✉s❛❞❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ✸✳✸✳✶✳

✶✳✷✳✶ ❆ t♦♣♦❧♦❣✐❛ ❞❡

C

(Ω)

❡ ♦ ❡s♣❛ç♦

D

(Ω)

❱❛♠♦s ❢❛❧❛r ❛❣♦r❛ ✉♠ ♣♦✉❝♦ s♦❜r❡ ❛ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ♦✉ ♥❛t✉r❛❧ ❞❡ C∞

c (Ω)✳ ❊st❛

t♦♣♦❧♦❣✐❛ é ❝❤❛♠❛❞❛ ❞❡ ❚♦♣♦❧♦❣✐❛ ❧✐♠✐t❡ ✐♥❞✉t✐✈♦ ❞♦s ❡s♣❛ç♦s C∞

c (K)✱K ⊂Ω❝♦♠♣❛❝t♦✳

❖ ♣ró①✐♠♦ t❡♦r❡♠❛ tr❛t❛ ❞❛ t♦♣♦❧♦❣✐❛ ❧✐♠✐t❡ ✐♥❞✉t✐✈♦ ❞❡ ♠❛♥❡✐r❛ ❣❡r❛❧✳

❚❡♦r❡♠❛ ✶✳✷✳✶✺✳ ❙❡❥❛ ❊ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ✭♦✉ r❡❛❧✮✳ ❙❡❥❛ {❊j}j∈N ✉♠❛

❢❛♠í❧✐❛ ❞❡ s✉❜❡s♣❛ç♦s ❝♦♠

❊j ⊂❊j+1✱ ❊=

[

j∈N

❊j✳

❙✉♣♦♥❤❛ q✉❡ ❝❛❞❛ ❊j ♣♦ss✉✐ ✉♠❛ t♦♣♦❧♦❣✐❛ ❧♦❝❛❧♠❡♥t❡ ❝♦♥✈❡①❛ ❡ q✉❡ ❛ t♦♣♦❧♦❣✐❛ ❞❡ ❊j

❝♦✐♥❝✐❞❡ ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ❊j+1✳ ❙❡❥❛ B ❛ ❝♦❧❡çã♦ ❞♦s ❝♦♥❥✉♥t♦s ❡q✉✐❧✐❜r❛❞♦s✱

❛❜s♦r✈❡♥t❡s ❡ ❝♦♥✈❡①♦s V ❡♠ ❊ t❛✐s q✉❡ V _∩❊j é ✉♠ ❛❜❡rt♦ ❡♠ ❊j ♣❛r❛ ❝❛❞❛ j ∈ N✳

❊♥tã♦✿

✶✳ B é ✉♠❛ ❜❛s❡ ❞❡ ✈✐③✐♥❤❛♥ç❛s ❞♦ ₀ ♣❛r❛ ✉♠❛ t♦♣♦❧♦❣✐❛ ❧♦❝❛❧♠❡♥t❡ ❝♦♥✈❡①❛✳

✷✳ ❆ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣♦r B é ❛ t♦♣♦❧♦❣✐❛ ❧♦❝❛❧♠❡♥t❡ ❝♦♥✈❡①❛ ♠❛✐s ✜♥❛ ❡♠ ❊ q✉❡ t♦r♥❛

✶✹ Pré✲r❡q✉✐s✐t♦s

✸✳ ❆ t♦♣♦❧♦❣✐❛ ✐♥❞✉③✐❞❛ ♣♦r ❊ ❡♠ ❊j ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❞❡ ❊j✳

❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✷✺❪✳

❖ ❡s♣❛ç♦ ❊ ❝♦♠ ❛ t♦♣♦❧♦❣✐❛ ❣❡r❛❞❛ ♣♦rBé ❝❤❛♠❛❞♦ ❞❡ ❧✐♠✐t❡ ✐♥❞✉t✐✈♦ ❞♦s ❡s♣❛ç♦s ❊_j✳

❆❣♦r❛✱ s❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❝♦♥❡①♦ ❞❡ Rn✳ ❙❡❥❛ _C∞

c (Ω) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ✐♥✜✲

♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ s✉♣♦rt❡ ❡♠ Ω✳ ❙❡❥❛ _{Kj}j∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❝♦♥❥✉♥t♦s

❝♦♠♣❛❝t♦s ❞❡ Ωt❛✐s q✉❡

Kj ⊂Kj+1✱

[

j∈N

Kj = Ω✳

P❛r❛ ❝❛❞❛ j _∈N ❝♦♥s✐❞❡r❡ ♦ ❡s♣❛ç♦ ❞❡ ❋ré❝❤❡t _C∞

c (Kj) ❝♦♠♦ ♥♦ ❡①❡♠♣❧♦ ✶✳✷✳✼✳ ❖❜s❡r✈❡

q✉❡ {C∞

c (Kj)}j∈N é ✉♠❛ ❢❛♠í❧❛ ❞❡ s✉❜❡s♣❛ç♦s ❞❡ C_c∞(Ω) q✉❡ s❛t✐s❢❛③ ❛s ❤✐♣ót❡s❡s ❞♦

t❡♦r❡♠❛ ✶✳✷✳✶✺ ❡ ❛ss✐♠ ❛ t♦♣♦❧♦❣✐❛ ❡♠ C∞

c (Ω) ❞❡✜♥✐❞❛ ♣❡❧♦ ❧✐♠✐t❡ ✐♥❞✉t✐✈♦ ❞♦s ❡s♣❛ç♦s

C∞

c (Kj) é ❛ q✉❡ ❝❤❛♠❛♠♦s ❞❡ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ❞❡ Cc∞(Ω)✳

❖❜s✳ ✶✳✷✳✶✻✳ ❆ t♦♣♦❧♦❣✐❛ ✉s✉❛❧ ❞❡ C∞

c (Ω) ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ s❡q✉ê♥❝✐❛ {Kj}j∈N ❞❡ ❝♦♠✲

♣❛❝t♦s ❞❡ Ω✳

❈r✐tér✐♦ ❞❡ ❈♦♥✈❡r❣ê♥❝✐❛ ❡♠ C∞

c (Ω)

❯♠❛ s❡q✉ê♥❝✐❛ (ϕj)j∈N ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦ ❡♠ C_c∞(Ω) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠

❝♦♠♣❛❝t♦ K _⊂Ωt❛❧ q✉❡✿

✶✳ supp(ϕj)⊂K✱ ♣❛r❛ t♦❞♦ j ∈N❀

✷✳ ♣❛r❛ t♦❞♦α _∈Zn

+✱ ❛ s❡q✉ê♥❝✐❛ (∂αϕj)j∈N ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ③❡r♦ ❡♠ K✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✼✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ ❡♠ C∞

c (Ω)✳

❉❡♥♦t❛♠♦s ♣♦rD′_(Ω)♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❞✐str✐❜✉✐çõ❡s ❡♠_Ω✳ ❆ ❛çã♦ ❞❡_u∈D′_(Ω)

s♦❜r❡ ϕ ∈C∞

c (Ω) é ✐♥❞✐❝❛❞❛ ♣♦r hu, ϕi✳

❚❡♦r❡♠❛ ✶✳✷✳✶✽✳ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡♠ C∞

c (Ω)é ❝♦♥tí♥✉♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ ❝❛❞❛

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✶✺

|hu, ϕ_{i| ≤}C X |α|≤m

sup

K |

∂αϕ_|✱

♣❛r❛ t♦❞❛ ϕ _∈C∞

c (K)✳

❉❡♠♦♥str❛çã♦✿

❙❡❥❛ u_∈ D′_(Ω)✳ ❈♦♠♦ _u é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦ ❡♠ _Cc∞_(Ω) ❡ ♣❛r❛ ❝❛❞❛ _{K ⊂ Ω}

❝♦♠♣❛❝t♦ ❛ ✐♥❝❧✉sã♦ C∞

c (K)−→Cc∞(Ω) é ❝♦♥tí♥✉❛✱ t❡♠♦s q✉❡ ❛ r❡str✐çã♦ ❞❡u ❛Cc∞(K)

é ❝♦♥tí♥✉❛✳ P♦rt❛♥t♦ ♣❡❧♦ ❝♦r♦❧ár✐♦ ✶✳✷✳✶✷ ❡①✐st❡♠ k1, ..., kN ∈Z+ ❡ C >0 t❛✐s q✉❡

|hu, ϕ_{i| ≤}Cmax_{Pk1(ϕ), ..., PkN(ϕ)}✱ ♣❛r❛ t♦❞❛ ϕ ∈C ∞

c (K)✱

♦♥❞❡ Pk✱ k ∈ Z+✱ ❞❡♥♦t❛♠ ❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐♥♦r♠❛s q✉❡ ❣❡r❛ ❛ t♦♣♦❧♦❣✐❛ ❞❡ Cc∞(K)

❞✐s❝✉t✐❞♦ ♥♦ ❡①❡♠♣❧♦ ✶✳✷✳✼✳ ❉❛í✱

|hu, ϕi| ≤C

N

X

j=1

Pkj(ϕ)≤C

X

|α|≤m

sup

K |

∂αϕ|✱

♣❛r❛ t♦❞❛ ϕ∈C∞

c (K)✱ ♦♥❞❡m = max{k1, ..., kN}✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❛❞❛ ❝♦♠♣❛❝t♦K ⊂Ω✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C ❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ m t❛✐s q✉❡

|hu, ϕ_{i| ≤}C X |α|≤m

sup

K |

∂α_ϕ|✱

♣❛r❛ t♦❞❛ ϕ _∈ C∞

c (K)✳ ❊♥tã♦ ♣❛r❛ ❝❛❞❛ K ⊂ Ω ❛ r❡str✐çã♦ ❞❡ u ❛ Cc∞(K) é ❝♦♥tí♥✉❛✳

▲♦❣♦ ué ❝♦♥tí♥✉♦ ❡♠ C∞

c (Ω)✳ ❖✉ s❡❥❛✱ u∈D′(Ω)✳

❱❛♠♦s ✈❡r ❛❣♦r❛ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❞✐str✐❜✉✐çõ❡s

❊①❡♠♣❧♦ ✶✳✷✳✶✾✳ ❙❡❥❛ f ∈L1

loc(Ω)✳ ❊♥tã♦

huf, ϕi=

Z

Ω

f(x)ϕ(x)dx✱ ϕ ∈C∞

c (Ω)✱

✶✻ Pré✲r❡q✉✐s✐t♦s

❖❜s✳ ✶✳✷✳✷✵✳ ❆ ❛♣❧✐❝❛çã♦ L1

loc(Ω) ∋ f 7→ uf ∈ D′(Ω) é ❧✐♥❡❛r ❡ ✐♥❥❡t♦r❛✳ ▲♦❣♦ L1loc(Ω)

♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ✉♠ s✉❜❡s♣❛ç♦ ❞❡ D′_{(Ω)✳ ❉❡✈✐❞♦ ❛ ✐ss♦✱ ❛ ❞✐str✐❜✉✐çã♦ u}

f s❡rá

❞❡♥♦t❛❞❛ ♣♦r f✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t♦❞❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡Ck_{(Ω)✱ k= 0,1,2, ...,∞ ❞❡✜♥❡ ✉♠❛}

❞✐str✐❜✉✐çã♦ ❡♠ Ω✳

❊①❡♠♣❧♦ ✶✳✷✳✷✶✳ ❆ ♠❡❞✐❞❛ ❞❡ ❉✐r❛❝ ♥♦ ♣♦♥t♦ a∈Ω ❞❡✜♥❡ ♣♦r ♠❡✐♦ ❞❡

hδa, ϕi=ϕ(a)✱ ϕ∈Cc∞(Ω)✱

✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω✳

❉✐③❡♠♦s q✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ u _∈ D′_(Ω) é ♥✉❧❛ ♥✉♠ ❛❜❡rt♦ _{V ⊂ Ω} s❡ _{hu, ϕi = 0}✱

♣❛r❛ t♦❞❛ ϕ∈C∞

c (V)✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✷✳ ❙❡ u ∈ D′_{(Ω) ❞❡✜♥✐♠♦s ♦ s✉♣♦rt❡ ❞❡ u✱ supp(u)✱ ❝♦♠♦ ❛ ✐♥t❡rs❡çã♦} ❞❡ t♦❞♦s ♦s ❢❡❝❤❛❞♦s ❞❡ Ω ❢♦r❛ ❞♦s q✉❛✐s u é ♥✉❧❛✳

❊①❡♠♣❧♦ ✶✳✷✳✷✸✳ supp(δa) ={a}✳

❊①❡♠♣❧♦ ✶✳✷✳✷✹✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❡♠ Ω✱ ❡♥tã♦ f ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω ❝♦♠♦ ❡❧❡♠❡♥t♦ ❞❡ L1

loc(Ω)✳ ❙❡✉ s✉♣♦rt❡✱ ❝♦♠♦ ❢✉♥çã♦✱ ❝♦✐♥❝✐❞❡ ❝♦♠ s❡✉ s✉♣♦rt❡✱

❝♦♠♦ ❞✐str✐❜✉✐çã♦✳ ■st♦ é✱ ♥❛ ♥♦t❛çã♦ ❞♦ ❡①❡♠♣❧♦ ✶✳✷✳✶✾✱

supp(f) = supp(uf)✳

❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ f é ❝♦♥tí♥✉❛ ❡♠ Ω ❡ supp(uf) t❡♠ ♠❡❞✐❞❛ ❞❡ ▲❡❜❡s❣✉❡ ♥✉❧❛

❡♠ Ω✱ ❡♥tã♦ f é ✉♠❛ ❢✉♥çã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛ ❡♠ Ω✳

■♥s♣✐r❛❞♦ ♥♦ ❡①❡♠♣❧♦ ✶✳✷✳✶✾✱ ✉♠❛ ❞✐str✐❜✉✐çã♦u_∈D′_(Ω) é ❞✐t❛_C∞ ♥♦ ❛❜❡rt♦_{U ⊂Ω}✱

s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ f _∈C∞_{(U) t❛❧ q✉❡}

hu, ϕ_i=

Z

U

f(x)ϕ(x)dx✱ ♣❛r❛ t♦❞❛ ϕ _∈C∞

c (U)✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✺✳ ❙❡ u_∈D′_(Ω) ❞❡✜♥✐♠♦s ♦ s✉♣♦rt❡ s✐♥❣✉❧❛r ❞❡ _u✱ _suppsing(u)✱ ❝♦♠♦ ❛

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✶✼

❉❡r✐✈❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❢✉♥çõ❡s C∞

❉❡✜♥✐çã♦ ✶✳✷✳✷✻✳ ❙❡ u_∈D′_{(Ω) ❡ α∈Z}n

+✱ ❞❡✜♥✐♠♦s ∂αu∈D′(Ω) ♣♦r

h∂α_{u, ϕi= (}. ₋₁₎|α|_{hu, ∂}α_{ϕi✱ ♣❛r❛ t♦❞❛ ϕ ∈C}∞

c (Ω)✳

❊①❡♠♣❧♦ ✶✳✷✳✷✼✳ ❙❡❥❛ H _∈L1

loc(R) ❛ ❢✉♥çã♦ ❞❡ ❍❡❛✈✐s✐❞❡ ❞❛❞❛ ♣♦r

H(x) =

 



1, s❡ x >0

0, s❡ x_≤0

❊♥tã♦ H′ _=δ✳

❉❡✜♥✐çã♦ ✶✳✷✳✷✽✳ ❙❡ u_∈D′_{(Ω) ❡ f ∈C}∞_{(Ω)✱ ❞❡✜♥✐♠♦s f u∈D}′_{(Ω) ♣♦r}

hf u, ϕ_i=. _hu, f ϕ_i✱ ♣❛r❛ t♦❞❛ ϕ _∈C∞

c (Ω)✳

❊①❡♠♣❧♦ ✶✳✷✳✷✾✳ ❙❡ 0∈Ω❡ f ∈C∞_{(Ω) é t❛❧ q✉❡ f(0) = 0✱ ❡♥tã♦ f δ}

0 = 0✳

❖❜s✳ ✶✳✷✳✸✵✳ ❱❛❧❡ ❛ ❘❡❣r❛ ❞❡ ▲❡✐❜♥✐③✿ s❡ u_∈D′_(Ω)✱ _{f ∈C}∞_(Ω) ❡ _{α ∈}Zn

+✱ ❡♥tã♦

∂α_{(f u) =}X β≤α

α β

✶✽ Pré✲r❡q✉✐s✐t♦s

✶✳✷✳✷ ▼✉❞❛♥ç❛ ❞❡ ❱❛r✐á✈❡✐s ❡♠ ❉✐str✐❜✉✐çõ❡s

❙❡❥❛♠ Ω1✱ Ω2 ⊂R2 ❛❜❡rt♦s✳

❙❡❥❛F : Ω1 −→Ω2 ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ C∞✳ ❉❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦

F∗ _{: C}∞_(Ω

2) −→ C∞(Ω1)

u _7−→ u_◦F

◆♦t❡ q✉❡ F∗ _{é ❜✐❥❡t♦r❛ ❡ ❧✐♥❡❛r✳ ❙❡ U ⊂ Ω}

1 ❡ V ⊂ Ω2 sã♦ ❛❜❡rt♦s t❛✐s q✉❡ F(U) = V✱

❡♥tã♦ ❛ r❡str✐çã♦ ❞❡ F∗ _{❛ C}∞_{(V) é ✉♠❛ ❜✐❥❡çã♦ ❧✐♥❡❛r ❞❡ C}∞_{(V) ❡♠ C}∞_{(U)✳ ❖❜s❡r✈❡} t❛♠❜é♠ q✉❡ s❡ f✱ g ∈C∞_(Ω

2)✱ ❡♥tã♦ F∗(f g) = F∗(f)F∗(g)✳ ❚❡♠♦s ❛ss✐♠ q✉❡F∗ é ✉♠

✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ❡s♣❛ç♦s ✈❡t♦r✐❛✐s C∞_(Ω

2) ❡ C∞(Ω1)✳ ❆ ❛♣❧✐❝❛çã♦ F∗ é ❝❤❛♠❛❞❛ ❞❡

♣✉❧❧❜❛❝❦✳

❚❡♦r❡♠❛ ✶✳✷✳✸✶✳ ❆ ❛♣❧✐❝❛çã♦ F∗ _{s❡ ❡st❡♥❞❡ ❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ❝♦♥tí♥✉❛ F}∗ _:

D′_(Ω2)−→D′_(Ω1)✳ ❆❧é♠ ❞✐ss♦ _F∗ é ✉♠❛ ❜✐❥❡çã♦ ❝♦♠ ✐♥✈❡rs❛ _(F−1₎∗✳

C∞_(Ω

2)

F∗

/

/_C∞_(Ω

1)

D′_(Ω2) F∗ //_D′_(Ω

1)

❆s s❡t❛s ✈❡rt✐❝❛✐s r❡♣r❡s❡♥t❛♠ ❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦✿

C∞_(Ω

j)∋f 7→uf ∈D′(Ωj)✱ j = 1✱ 2✳

❉❡♠♦♥str❛çã♦✿

❆ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ F∗ _:D′_(Ω

2)−→D′(Ω1) é ❡♠ r❡❧❛çã♦ à ❝♦♥✈❡r❣ê♥❝✐❛ ❢r❛❝❛✳

❆ ✜♠ ❞❡ ♦❜t❡r ✉♠❛ ❡①t❡♥sã♦ ❞❡ F∗ _{❝♦♠♦ ❛ ❞♦ ❡♥✉♥❝✐❛❞♦ ❞♦ t❡♦r❡♠❛✱ s❡❥❛♠ u ∈} C∞_(Ω

2) ❡ ϕ∈Cc∞(Ω1)✳ ❖❧❤❛♥❞♦ ♣❛r❛ u ❝♦♠♦ ✉♠❛ ❞✐str✐❜✉✐çã♦✱ t❡♠♦s

D

F∗_{(u), ϕ}E₌

Z

u(F(x))ϕ(x)dx=

Z

✶✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ❚♦♣♦❧ó❣✐❝♦s ✶✾

=Du,|det[(F−1₎′_]|(ϕ◦F−1₎E_✳

❉❛❞❛ ❡♥tã♦ u_∈D′_(Ω2) ❞❡✜♥✐♠♦s

F∗_{(u) :C}∞

c (Ω1)−→C

❞❛❞❛ ♣♦r

F∗_{(u)(ϕ) = u|det[(F}−1₎′_]|(ϕ◦F−1_)✱

ϕ _∈ C∞

c (Ω1)✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ F∗(u) é ❧✐♥❡❛r✳ ❱❛♠♦s ♠♦str❛r q✉❡ F∗(u) é ❝♦♥tí♥✉♦ ✭✐✳❡✳✱

F∗_{(u) ∈ D}′_(Ω

1)✮✳ P❛r❛ ✐ss♦✱ s❡❥❛ (ϕj)j∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ C_c∞(Ω₁) t❛❧ q✉❡ ϕ_j −→ 0

❡♠ C∞

c (Ω1)✳ ■st♦ é✱ ❡①✐st❡ ✉♠ ❝♦♠♣❛❝t♦ K ⊂ Ω1 t❛❧ q✉❡ supp(ϕj) ⊂ K✱ ♣❛r❛ t♦❞♦

j ∈ N ❡ _∂α_{ϕj −→ 0} ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ t♦❞♦ _{α ∈} Zn

+✳ ❈♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ (φj)j∈N

❡♠ C∞

c (Ω2)✱ ♦♥❞❡ φj =. |det[(F−1)′]|(ϕj ◦ F−1)✱ j ∈ N✳ ◆♦t❡ q✉❡ F∗(u)(ϕj) = u(φj)✱

♣❛r❛ t♦❞♦ j ∈ N✳ ❱❛♠♦s ♠♦str❛r q✉❡ _{φj −→ 0} ❡♠ _C∞

c (Ω2)✳ P❛r❛ ✐ss♦✱ s❡❥❛ α ∈ Zn+✳

❚❡♠♦s q✉❡ ∂α_φ

j é ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❡ ♣r♦❞✉t♦s ❞❡ ❞❡r✐✈❛❞❛s ❞❛s ❢✉♥çõ❡s✱ |det[(F−1)′]|✱

ϕj ◦F−1 ❡ ❞❡r✐✈❛❞❛s ❞❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❞❡ F−1✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝❛❞❛ t❡r♠♦ ❞❡ss❛

s♦♠❛ ♣♦ss✉✐ ❡♠ s✉❛ ❞❡❝♦♠♣♦s✐çã♦ ❡♠ ♣r♦❞✉t♦✱ ✉♠ t❡r♠♦ ❞❛ ❢♦r♠❛ (∂β_ϕ

j)◦F−1✱ β ≤α✳

❉❛í✱ ❝❛❞❛ t❡r♠♦ ❞❡ss❛ s♦♠❛ t❡♠ s✉♣♦rt❡ ❡♠ F(K) ❡ ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ ③❡r♦

❡♠ F(K)✳ ❆ss✐♠ supp(φj) ⊂ F(K) ❡ ♥♦✈❛♠❡♥t❡✱ ❝♦♠♦ s❡ tr❛t❛ ❞❡ ✉♠❛ s♦♠❛ ✜♥✐t❛✱

t❡♠♦s q✉❡ ∂α_φ

j −→ 0 ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ F(K)✳ P♦rt❛♥t♦✱ φj −→ 0 ❡♠ Cc∞(Ω2)✳ ❈♦♠♦

u _∈ D′_(Ω2)✱ t❡♠♦s _{u(φj) −→ 0} ✭❡♠ C✮✳ ❖✉ s❡❥❛✱ _F∗_{(u)(ϕj) −→ 0}✳ P♦rt❛♥t♦ _F∗_(u)

é ❝♦♥tí♥✉❛✱ ✐✳❡✳✱ F∗_{(u) ∈ D}′_(Ω

1)✳ ❈♦♥❝❧✉í♠♦s ❛ss✐♠ q✉❡ F∗ : D′(Ω2) −→ D′(Ω1) ❡stá

❜❡♠ ❞❡✜♥✐❞❛✳ ❆❣♦r❛ ✈❛♠♦s ♠♦str❛r s✉❛ ❝♦♥t✐♥✉✐❞❛❞❡✳ ❙❡❥❛ uj ∈ D′(Ω2)✱ j ∈ N t❛❧

q✉❡ uj −→ 0 ❡♠ D′(Ω2)✳ ❉❛❞❛ ϕ ∈ Cc∞(Ω1)✱ t❡♠♦s |det[(F−1)′]|(ϕ◦F−1) ∈ Cc∞(Ω2)✳

▲♦❣♦ uj

|det[(F−1₎′_{]|(ϕ ◦F}−1_{) −→ 0 ✭❡♠ C✮✳ ❖✉ s❡❥❛✱ F}∗_(u

j)(ϕ) −→ 0 ♣❛r❛ t♦❞❛

ϕ ∈C∞

c (Ω1)✳ ❆ss✐♠ F∗(uj)−→0❡♠ D′(Ω1)✳ ❙❡❣✉❡ ❞❛í q✉❡F∗ é ❝♦♥tí♥✉❛✳

P♦r ✜♠✱ ❝♦♠♦ F é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦✱ t❡♠♦s ❛ss✐♠ q✉❡ (F−1₎∗ _{: D}′_(Ω

1) −→ D′(Ω2)

❡stá ❜❡♠ ❞❡✜♥✐❞♦✱ ❡ é ❢á❝✐❧ ✈❡r q✉❡ (F−1₎∗ _{é ❛ ✐♥✈❡rs❛ ❞❡F✳}

✷✵ Pré✲r❡q✉✐s✐t♦s

✶✳✸ ❖♣❡r❛❞♦r❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ▲✐♥❡❛r❡s

❯♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ✭❖❉P▲✮ ❡♠ ✉♠ ❛❜❡rt♦Ω❞❡Rné ✉♠ ♦♣❡r❛❞♦r

❞❛ ❢♦r♠❛

P(x, D) = X

|α|≤m

aα(x)Dα✱

♦♥❞❡ ♦s ❝♦❡✜❝✐❡♥t❡saα(x) (|α| ≤m)sã♦ ❢✉♥çõ❡sC∞❛ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❞❡✜♥✐❞❛s ❡♠Ω✳

◗✉❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s aα(x) sã♦ ❝♦♥st❛♥t❡s✱ ❞❡♥♦t❛r❡♠♦s ♦ ♦♣❡r❛❞♦r ♣♦rP(D)✳ ❙❡ ♣❛r❛

❛❧❣✉♠ |α_|=m❡①✐st✐r ✉♠ ❝♦❡✜❝✐❡♥t❡aα ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❡♠Ω✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡

m é ❛ ♦r❞❡♠ ❞❡ P(x, D)✳ ❆ ♣❛rt❡ ♣r✐♥❝✐♣❛❧ ❞❡ P(x, D) é ♦ ♦♣❡r❛❞♦r ♦❜t✐❞♦ ♦♠✐t✐♥❞♦ ♦s

t❡r♠♦s ❞❡ ♦r❞❡♠ ♠❡♥♦r q✉❡ m✿

Pm(x, D) =

X

|α|=m

aα(x)Dα✳

❖ ♣♦❧✐♥ô♠✐♦

Pm(x, ξ) =

X

|α|=m

aα(x)ξα

❞❡ ❣r❛✉ m ❝♦♠ r❡s♣❡✐t♦ ❛ ξ_∈Rn✱ é ❝❤❛♠❛❞♦ ❞❡ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡_P✳

◆♦ s❡♥t✐❞♦ ❝❧áss✐❝♦ ✉♠ ❖❉P▲ ❡♠ Ω ❞❡ ♦r❞❡♠ m ❡stá ❞❡✜♥✐❞♦ ♥♦ ❡s♣❛ç♦ Cm_(Ω)✳

▼❛s✱ ✉s❛♥❞♦ ❛ ❞❡r✐✈❛çã♦ ❞❡ ❞✐str✐❜✉✐çõ❡s ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ♣♦r ✉♠❛ ❢✉♥çã♦ C∞_{✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ❖❉P▲ ❡♠Ω❝♦♠♦ s❡♥❞♦ ✉♠ ♦♣❡r❛❞♦r ❞❡ D}′_(Ω)❡♠

D′_(Ω)✳

❉❡✜♥✐çã♦ ✶✳✸✳✶✳ ❉✐③❡♠♦s q✉❡P(x, D)é ❞❡ t✐♣♦ ♣r✐♥❝✐♣❛❧ s❡ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿

P❛r❛ t♦❞♦ x∈Ω✱ ❡ ξ ∈Rn✱ _{ξ6= 0}✱

dξPm(x, ξ)6= 0✳

❊①❡♠♣❧♦ ✶✳✸✳✷✳ ❯♠ ❖❉P▲ ❞❡ ♦r❞❡♠ ✶

P(x, D) =

n

X

j=1

✶✳✸ ❖♣❡r❛❞♦r❡s ❉✐❢❡r❡♥❝✐❛✐s P❛r❝✐❛✐s ▲✐♥❡❛r❡s ✷✶

é ❞❡ t✐♣♦ ♣r✐♥❝✐♣❛❧ ❡♠ Ω s❡✱ ❡ s♦♠❡♥t❡ s❡✱

n

X

j=1

|aj(x)| 6= 0

♣❛r❛ t♦❞♦ x∈Ω✳

❉❡✜♥✐çã♦ ✶✳✸✳✸✳ ❯♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r P ❞❡ ♦r❞❡♠ mé ❝❤❛♠❛❞♦ ❡❧ít✐❝♦ ❞❡ ♦r❞❡♠ m ❡♠ x s❡✱ ❡ s♦♠❡♥t❡ s❡✱

Pm(x, ξ)6= 0 s❡ ξ ∈Rn ❡ ξ 6= 0✳

P é ❝❤❛♠❛❞♦ ❞❡ ❡❧ít✐❝♦ ❡♠ Ω s❡ é ❡❧ít✐❝♦ ❡♠ t♦❞♦ x_∈Ω✳

❙❡P é ❡❧ít✐❝♦ ❡♠Ω✱ ❡♥tã♦ P é ❞❡ t✐♣♦ ♣r✐♥❝✐♣❛❧ ❡♠ Ω✳

❉❡✜♥✐çã♦ ✶✳✸✳✹✳ ❯♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r P é ❝❤❛♠❛❞♦ ❤✐♣♦❡❧ít✐❝♦ ❡♠ Ω

s❡✱ ❡ s♦♠❡♥t❡ s❡✱

suppsing(P u) = suppsing(u)✱ _∀u_∈D′_(Ω)✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ P é ❤✐♣♦❡❧ít✐❝♦ ❡♠ Ω✱ q✉❛♥❞♦ ♣❛r❛ t♦❞❛ u ∈ D′_{(Ω) ❡ U ⊂ Ω} ❛❜❡rt♦ s❡ t❡♠✿

P u_∈C∞_{(U)✱ ❡♥tã♦ u∈C}∞_(U)✳

❚❡♦r❡♠❛ ✶✳✸✳✺✳ ❙❡ P(x, D) é ✉♠ ♦♣❡r❛❞♦r ❡❧ít✐❝♦ ❡♠ Ω✱ ❡♥tã♦ P(x, D) é ✉♠ ♦♣❡r❛❞♦r

❤✐♣♦❡❧ít✐❝♦ ❡♠ Ω✳

❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✾❪✳

❉❡✜♥✐çã♦ ✶✳✸✳✻✳ ❙❡❥❛ P(x, D) ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ❞❡✜♥✐❞♦ ♥♦ ❛❜❡rt♦ Ω⊂Rn_{✳ ❉❡✜♥✐♠♦s ♦ ❚r❛♥s♣♦st♦ ❞❡ P(x, D)✱ ❝♦♠♦ ♦ ú♥✐❝♦ ♦♣❡r❛❞♦r}

t_{P(x, D) :D}′_{(Ω) −→D}′_(Ω)✱ q✉❡ s❛t✐s❢❛③✿

❉❛❞❛ u_∈D′_(Ω)✱

D

✷✷ Pré✲r❡q✉✐s✐t♦s

♣❛r❛ t♦❞❛ ϕ _∈C∞

c (Ω)✳

❉❛❞❛u∈D′_{(Ω)✱ t❡♠♦s}

D

t_{P(x, D)u, ϕ}E₌D_u, X

|α|≤m

aα(x)Dαϕ

E

= X

|α|≤m

D

u, aα(x)Dαϕ

E

= X

|α|≤m

D

aα(x)u, Dαϕ

E

= X

|α|≤m

D

(−1)|α|Dα(aα(x)u), ϕ

E

=D X

|α|≤m

(−1)|α|Dα(aα(x)u), ϕ

E

✱

♣❛r❛ t♦❞❛ ϕ_∈C∞

c (Ω)✳ P♦rt❛♥t♦✱

t_{P(x, D) =} X

|α|≤m

(₋1)|α|Dα(aα(x) ·)✳

❊①❡♠♣❧♦ ✶✳✸✳✼✳ ◆♦ ❝❛s♦ ❞❡ ✉♠ ❖❉P▲ ❞❡ ♦r❞❡♠ ✶

P(x, D) =

n

X

j=1

aj(x)Dxj+a0(x) ❚❡♠♦s

t_{P(x, D) =−P(x, D)−} n

X

j=1

Dxj(aj)(x) + 2a0(x)✳

❖❜s✳ ✶✳✸✳✽✳ ❊♠ r❡❧❛çã♦ ❛♦ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧✱ ✉♠ ❖❉P▲ é s❡♠❡❧❤❛♥t❡ ❛♦ s❡✉ tr❛♥s♣♦st♦✳ ❙✉♣♦♥❤❛ P(x, D) = X

|α|≤m

aα(x)Dα ❡♠ Ω✳ ❉❛❞♦u∈D′(Ω)✱ ♣♦r ▲❡✐❜♥✐③

Dα_(a αu) =

X

β≤α

α β

Dα−β(aα)Dβ(u) = aαDα(u) +

X β<α α β

Dα−β(aα)Dβ(u)✱

♣❛r❛ t♦❞♦ α ∈Zn

+✳ ❙❡❣✉❡ ❞❛í✱

t_{P(x, D) = (−1)}m_P

m(x, D) +

X

|α|<m

(−1)|α|n X

β≤α

α β

Dα−β(aα)Dβ( · )

o

✳

❊ ❛ss✐♠ ❝♦♥❝❧✉í♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❡♥tr❡ ♦s sí♠❜♦❧♦s ♣r✐♥❝✐♣❛✐s ❞❡ P ❡ Pt_✿ t_P

m(x, ξ) = (−1)mPm(x, ξ) ❡♠ Ω×Rn✳

❊♠ ♣❛rt✐❝✉❧❛r✱ P é ❡❧ít✐❝♦ ✭r❡s♣✳ t✐♣♦ ♣r✐♥❝✐♣❛❧✮ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t_{P é ❡❧ít✐❝♦ ✭r❡s♣✳ t✐♣♦}