Regularidade analítica para estruturas
de coposto um
Regularidade analítica para estruturas de coposto um
Érik Fernando de Amorim
Orientador: Prof. Dr. Sérgio Luís Zani
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 2014
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)
A524r
Amorim, Érik Fernando
Regularidade analítica para estrutura de coposto um / Érik Fernando Amorim; orientador Sérgio Luís Zani. -- São Carlos, 2014.
112 p.
Dissertação (Mestrado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2014.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ▼❛t❡♠át✐❝❛s ❡ ❞❡ ❈♦♠♣✉t❛çã♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ♣♦r ♠❡ ♣r♦♣♦r❝✐♦♥❛r t♦❞♦s ♦s r❡❝✉rs♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛ ❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞♦ ■♥st✐t✉t♦❀ t♦❞♦s tê♠ s✉❛ ❝♦♥tr✐❜✉✐çã♦ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛ ❡♠ ♠✐♥❤❛ ❡❞✉❝❛çã♦✳
❆♦ ♣r♦❢❡ss♦r ❙ér❣✐♦ ▲✉ís ❩❛♥✐✱ ♣♦r t♦❞♦ ♦ ❡♠♣❡♥❤♦ q✉❡ ❞❡❞✐❝♦✉ ❛♦ ♣r♦❥❡t♦✳ ❖r✐❡♥t❛❞♦r ❞✉r❛♥t❡ ♦s ❞♦✐s ❛♥♦s ❞❡ss❡ ♣r♦❥❡t♦ ❡ ♦s três ❛♥♦s ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛ q✉❡ ♦ ♣r❡❝❡❞❡r❛♠✱ t❡✈❡ ♣❛♣❡❧ ❞❡❝✐s✐✈♦ ♥♦ ❞❡s♣❡rt❛r ❞❡ ♠❡✉ ✐♥t❡r❡ss❡ ♣♦r ❆♥á❧✐s❡✳
❆♦ ♣r♦❢❡ss♦r ❆❞❛❧❜❡rt♦ ❇❡r❣❛♠❛s❝♦✱ q✉❡ ❡♠ ❞✐✈❡rs❛s ♦♣♦rt✉♥✐❞❛❞❡s ❛✉①✐❧✐♦✉ ♥❛ r❡s♦✲ ❧✉çã♦ ❞❡ ❡♠♣❡❝✐❧❤♦s ♠❛t❡♠át✐❝♦s q✉❡ s✉r❣✐r❛♠ ♣❡❧♦ ❝❛♠✐♥❤♦✳
➚ ❋❆P❊❙P✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❝♦♥s✐❞❡r❛♠♦s s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❧✐♥❡❛r❡s ❞❡ ♣r✐✲ ♠❡✐r❛ ♦r❞❡♠✱ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❛♥❛❧ít✐❝♦s✱ ❞❡✜♥✐❞♦s ❡♠ ✈❛r✐❡❞❛❞❡s ❛♥❛❧ít✐❝❛s r❡❛✐s✱ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ s❡✉ ❝♦♣♦st♦ é ✐❣✉❛❧ ❛ ✉♠✳ ❉❡♠♦♥str❛♠♦s q✉❡ ❡ss❡ t✐♣♦ ❞❡ s✐st❡♠❛ ❛❞♠✐t❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❧♦❝❛✐s✱ ❡ ❜✉s❝❛♠♦s ❝❛r❛❝t❡r✐③❛r s✉❛ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ❛♥❛❧ít✐❝❛ ❧♦❝❛❧ ❡ ❣❧♦❜❛❧ ❡♠ t❡r♠♦s ❞❡ ♣r♦♣r✐❡❞❛❞❡s t♦♣♦❧ó❣✐❝❛s ❞❛s ♠❡s♠❛s✳ ❚❛♠❜é♠ ♣r♦✈❛♠♦s ❛ ❋ór♠✉❧❛ ❞❡ ❆♣r♦①✐♠❛çã♦ ❞❡ ❇❛♦✉❡♥❞✐✲❚rè✈❡s✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✇❡ ❝♦♥s✐❞❡r s②st❡♠s ♦❢ ✜rst✲♦r❞❡r ❧✐♥❡❛r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ✇✐t❤ ❛♥❛❧②t✐❝ ❝♦❡✣❝✐❡♥ts✱ ❞❡✜♥❡❞ ♦♥ r❡❛❧✲❛♥❛❧②t✐❝ ♠❛♥✐❢♦❧❞s✱ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡ ❝♦r❛♥❦ ✐s ❡q✉❛❧ t♦ ♦♥❡✳ ❲❡ ♣r♦✈❡ t❤❛t t❤✐s t②♣❡ ♦❢ s②st❡♠s ❛❞♠✐ts ❧♦❝❛❧ ✜rst ✐♥t❡❣r❛❧s✱ ❛♥❞ ✇❡ s❡❡❦ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡✐r ❧♦❝❛❧ ❛♥❞ ❣❧♦❜❛❧ ❛♥❛❧②t✐❝ ❤②♣♦❡❧❧✐♣t✐❝✐t② ✐♥ t❡r♠s ♦❢ t♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡s❡ ✜rst ✐♥t❡❣r❛❧s✳ ❲❡ ❛❧s♦ ♣r♦✈❡ t❤❡ ❇❛♦✉❡♥❞✐✲❚rè✈❡s ❆♣♣r♦①✐♠❛t✐♦♥ ❋♦r♠✉❧❛✳
■♥tr♦❞✉çã♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✱ s❡♥❞♦ q✉❡ ♦ ♣r✐♠❡✐r♦ ❛❜♦r❞❛ ❛ ❜❛s❡ ❞❛ t❡♦r✐❛ ❛ s❡r ✉t✐❧✐③❛❞❛✱ ♦s ❞♦✐s s❡❣✉✐♥t❡s ❝♦♥tê♠ ♦s r❡s✉❧t❛❞♦s ♣r✐♥❝✐♣❛✐s✱ ❡ ♦ q✉❛rt♦ tr❛t❛ ❞♦ tó♣✐❝♦ ❛ ♣❛rt✐r ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠♦❞❡r♥♦✳
◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡s❡♥✈♦❧✈❡♠♦s ❛s ❢❡rr❛♠❡♥t❛s ❜ás✐❝❛s ❞❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ✐♥✈♦✲ ❧✉t✐✈♦s q✉❡ s❡rã♦ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❢♦r♠✉❧❛çã♦ ❡ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ♣r❡s❡♥t❡s ♥♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✳ ■♥tr♦❞✉③✐♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s✱ ✈❡t♦r❡s t❛♥✲ ❣❡♥t❡s ❡ ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s✱ ♣❡r♠✐t✐♥❞♦✲♥♦s ❛❜♦r❞❛r q✉❡stõ❡s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❧✐♥❡❛r❡s s♦❜r❡ ✈❛r✐❡❞❛❞❡s s✉❛✈❡s ❝♦♠♦ q✉❡stõ❡s ♣❡rt✐♥❡♥t❡s ❛ ❡ss❛ t❡♦r✐❛✳ ❉❡✲ ♠♦♥str❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s ♥❡ss❡ ❝♦♥t❡①t♦✱ q✉❡ ♣♦st❡r✐♦r♠❡♥t❡ é ❡♠♣r❡❣❛❞♦ ♥❛ ♣r♦✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❧♦❝❛✐s ♣❛r❛ ❝❡rt♦s t✐♣♦s ❞❡ ❡str✉t✉r❛s ✐♥✈♦❧✉t✐✲ ✈❛s✳ ❊st❛s✱ ♣♦r s✉❛ ✈❡③✱ ❞❡s❡♠♣❡♥❤❛♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ❞✉r❛♥t❡ ♦ r❡st❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ ▼♦str❛♠♦s ❝♦♠♦ ❡♠♣r❡❣❛r ♠✉❞❛♥ç❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝♦♠ ♦ ♣r♦♣ós✐t♦ ❞❡ s✐♠♣❧✐✜❝❛r ♦s s✐s✲ t❡♠❛s ❡st✉❞❛❞♦s s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✳ ❚❛♠❜é♠ ❛♣r❡s❡♥t❛♠♦s ♥♦ss❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❉✉❤❛♠❡❧ ♣❛r❛ s✐st❡♠❛s❀ tr❛t❛✲s❡ ❞❡ ✉♠❛ ❢ór♠✉❧❛ ✐♠♣♦rt❛♥t❡ ♥♦ ❛rs❡♥❛❧ ❞❡ q✉❛❧q✉❡r ❛♥❛❧✐st❛ ❡ q✉❡ ♥♦ ❡♥t❛♥t♦ é r❛r❛♠❡♥t❡ ♠❡♥❝✐♦♥❛❞❛ ♥♦s ❧✐✈r♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ❢❡❝❤❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ◆✐r❡♥✲ ❜❡r❣✱ ❛ r❡s♣❡✐t♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❧♦❝❛✐s q✉❛♥❞♦ ❛ ❡str✉t✉r❛ ❡♠ q✉❡stã♦ é ❡❧í♣t✐❝❛✳
❖ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ❛♥❛❧ít✐❝❛ ❧♦❝❛❧✱ t❛❧ ❝♦♠♦ ❢❡✐t♦ ❡♠ ❬❇❚❪✳ ❖ r❡s✉❧t❛❞♦ ♣r✐♥❝✐♣❛❧ ❛✜r♠❛ q✉❡✱ ♣❛r❛ ✉♠❛ ❡str✉t✉r❛ ✐♥✈♦❧✉t✐✈❛ ❛♥❛❧ít✐❝❛ ❞❡ ❝♦♣♦st♦ ✉♠ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛♥❛❧ít✐❝❛✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ❛♥❛❧ít✐❝❛ ❡♠ ✉♠ ♣♦♥t♦ é ❡q✉✐✈❛❧❡♥t❡ à ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❛❜❡rt❛ ❡♠ t♦r♥♦ ❞❡st❡ ♣♦♥t♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦ ❡♥✈♦❧✈❡rá ♦ ✉s♦ ❞♦ ❝❤❛♠❛❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝♦♥stâ♥❝✐❛ ❧♦❝❛❧ ❞❛s s♦❧✉çõ❡s ❞❛ ❡str✉t✉r❛ ❡♠ q✉❡stã♦✳ ❍♦❥❡ s❡ ❝♦♥❤❡❝❡♠ ✈❡rsõ❡s ♠❛✐s ❢♦rt❡s ❞❡ss❡ ♣r✐♥❝í♣✐♦✱ ✐♥❝❧✉s✐✈❡ s❡♠ r❡str✐çõ❡s s♦❜r❡ ♦ ♣♦st♦ ❞❛ ❡str✉t✉r❛✱ ♠❛s ❛ ✈❡rsã♦ ❡ ❛ ❞❡♠♦♥str❛çã♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s sã♦ ❛ ❢♦r♠❛ ❝♦♠♦ ❡ss❡ ✐♠♣♦rt❛♥t❡ r❡s✉❧t❛❞♦ ❢♦✐ ♦r✐❣✐♥❛❧✲ ♠❡♥t❡ ❞❡s❝♦❜❡rt♦✳ ❖ ♣r✐♠❡✐r♦ ❡♥✉♥❝✐❛❞♦ ❞♦ ♣r✐♥❝í♣✐♦ ❛ s❡r ❛♣r❡s❡♥t❛❞♦ ❛q✉✐✱ tr❛t❛♥❞♦
✈✐✐✐
❞❡ s♦❧✉çõ❡s ❝♦♥tí♥✉❛s✱ ❡♥✈♦❧✈❡ ✉♠ ❡st✉❞♦ ♠❡t✐❝✉❧♦s♦ ❞❛s ✜❜r❛s ❞❡ ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦✱ ❡ s✉❛ ❞❡♠♦♥str❛çã♦ ❞❡♣❡♥❞❡rá ❞❡❝✐s✐✈❛♠❡♥t❡ ❞♦ ❢❛t♦ ❞♦ ❝♦♣♦st♦ s❡r ✐❣✉❛❧ ❛ ✉♠✳ ❖ s❡❣✉♥❞♦ ❡♥✉♥❝✐❛❞♦✱ ❛ r❡s♣❡✐t♦ ❞❡ s♦❧✉çõ❡s ❞✐str✐❜✉❝✐♦♥❛✐s✱ ❡♥✈♦❧✈❡ ✉♠ ❡st✉❞♦ ❞❛ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ♣❛r❝✐❛❧ ❞♦s ♦♣❡r❛❞♦r❡s ❡♠ q✉❡stã♦ ✭❛♣ós ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦✲ ♦r❞❡♥❛❞❛s ❝♦♥✈❡♥✐❡♥t❡✮✱ ❡ ♣❛r❛ t❛♥t♦ ❢❡③✲s❡ ♥❡❝❡ssár✐♦ ❡①♣❧♦r❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡s♣❛ç♦s ✈❡t♦r✐❛✐s q✉❡ ❛q✉✐ ❝❤❛♠❛♠♦s ❞❡ ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♣❛r❝✐❛✐s✳ ❚r❛t❛✲s❡ ❞❡ ✉♠ ❝♦♥❝❡✐t♦ q✉❡ ♥ã♦ é ❡st✉❞❛❞♦ ❡♠ ❞❡t❛❧❤❡s ♥❛ ❜✐❜❧✐♦❣r❛✜❛✱ ❡ q✉❡ ♣♦r ✐ss♦ ❣❡r♦✉ ❛s ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ ♣r♦❥❡t♦✳
❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ tr❛t❛ ♥♦✈❛♠❡♥t❡ ❞❡ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ❛♥❛❧ít✐❝❛✱ ♠❛s ❛❣♦r❛ ♥♦ ❝❛s♦ ❣❧♦❜❛❧✱ s❡❣✉✐♥❞♦ ❬❇❩❪✳ ❉❛❞❛ ✉♠❛ ❡str✉t✉r❛ ✐♥✈♦❧✉t✐✈❛ ❛♥❛❧ít✐❝❛ ❞❡ ❝♦♣♦st♦ ✉♠ s♦❜r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❛♥❛❧ít✐❝❛✱ ❞❡✜♥✐♠♦s ♦s ❝❤❛♠❛❞♦s ❝♦♥❥✉♥t♦s ❞❡ ♥í✈❡❧✱ q✉❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❝♦♥s✐s✲ t❡♠ ❞❛s ✜❜r❛s ❧♦❝❛✐s ❞❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❝♦❧❛❞❛s ✉♠❛s às ♦✉tr❛s✳ ❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❝❛r❛❝t❡r✐③❛ ❛ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ❛♥❛❧ít✐❝❛ ❣❧♦❜❛❧ ❞❛ ❡str✉t✉r❛ ❡♠ t❡r♠♦s ❞❛ ❡①✐stê♥❝✐❛✱ ❡♠ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❞❡ ♥í✈❡❧✱ ❞❡ ✉♠ ♣♦♥t♦ ❡♠ t♦r♥♦ ❞♦ q✉❛❧ ❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s s❡❥❛♠ ❛❜❡r✲ t❛s✳ ❘❡ss❛❧t❛♠♦s q✉❡ ❡ss❛ ❝❛r❛❝t❡r✐③❛çã♦ é ✈á❧✐❞❛ ❛♣❡♥❛s s♦❜ ❝❡rt❛s ❤✐♣ót❡s❡s t♦♣♦❧ó❣✐❝❛s ❛ r❡s♣❡✐t♦ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ♥í✈❡❧✱ ❡ ♠❡♥❝✐♦♥❛♠♦s ✉♠ ❡①❡♠♣❧♦ ❡♠ q✉❡ ❛s ❤✐♣ót❡s❡s ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ sã♦ ✈✐♦❧❛❞❛s✳
❋✐♥❛❧♠❡♥t❡✱ ♥♦ q✉❛rt♦ ❝❛♣ít✉❧♦ r❡♣r♦❞✉③✐♠♦s ❡♠ ❞❡t❛❧❤❡s ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❋ór✲ ♠✉❧❛ ❞❡ ❆♣r♦①✐♠❛çã♦ ❞❡ ❇❛♦✉❡♥❞✐✲❚rè✈❡s✱ ❜❛s❡❛❞❛ ❡♠ ❬❇❈❍❪✳ ❊♠ ❧✐♥❤❛s ❣❡r❛✐s✱ tr❛t❛✲ s❡ ❞❡ ✉♠ t❡♦r❡♠❛ q✉❡ ❛✜r♠❛ q✉❡ t♦❞❛ s♦❧✉çã♦ s✉❛✈❡ ❞❡ ✉♠❛ ❡str✉t✉r❛ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✲ ✈❡❧ ♣♦❞❡ s❡r ❧♦❝❛❧♠❡♥t❡ ❛♣r♦①✐♠❛❞❛ ♣♦r ♣♦❧✐♥ô♠✐♦s ♥❛s ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s✳ ❊st❡ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ♦ r❡s✉❧t❛❞♦ ♠❛✐s ✐♠♣♦rt❛♥t❡ ❞❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ✐♥✈♦❧✉t✐✈♦s✱ s❡♥❞♦ ❡♠♣r❡❣❛❞♦ ♣❛r❛ ❛s ♠❛✐s ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s✳ ❖ ♣ró♣r✐♦ ♣r✐♥❝í♣✐♦ ❞❛ ❝♦♥stâ♥❝✐❛ ❧♦❝❛❧ t❛♠❜é♠ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ s✐♠♣❧❡s ❞❛ ❛♣r♦①✐♠❛çã♦ ❢♦r♥❡❝✐❞❛ ❢ór♠✉❧❛✳ ❊♥tr❡t❛♥t♦✱ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ tó♣✐❝♦ ♥ã♦ ♥♦s ♣❡r♠✐t✐✉ ❡st✉❞❛r ❛♣r♦①✐♠❛çã♦ ❞❡ s♦❧✉çõ❡s ❞✐str✐❜✉❝✐♦♥❛✐s ♣♦r ❡ss❡ ♠ét♦❞♦✱ ❡ ♣♦r ✐ss♦ ❢♦✐ ♥❡❝❡ssár✐♦ ❢❛③ê✲❧♦ ❝♦♠♦ ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✳
❆s s❡❣✉✐♥t❡s ❝♦♥✈❡♥çõ❡s ♥♦t❛❝✐♦♥❛✐s s❡rã♦ ✉t✐❧✐③❛❞❛s✿
✎ ❙❡ f : U ⊆ RN → C é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧✱ U ❛❜❡rt♦✱ ❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ ❞❡
f ❝♦♠ r❡❧❛çã♦ à ❝♦♦r❞❡♥❛❞❛ xj s❡rá ❞❡♥♦t❛❞❛ às ✈❡③❡s ♣♦r
∂f
∂xj✱ às ✈❡③❡s ♣♦r
∂xjf✳
❉✐r❡♠♦s q✉❡ f é s✉❛✈❡ s❡ é ❞❡ ❝❧❛ss❡C∞
❡♠U✳
✎ ❆♥❛❧ít✐❝♦ s❡♠♣r❡ s❡ r❡❢❡r❡ ❛ ❛♥❛❧ít✐❝♦ r❡❛❧✱ ❡ ❤♦❧♦♠♦r❢♦ é ✉s❛❞♦ ♣❛r❛ ❛♥❛❧ít✐❝♦
❝♦♠♣❧❡①♦✳ ❙❡ U⊆RN é ❛❜❡rt♦✱ Cω(U) ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❛♥❛❧ít✐❝❛s ❡♠
U✳ ❙❡ V ⊆CN é ❛❜❡rt♦✱ O(V) ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❤♦❧♦♠♦r❢❛s ❡♠ V✳
✎ ❖ s✉♣♦rt❡ s✐♥❣✉❧❛r ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ u✱ ❞❡♥♦t❛❞♦ ♣♦r s✐♥❣s✉♣♣u✱ é ♦ ❝♦♠♣❧❡✲
♠❡♥t❛r ❞♦ ♠❛✐♦r ❛❜❡rt♦ ♦♥❞❡u é ✉♠❛ ❢✉♥çã♦ C∞
✳
✎ ❙❡ v1, . . . , vn sã♦ ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✱ ♦ ❡s♣❛ç♦ ❣❡r❛❞♦ ♣♦r ❡❧❡s s❡rá
✐①
✎ Mm(K)❞❡♥♦t❛ ♦ ❡s♣❛ç♦ ❞❛s ♠❛tr✐③❡s q✉❛❞r❛❞❛sm×m ❝♦♠ ❡❧❡♠❡♥t♦s ❡♠ ✉♠ ❝♦r♣♦
K✳
✎ ◗✉❛♥❞♦ γ: [a, b]→X é ✉♠❛ ❝✉r✈❛ ♥✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ❞❡♥♦t❛♠♦s t❛♠❜é♠ ♣♦r
γ s✉❛ ✐♠❛❣❡♠✳
✎ ❙❡ pé ✉♠ ♣♦♥t♦ ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦✱ ❞✐③❡r q✉❡ ✉♠❛ ❝❡rt❛ ♣r♦♣r✐❡❞❛❞❡ ✈❛❧❡ ❡♠
t♦r♥♦ ❞❡ ps✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ p♦♥❞❡ ❡❧❛ é ✈❡r❞❛❞❡✐r❛✳
✎ ❉✐③❡♠♦s q✉❡ ✉♠ ❛❜❡rt♦ A ❞❡ ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ❡stá ❝♦♠♣❛❝t❛♠❡♥t❡ ❝♦♥t✐❞♦
❡♠ ✉♠ ❛❜❡rt♦ B✱ ❡ ❡s❝r❡✈❡♠♦s A⊂⊂B✱ q✉❛♥❞♦ A⊂B✳
✎ ◆♦ ❝♦♥t❡①t♦ ❞❡ ✈❛r✐❡❞❛❞❡s✱ ♦ t❡r♠♦ ❝❛rt❛ ❧♦❝❛❧ ❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦ p ♣♦❞❡ s❡
r❡❢❡r✐r ❛ ✉♠ ❛❜❡rt♦ tr✐✈✐❛❧✐③❛♥t❡ U✱ ❛ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❡✉❝❧✐❞✐❛♥❛s ❧♦❝❛✐s
❮♥❞✐❝❡
✶ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s ✶
✶✳✶ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
✶✳✷ ❱❡t♦r❡s t❛♥❣❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸ ❋♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❈❛♠♣♦s ❡ ❢♦r♠❛s r❡❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❚❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✻ ❊str✉t✉r❛s ❤♦❧♦♠♦r❢❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✼ ❈♦♥❥✉♥t♦ ❝❛r❛❝t❡ríst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✽ ❚✐♣♦s ❞❡ ❡str✉t✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✾ ❊str✉t✉r❛s ❛♥❛❧ít✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✶✵ ●❡r❛❞♦r❡s ❧♦❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✶✶ ❊str✉t✉r❛s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✶✳✶✷ ❚❡♦r❡♠❛ ❞❡ ◆✐r❡♥❜❡r❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
✷ ❘❡❣✉❧❛r✐❞❛❞❡ ❆♥❛❧ít✐❝❛ ▲♦❝❛❧ ✹✺
✷✳✶ Pr✐♥❝í♣✐♦ ❞❛ ❝♦♥stâ♥❝✐❛ ❧♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✷✳✷ ❙♦❧✉çõ❡s ❞✐str✐❜✉❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✷✳✸ ❈♦♥s❡q✉ê♥❝✐❛s ♣❛r❛ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✹ ✷✳✹ ❍✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ♣❛r❝✐❛❧ ❞❛s s♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✸ ❘❡❣✉❧❛r✐❞❛❞❡ ❆♥❛❧ít✐❝❛ ●❧♦❜❛❧ ✼✾
①✐✐ ❮◆❉■❈❊
✸✳✶ ❈♦♥❥✉♥t♦s ❞❡ ♥í✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✸✳✷ ❍✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ❛♥❛❧ít✐❝❛ ❣❧♦❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹
❈❛♣ít✉❧♦
✶
❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
❙✐st❡♠❛s ✐♥✈♦❧✉t✐✈♦s ❞❡ ❝❛♠♣♦s ✈❡t♦r✐❛✐s ❝♦♠♣❧❡①♦s sã♦ ♦❜❥❡t♦s ❛❜str❛t♦s ❞❡✜♥✐❞♦s ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ ❝❛r❛❝t❡r✐③❛❞♦s ♣♦r ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦s t♦r♥❛♠ ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❧✐♥❡❛r❡s ❤♦♠♦❣ê♥❡♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ♦ q✉❡ ♥♦s ♣❡r♠✐t❡ ❡st✉❞❛r s✐st❡♠❛s ❞❡st❡s ú❧t✐♠♦s ❞❡✜♥✐❞♦s s♦❜r❡ ✈❛r✐❡❞❛❞❡s ❞✐❢❡r❡♥❝✐á✈❡✐s✳
❆♦ ❧♦♥❣♦ ❞❡st❡ ❝❛♣ít✉❧♦✱Ω r❡♣r❡s❡♥t❛rá ✉♠❛ ✈❛r✐❡❞❛❞❡ N✲❞✐♠❡♥s✐♦♥❛❧✱ N=1, 2, . . .
✶✳✶ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s
◆❛ ●❡♦♠❡tr✐❛✱ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s s✉❛✈❡ ❡♠ Ω é ✉♠❛ ❛♣❧✐❝❛çã♦ L : Ω → TΩ ❞❡
❝❧❛ss❡ C∞
q✉❡ ❛ ❝❛❞❛ ♣♦♥t♦ p ∈ Ω ❛ss♦❝✐❛ ✉♠ ✈❡t♦r Lp ∈ TpΩ ✭❛q✉✐ TΩ r❡♣r❡s❡♥t❛ ♦
✜❜r❛❞♦ t❛♥❣❡♥t❡ ❞❛ ✈❛r✐❡❞❛❞❡✱ ❡ TpΩ é ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❡♠ p✮✳ ❊♥tr❡t❛♥t♦✱ t❡♥❞♦✲s❡
❡♠ ✈✐st❛ ❛♣❧✐❝❛r ❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ✐♥✈♦❧✉t✐✈♦s às ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✱ é ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ❢♦r♠❛❧✐③❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ✭s✉❛✈❡✱ ❝♦♠♣❧❡①♦✮ ❡♠ Ω é ✉♠❛ ❛♣❧✐❝❛çã♦
C✲❧✐♥❡❛r
L:C∞
(Ω)→C∞
(Ω)
✭♦♥❞❡ C∞
(Ω)é ♦ C✲♠ó❞✉❧♦ ❞❛s ❢✉♥çõ❡s s✉❛✈❡s ❞❡ Ω❛ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s✮ q✉❡ s❛t✐s❢❛③
❛ ❘❡❣r❛ ❞❡ ▲❡✐❜♥✐③✿
L(fg) =fLg+gLf , f, g∈C∞
(Ω)
❉❡♥♦t❛✲s❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞❡ Ω ♣♦r X(Ω)✳
✷ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
❖ ❡①❡♠♣❧♦ ❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♠❛✐s tr✐✈✐❛❧✱ ❡ ♦ q✉❡ ♥♦s s❡rá ♠❛✐s ✐♠♣♦rt❛♥t❡✱ é ♦ ❞❡
✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ❤♦♠♦❣ê♥❡♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✱ ❛ ❝♦❡✜❝✐❡♥t❡sC∞
✳
❙❡❣✉❡ ❞❛ ❘❡❣r❛ ❞❡ ▲❡✐❜♥✐③ q✉❡ s❡ f é ✉♠❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ ❡♥tã♦ Lf =0 ♣❛r❛ t♦❞♦
❝❛♠♣♦ L✳ ❉❡ ❢❛t♦✱ ♣♦r C✲❧✐♥❡❛r✐❞❛❞❡ ❜❛st❛ ✈❡r♠♦s q✉❡ L1 = 0✱ ❡ ✐st♦ é ❝♦♥s❡q✉ê♥❝✐❛ ❞❡
L1=L(1·1) =1(L1) + (L1)1=2(L1)✳ ❊st❛ ♣r♦♣r✐❡❞❛❞❡ s❡rá ✉s❛❞❛ ♣❛r❛ ❞❡♠♦♥str❛r♠♦s✿
Pr♦♣♦s✐çã♦ ✶✳✷✳ ❙❡ f∈C∞
(Ω) ❡ L∈X(Ω)✱ ❡♥tã♦ s✉♣♣Lf⊆s✉♣♣f✳
❖ sí♠❜♦❧♦ s✉♣♣ ✐♥❞✐❝❛ ♦ s✉♣♦rt❡ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ✐st♦ é✱ ♦ ❢❡❝❤♦ ❞♦ ❝♦♥❥✉♥t♦ ♦♥❞❡ ❡❧❛ ♥ã♦ s❡ ❛♥✉❧❛✳
❉❡♠♦♥str❛çã♦✿ ❉❡✈❡✲s❡ ♣r♦✈❛r q✉❡ s❡ f s❡ ❛♥✉❧❛ ❡♠ ✉♠ ❛❜❡rt♦ V ❡♥tã♦ Lf t❛♠❜é♠ s❡
❛♥✉❧❛ ❛❧✐✳ ❉❛❞♦ p∈V✱ s❡❥❛ g ∈C∞
(Ω) t❛❧ q✉❡ f= (1−g)f❡ g(p) =1✳ ❯♠❛ t❛❧ ❢✉♥çã♦
r❡❛❧♠❡♥t❡ ❡①✐st❡✿ ❜❛st❛ ❡s❝♦❧❤❡r ✉♠ ❛❜❡rt♦ U✱ ❝♦♠ p ∈ U ⊂⊂ V✱ ❡ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ φ
❝♦♠ s✉♣♦rt❡ ❡♠ U t❛❧ q✉❡ φ(p) =1✱ ❡ ❡♥tã♦ ❞❡✜♥✐r ❛ ❢✉♥çã♦ ✭s✉❛✈❡✮ g=φ ❡♠ U ❡ g≡0
❡♠Ω\U✳ ❉❡ss❛ ❢♦r♠❛✱ f= (1−g)f❡♠V ♣♦rq✉❡ ❛♠❜♦s ♦s ♠❡♠❜r♦s sã♦ ♥✉❧♦s✱ ❡ ❛ ♠❡s♠❛
✐❣✉❛❧❞❛❞❡ ✈❛❧❡ ❢♦r❛ ❞❡ V ♣♦rq✉❡ ❛❧✐ 1−g≡1✳ ❚❡♠♦s ♣♦rt❛♥t♦
Lf(p) =L((1−g)f)(p) = (1−g(p))Lf(p) +f(p)L(1−g)(p) =0
♣♦✐s 1−g(p) =f(p) =0✳
❯♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡st❡ r❡s✉❧t❛❞♦ ❡ ❞❛ ❧✐♥❡❛r✐❞❛❞❡ ❞♦s ❝❛♠♣♦s é q✉❡✱ ❛ss✐♠ ❝♦♠♦ é ♦
❝❛s♦ ❝♦♠ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♦ ✈❛❧♦r ❞❡ Lf♥✉♠ ♣♦♥t♦ só ❞❡♣❡♥❞❡ ❞♦s ✈❛❧♦r❡s ❞❡ f❡♠
q✉❛❧q✉❡r ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ss❡ ♣♦♥t♦✳ ❈♦♠ ❡st❛ ♣r♦♣r✐❡❞❛❞❡✱ ♠♦str❛✲s❡ ❛ ❜♦❛✲❞❡✜♥✐çã♦
❞♦ ❝♦♥❝❡✐t♦ ❞❡ r❡str✐çã♦ ❞❡ ✉♠ ❝❛♠♣♦ ❛ ✉♠ ❛❜❡rt♦✿ s❡ U ⊆ Ω é ❛❜❡rt♦ ✭❧♦❣♦ é ✉♠❛
✈❛r✐❡❞❛❞❡ N✲❞✐♠❡♥s✐♦♥❛❧✮ ❡ L∈X(Ω)✱ ❞❡✜♥❡✲s❡ ♦ ❝❛♠♣♦ L|U ∈X(U) ♣♦r
L|Uf(p) =Lf✄(p) , f∈C
∞
(U) , p∈U
♦♥❞❡ ✄f∈C∞
(Ω)é q✉❛❧q✉❡r ❢✉♥çã♦ q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♠f♣❡rt♦ ❞❡p✱ ❝♦♥str✉í❞❛✱ ♣♦r ❡①❡♠♣❧♦✱
❛tr❛✈és ❞❡ ❢✉♥çõ❡s ❞❡ ❝♦rt❡✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❞✐❛❣r❛♠❛ ❛❜❛✐①♦ t♦r♥❛✲s❡ ❝♦♠✉t❛t✐✈♦ ✭s❡t❛s ✈❡rt✐❝❛✐s ✐♥❞✐❝❛♠ ❛ r❡str✐çã♦ ✉s✉❛❧ ❞❡ ❢✉♥çõ❡s✮✿
C∞
(Ω) L //
C∞
(Ω)
C∞
(U)
L|U //C
∞
(U)
❉❡✜♥✐çã♦ ✶✳✸✳ ❯♠❛ s♦❧✉çã♦ ❞❡ ✉♠ ❝❛♠♣♦ L ❡♠ ✉♠ ❛❜❡rt♦ U ⊆ Ω é ✉♠❛ ❢✉♥çã♦
s✉❛✈❡ u∈C∞
(U) t❛❧ q✉❡ LU(u) =0✳
❉✉❛s ♦♣❡r❛çõ❡s ✐♠♣♦rt❛♥t❡s ❝♦♠ ❝❛♠♣♦s sã♦ ♦ ♣r♦❞✉t♦ ♣♦r ❢✉♥çõ❡s✱ q✉❡ tr❛♥s❢♦r♠❛
X(Ω)❡♠ ✉♠ C∞
(Ω)✲♠ó❞✉❧♦✱ ❡ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❞❛❞♦sg∈C∞
✶✳✶ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s ✸
❡ L, M∈X(Ω)✱ ❞❡✜♥✐♠♦s gL∈X(Ω) ❡ [L, M]∈X(Ω) ♣♦r
(gL)f=g·(Lf) , [L, M]f=L(Mf) −M(Lf) , f∈C∞
(Ω)
❱❡r✐✜❝❛r q✉❡ r❡❛❧♠❡♥t❡ s❡ tr❛t❛ ❞❡ ❝❛♠♣♦s r❡q✉❡r ❛♣❡♥❛s ❝á❧❝✉❧♦s ❞✐r❡t♦s✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❘❡❣r❛ ❞❡ ▲❡✐❜♥✐③ ✈❛❧❡ ♣❛r❛ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡✿
[L, M](fg) =L(M(fg)) −M(L(fg)) =L(fMg+gMf) −M(fLg+gLf)
=f(L(Mg)) + (Lf)(Mg) +g(L(Mf)) + (Lg)(Mf) −
−f(M(Lg)) − (Mf)(Lg) −g(M(Lf)) − (Mg)(Lf)
=f([L, M]g) +g([L, M]f)
❖❜s❡r✈❛çã♦ ✶✳✹✳ ❆ss✐♠ ❝♦♠♦ q✉❛❧q✉❡r ♦❜❥❡t♦ ❞❡✜♥✐❞♦ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡✱ t❛♠❜é♠ ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s tê♠ ✉♠❛ ❢♦r♠❛ ♣❛❞rã♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✳ ❙❡❥❛
p∈Ω❡ s❡❥❛(U, x)✉♠❛ ❝❛rt❛ ❧♦❝❛❧✱ ✐st♦ é✱U⊆Ωé ❛❜❡rt♦✱p∈U✱ ❡x = (x1, . . . , xN) :U→
RN é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ s✉❛✈❡✳ ❆sxjsã♦ ❛s ❝❤❛♠❛❞❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✳ ❉❡♥♦t❡
♣♦r a= (a1, . . . , aN) ♦ ♣♦♥t♦ x(p) ∈RN✳ ❊♥tã♦ f◦x−1 :x(U)→ C é ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✱ ❡
❛ss✐♠ ❡①✐st❡ ✉♠ ❛❜❡rt♦ V ⊆RN✱ ❝♦♠ a∈ V✱ ❡ ❡①✐st❡♠ ❢✉♥çõ❡s s✉❛✈❡s h
1, . . . , hN :V →C
t❛✐s q✉❡
(f◦x−1)(y) = (f◦x−1)(a) +
N X
j=1
hj(y)(yj−aj) , y∈V ✭✶✳✺✮
❉❡ ❢❛t♦✱ ❡st❛ é ✉♠❛ ❛♣❧✐❝❛çã♦ ♣❛❞rã♦ ❞♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦ ❝á❧❝✉❧♦✳ ❆s ❢✉♥çõ❡s hj
sã♦ ❞❛❞❛s ♣♦r
hj(y) = Z1
0
∂(f◦x−1)
∂yj
(a1+t(y1−a1), . . . , aN+t(yN−aN))dt
❚r❛♥s❢❡r✐♥❞♦ ♦s ♣♦♥t♦s ❛ U⊆Ω ❛tr❛✈és ❞♦ ❞✐❢❡♦♠♦r✜s♠♦x✱ ♦❜t❡♠♦s ❞❡ ✭✶✳✺✮ ❛❝✐♠❛ q✉❡
f(q) =f(p) +
N X
j=1
(hj◦x)(q)(xj(q) −xj(p)) , q∈x−1(V) ✭✶✳✻✮
▲♦❣♦✱ s❡ L∈X(Ω)✿
Lf(q) =0+
N X
j=1
L(hj◦x)(q)(xj(q) −xj(p)) + (hj◦x)(q)Lxj(q)
❋✐♥❛❧♠❡♥t❡✱ ❢❛③❡♥❞♦ q=p✿
Lf(p) =
N X
j=1
(hj◦x)(p)Lxj(p) =
N X
j=1
∂(f◦x−1)
∂yj
(x(p))Lxj(p) ✭✶✳✼✮
❆s N❛♣❧✐❝❛çõ❡s
C∞
(U)∋f7→ ∂(f◦x
−1)
∂yj ◦
x∈C∞
✹ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞❡ U✱ ❞❡♥♦t❛❞♦s ♣♦r ∂
∂xj
♦✉ s✐♠♣❧❡s♠❡♥t❡ ∂xj✳ ➱ s✐♠♣❧❡s ✈❡r✱
❛♣❧✐❝❛♥❞♦✲♦s s❡♣❛r❛❞❛♠❡♥t❡ ❛ ❢✉♥çõ❡s q✉❡ ❞❡♣❡♥❞❡♠ ❛♣❡♥❛s ❞❡ ❝❛❞❛ ✈❛r✐á✈❡❧ yj✱ q✉❡ ❡s✲
s❡s ❝❛♠♣♦s sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s s♦❜r❡ ♦ ❛♥❡❧ C∞
(U)✳ ❯t✐❧✐③❛♥❞♦✲♦s✱ ♣♦❞❡♠♦s
❡s❝r❡✈❡r ❛ ❡q✉❛çã♦ ✭✶✳✼✮ ❛❝✐♠❛ ❝♦♠♦
L=
N X
j=1
(Lxj)
∂ ∂xj
▲♦❣♦ {∂xj ; j=1, . . . , N} ❝♦♥st✐t✉✐ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ C ∞
(U)✲♠ó❞✉❧♦X(U)✳
❚❡♦r❡♠❛ ✶✳✽✳ ✭❘❡❣r❛ ❞❛ ❈❛❞❡✐❛✮ ❙❡❥❛♠ L ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s s♦❜r❡ Ω ❡ u ∈
C∞
(Ω)✳ ❙❡ f é ❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ ✉♠ ❛❜❡rt♦ ❝♦♥t✐❞♦ ❡♠ u(Ω)⊆C ✭❝♦♥s✐❞❡r❛♥❞♦✲s❡ f
❝♦♠♦ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ❡♠ ✉♠ ❛❜❡rt♦ ❞❡ R2✮✱ ❡♥tã♦
L(f◦u)(p) = ∂f
∂ζ(u(p))Lu(p) + ∂f
∂ζ(u(p))Lu(p)
❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ u é s♦❧✉çã♦ ❞❡ L ❡ f é ❤♦❧♦♠♦r❢❛✱ ❡♥tã♦ f◦u t❛♠❜é♠ é s♦❧✉çã♦ ❞❡
L✳
❉❡♠♦♥str❛çã♦✿ P♦❞❡♠♦s ♣r♦✈❛r ♦ r❡s✉❧t❛❞♦ ❧♦❝❛❧♠❡♥t❡✱ ❡ ♣♦rt❛♥t♦ ❛ss✉♠✐r q✉❡ L é
✉♠❛ ❝♦♠❜✐♥❛çã♦ C∞✲❧✐♥❡❛r ❞❡ ❞❡r✐✈❛❞❛s
∂xj✳ ▼❛✐s ❞♦ q✉❡ ✐ss♦✱ ❞❛❞❛ ❛ C
∞✲❧✐♥❡❛r✐❞❛❞❡ ❞❛
❡q✉❛çã♦ ❛ s❡r ♣r♦✈❛❞❛✱ ♥ã♦ ❤á ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ❡♠ ❛ss✉♠✐r q✉❡ L=∂xj ♣❛r❛ ❛❧❣✉♠
j = 1, . . . , N✳ ■❞❡♥t✐✜q✉❡ ❛ ✐♠❛❣❡♠ ❞❡ u ❡♠ C ❝♦♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ R2✱ ♠❛s ❝♦♠
✏❝♦♦r❞❡♥❛❞❛s✑ (ζ, ζ)✳ ❆ r✐❣♦r✱ ♥ã♦ s❡ tr❛t❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ♣♦✐s ζ ❡ ζ ♥ã♦ sã♦ ♥ú♠❡r♦s
r❡❛✐s✱ ♠❛s t♦❞♦s ♦s ♣r♦❝❡❞✐♠❡♥t♦s ♣❛❞rõ❡s ❞♦ ❈á❧❝✉❧♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ✉s✉❛❧✱ ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s ❛ (R2,(ζ, ζ)) ❝♦♠♦ s❡ ❢♦ss❡♠✳ ❉❡ss❛ ❢♦r♠❛ ❛s ❢✉♥çõ❡s
❝♦♦r❞❡♥❛❞❛s ❞❡ u sã♦ u ❡ u✳ ■❞❡♥t✐✜q✉❡ t❛♠❜é♠ ❛ ✐♠❛❣❡♠ ❞❡ f ❡♠ C ❝♦♠ R2✱ ❝♦♠ ❛s
❝♦♦r❞❡♥❛❞❛s ✉s✉❛✐s (ℜζ,ℑζ)❀ ♣♦rt❛♥t♦ ❛s ❢✉♥çõ❡s ❝♦♦r❞❡♥❛❞❛s ❞❡ f sã♦ ℜf ❡ ℑf✳ ◆❡ss❛s
❝♦♥❞✐çõ❡s✱ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ✉s✉❛❧ ❛✜r♠❛ q✉❡ ∂xj(f◦u) é ❞❛❞♦ ♣❡❧♦ ✈❡t♦r (v1, v2)✱ ♦♥❞❡
v1=
∂ℜf ∂ζ ◦u
∂u ∂xj
+
∂ℜf ∂ζ ◦u
∂u ∂xj
v2=
∂ℑf ∂ζ ◦u
∂u ∂xj
+
∂ℑf ∂ζ ◦u
∂u ∂xj
❉❡ ❛❝♦r❞♦ ❝♦♠ ♥♦ss❛ ú❧t✐♠❛ ✐❞❡♥t✐✜❝❛çã♦✱(v1, v2)❝♦rr❡s♣♦♥❞❡ ❛♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦v1+iv2✱
✐st♦ é✱ ❛♦ ♥ú♠❡r♦
∂ℜf ∂ζ ◦u
∂u ∂xj
+
∂ℜf ∂ζ ◦u
∂u ∂xj + +i
∂ℑf ∂ζ ◦u
∂u ∂xj
+
∂ℑf ∂ζ ◦u
∂u ∂xj = = ∂f ∂ζ ◦u
∂u ∂xj + ∂f ∂ζ ◦u
∂u ∂xj
✶✳✷ ❱❡t♦r❡s t❛♥❣❡♥t❡s ✺
✶✳✷ ❱❡t♦r❡s t❛♥❣❡♥t❡s
❆ss✐♠ ❝♦♠♦ ❢♦✐ ❢❡✐t♦ ❝♦♠ ♦s ❝❛♠♣♦s✱ é ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡ ❞❡✜♥✐r ✈❡t♦r t❛♥❣❡♥t❡ ♥ã♦ ❝♦♠♦ ♥❛ ●❡♦♠❡tr✐❛✱ ♠❛s ❝♦♠♦ ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❛❣❡ ❡♠ ❢✉♥çõ❡s✱ ♦✉ ♥❡st❡ ❝❛s♦ ❡♠
❣❡r♠❡s ❞❡ ❢✉♥çõ❡s✳ ❯♠ ❣❡r♠❡ ❞❡ ❢✉♥çõ❡s s✉❛✈❡s ❡♠ ✉♠ ♣♦♥t♦ p ∈Ω é ✉♠❛ ❝❧❛ss❡ ❞❡
❡q✉✐✈❛❧ê♥❝✐❛ ❞❛ r❡❧❛çã♦
f∼g ⇐⇒ f=g ❡♠ ❛❧❣✉♠ ❛❜❡rt♦ U∋p , f, g∈C∞
(Ω)
❖ ❣❡r♠❡ ❞❡ ✉♠❛ ❢✉♥çã♦ f s❡rá ❞❡♥♦t❛❞♦ t❛♠❜é♠ ♣♦rf✳ ❖ ❡s♣❛ç♦ ❞♦s ❣❡r♠❡s ❡♠ p é ✉♠
❛♥❡❧ q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r C∞
(p)✳
❉❡✜♥✐çã♦ ✶✳✾✳ ❯♠ ✈❡t♦r t❛♥❣❡♥t❡ ✭❝♦♠♣❧❡①♦✮ ❛ Ω ❡♠ p ∈ Ω é ✉♠❛ ❛♣❧✐❝❛çã♦
C✲❧✐♥❡❛r
ν:C∞
(p)→C
t❛❧ q✉❡
ν(fg) =f(p)ν(g) +g(p)ν(f) , f, g∈C∞
(p)
❖ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ✭❝♦♠♣❧❡①♦✮ ❛ Ω ❡♠p✱ ❞❡♥♦t❛❞♦ ♣♦r CTpΩ✱ é ♦C✲❡s♣❛ç♦ ✈❡t♦r✐❛❧
❞♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❡♠ p✳
❉❛❞♦ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s L s❡❣✉♥❞♦ ❛ ♥♦çã♦ ❣❡♦♠étr✐❝❛✱ ♣♦❞❡♠♦s ❢❛❧❛r ❡♠ s❡✉
✈❛❧♦r ♥✉♠ ♣♦♥t♦ p∈ Ω✿ é ♦ ✈❡t♦r t❛♥❣❡♥t❡ L(p)✳ ❆ ❛♥❛❧♦❣✐❛ ❝♦♠ ♦s ❝❛♠♣♦s ❝♦♠♦ ❢♦r❛♠
❞❡✜♥✐❞♦s ❛q✉✐ tr❛❞✉③ ❡ss❡ ❝♦♥❝❡✐t♦ ♥♦ s❡❣✉✐♥t❡✿ ❛ ❧♦❝❛❧✐③❛çã♦ ❞❡L❡♠pé ♦ ✈❡t♦r t❛♥❣❡♥t❡
Lp ❞❡✜♥✐❞♦ ♣♦r
Lp(f) =Lf(p) , f∈C
∞
(p)
❊ss❛ ♦♣❡r❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❛r❛ ❣❡r♠❡s ♣♦rq✉❡ ♦ ✈❛❧♦r ❞❡ Lf ❡♠ p ❞❡♣❡♥❞❡ ❞❡
f ❛♣❡♥❛s ❡♠ ❛❧❣✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ p✳ ❚❛♠❜é♠ é ❝❧❛r❛♠❡♥t❡ C∞
(Ω)✲❧✐♥❡❛r✱ ♥♦
s❡❣✉✐♥t❡ s❡♥t✐❞♦✿
(L1+fL2)p = (L1)p+f(p)(L2)p , L1, L2 ∈X(Ω), f ∈C
∞
(Ω)
❯t✐❧✐③❛♥❞♦✲s❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❧♦❝❛❧ ❞❡ ❝❛♠♣♦s✱ é s✐♠♣❧❡s ♣r♦✈❛r ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
Pr♦♣♦s✐çã♦ ✶✳✶✵✳ CTpΩ={Lp ; L∈X(Ω)} ♣❛r❛ t♦❞♦ p∈Ω✳
❙❡(U, x) é ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❞❡Ω❡♠ t♦r♥♦ ❞❡ p✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❡q✉❛çã♦ ✭✶✳✻✮ ❛❝✐♠❛ ❡
♦❜t❡r ❛ r❡♣r❡s❡♥t❛çã♦ ❧♦❝❛❧ ❞❡ ✉♠ ✈❡t♦r t❛♥❣❡♥t❡✳ ❉❡ ❢❛t♦✱ ❛q✉❡❧❛ ❡q✉❛çã♦ ❞✐③✐❛ q✉❡ ❡①✐st❡
✉♠ ❛❜❡rt♦ V ⊆U✱ ❝♦♥t❡♥❞♦p✱ ♦♥❞❡ q✉❛❧q✉❡r f∈C∞
(Ω) é ❞❛❞❛ ♣♦r
f(q) =f(p) +
N X
j=1
✻ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
❝♦♠ ❛s hj ❞❡✜♥✐❞❛s ❝♦♠♦ ❛♥t❡s✳ ▲♦❣♦✱ s❡ ν∈CTpΩ✱ t❡♠♦s
ν(f) =0+
N X
j=1
ν
(hj◦x)(·)(xj(·) −xj(p))
=
N X
j=1
ν
∂xjf(·)(xj(·) −xj(p))
=
N X
j=1
∂xjf(p)ν(xj) + (xj(p) −xj(p))ν(∂xjf)
=
N X
j=1
∂xjf(p)ν(xj) =
N X
j=1
ν(xj)(∂xj)p(f)
❈♦♠♦ fé ❛r❜✐trár✐❛✱ ❝♦♥❝❧✉í♠♦s q✉❡
ν=
N X
j=1
ν(xj)
∂ ∂xj
q
❆ss✐♠✱ ❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ♣❛r❛ CTpU s♦❜r❡ C é {(∂x1)p, . . . ,(∂xN)p}✳
❖ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ✭❝♦♠♣❧❡①✐✜❝❛❞♦✮ ❞❡ Ω é ❛ r❡✉♥✐ã♦ ❞✐s❥✉♥t❛ ❞♦s ❡s♣❛ç♦s t❛♥✲
❣❡♥t❡s ❛ ❝❛❞❛ ♣♦♥t♦ ❞❡ Ω✱ ❞❡♥♦t❛❞♦ ♣♦r CTΩ✳ ◆❛ ●❡♦♠❡tr✐❛✱ ✉♠ s✉❜✜❜r❛❞♦ ✭r❡❛❧✮ ❞❡st❡
✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♥s✐st❡ ❡♠ ✉♠❛ r❡✉♥✐ã♦ ❞✐s❥✉♥t❛ ❧♦❝❛❧♠❡♥t❡ tr✐✈✐❛❧ ❞❡ s✉❜❡s♣❛ç♦s ❞❡
❝❛❞❛ ❡s♣❛ç♦ t❛♥❣❡♥t❡✱ t♦❞♦s ❝♦♠ ♠❡s♠❛ ❞✐♠❡♥sã♦ r❡❛❧ n✳ ❆ tr✐✈✐❛❧✐❞❛❞❡ ❧♦❝❛❧ s✐❣♥✐✜❝❛
q✉❡✱ ❡♠ t♦r♥♦ ❞❡ ❝❛❞❛ ♣♦♥t♦ ❞❛ ✈❛r✐❡❞❛❞❡✱ ❡①✐st❡ ✉♠ ❛❜❡rt♦Ut❛❧ q✉❡ ♦ s✉❜✜❜r❛❞♦ r❡str✐t♦
❛ Ué ❞✐❢❡♦♠♦r❢♦ ❛ U×Rn✳ ◆❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ✐♥✈♦❧✉t✐✈♦s✱ ❡st❡ ❝♦♥❝❡✐t♦ s❡ tr❛♥s❢♦r♠❛
♥♦ s❡❣✉✐♥t❡✿
❉❡✜♥✐çã♦ ✶✳✶✶✳ ❯♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ✭❝♦♠♣❧❡①♦✮ ❞❡ CTΩ ❞❡ ♣♦st♦ ♥ ❡ ❝♦♣♦st♦
◆−♥ é ✉♠❛ r❡✉♥✐ã♦ ❞✐s❥✉♥t❛
V= [
p∈Ω
Vp ⊆CTΩ
♦♥❞❡ ❝❛❞❛ Vp é ✉♠ s✉❜❡s♣❛ç♦ ❞❡ CTpΩ ❞❡ ❞✐♠❡♥sã♦ n✱ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡
tr✐✈✐❛❧✐❞❛❞❡ ❧♦❝❛❧✿ ♣❛r❛ ❝❛❞❛ p ∈ Ω✱ ❡①✐st❡♠ ✉♠ ❛❜❡rt♦ U ∋ p ❡ ❝❛♠♣♦s L1, . . . , Ln ∈
X(U) t❛✐s q✉❡✱ ♣❛r❛ t♦❞♦ q∈ U✱ Vq é ❣❡r❛❞♦ ♣❡❧♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s (L1)q, . . . ,(Ln)q✳
❚❛♠❜é♠ ♣♦❞❡♠♦s ♥♦s r❡❢❡r✐r ❛ V ❝♦♠♦ ✉♠ s✉❜✜❜r❛❞♦ t❛♥❣❡♥t❡✳ ❉✐③✲s❡ q✉❡ ♦s
❝❛♠♣♦s Lj ❣❡r❛♠ V ❡♠ t♦r♥♦ ❞❡ ♣✳ ❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ Vp é ❞✐t♦ ❛ ✜❜r❛ ❞❡ V ❡♠ p✳
❯♠❛ s❡çã♦ ❞♦ s✉❜✜❜r❛❞♦ Vs♦❜r❡ ✉♠ ❛❜❡rt♦ W ⊆Ωé ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s L∈X(W)
t❛❧ q✉❡ Lp ∈Vp ♣❛r❛ ❝❛❞❛ p∈W✳
✶✳✷ ❱❡t♦r❡s t❛♥❣❡♥t❡s ✼
❉❡✜♥✐çã♦ ✶✳✶✷✳ ❯♠❛ ❡str✉t✉r❛ ✐♥✈♦❧✉t✐✈❛ s♦❜r❡ Ω ❞❡ ♣♦st♦ ♥✱ t❛♠❜é♠ ❝❤❛♠❛❞❛
❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ é ✉♠ s✉❜✜❜r❛❞♦ V ❞❡ ♣♦st♦ n s❛t✐s❢❛③❡♥❞♦ ❛
❝♦♥❞✐çã♦ ✐♥✈♦❧✉t✐✈❛ ❞❡ ❋r♦❜❡♥✐✉s✿ s❡ W ⊆ Ω é ❛❜❡rt♦ ❡ L, M ∈X(W) sã♦ s❡çõ❡s ❞❡ V
s♦❜r❡ W✱ ❡♥tã♦ [L, M] t❛♠❜é♠ é s❡çã♦ ❞❡ V s♦❜r❡ W✳
❯♠❛ s♦❧✉çã♦ ❝❧áss✐❝❛ ❞❛ ❡str✉t✉r❛ ✐♥✈♦❧✉t✐✈❛Vs♦❜r❡ ✉♠ ❛❜❡rt♦W ⊆Ωé ✉♠❛ ❢✉♥çã♦
s✉❛✈❡ u∈ C∞
(W) t❛❧ q✉❡ Lu=0 ♣❛r❛ t♦❞❛ s❡çã♦ L ❞❡ V s♦❜r❡ W✳ P♦❞❡✲s❡ ❞❡♥♦t❛r ❡st❡
❢❛t♦ ❡s❝r❡✈❡♥❞♦✲s❡ q✉❡ Vu =0 ❡♠ W✳ ❯♠❛ s♦❧✉çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ V s♦❜r❡ W é ✉♠❛
❞✐str✐❜✉✐çã♦ u∈ D′(W)t❛❧ q✉❡Lu=0 ♣❛r❛ t♦❞❛ s❡çã♦L❞❡Vs♦❜r❡W✳ ◆ã♦ ❛❜♦r❞❛r❡♠♦s
❞✐str✐❜✉✐çõ❡s ❡♠ ✈❛r✐❡❞❛❞❡s ♥❡st❡ tr❛❜❛❧❤♦✱ ♠❛s✱ ❛♦ ❧♦♥❣♦ ❞♦ ♠❡s♠♦✱ ❡♥❝♦♥tr❛r❡♠♦s ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❜✉s❝❛r ♣♦r s♦❧✉çõ❡s ❣❡♥❡r❛❧✐③❛❞❛s ❞❡ ❡str✉t✉r❛s ✐♥✈♦❧✉t✐✈❛s✳ ❊♥tr❡t❛♥t♦✱ ❡ss❡s ❝❛s♦s s❡♠♣r❡ s❡rã♦ ❧✐❞❛❞♦s s♦❜r❡ ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✱ ❡ ❛❧✐ ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ♥ã♦
♣❛ss❛♠ ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❧✐♥❡❛r❡s✱ ❡ ♥❡ss❡ ❝♦♥t❡①t♦ ❛ ❡①♣r❡ssã♦ Lu t❡♠
s❡♥t✐❞♦ ♣❛r❛ ✉♠❛ ❞✐str✐❜✉✐çã♦ u✳
❖❜s❡r✈❡ q✉❡✱ ❝❛s♦ ✉♠❛ ❡str✉t✉r❛ V s❡❥❛ ❣❡r❛❞❛ ♣♦r ❝❛♠♣♦s L1, . . . , Ln✱ ❡♥tã♦ V s❡rá
✐♥✈♦❧✉t✐✈❛ s❡ ❡ s♦♠❡♥t❡ s❡ ❝❛❞❛ [Lj, Lk] ❢♦r ✉♠❛ s❡çã♦ ❞❡ V s♦❜r❡ Ω✱ ❡ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡
u∈C∞
(W) s❡rá ✉♠❛ s♦❧✉çã♦ ❞❡V s❡ ❡ s♦♠❡♥t❡ s❡ Lju=0 ❡♠ W ♣❛r❛ ❝❛❞❛ j=1, . . . , n✳
❚❡♦r❡♠❛ ✶✳✶✸✳ ❙❡ V é ✉♠❛ ❡str✉t✉r❛ ✐♥✈♦❧✉t✐✈❛ ❞❡ ♣♦st♦ n ❡ s❡ ♦s ❝❛♠♣♦s L1, . . . , Ln
❣❡r❛♠ V ❡♠ t♦r♥♦ ❞❡ p ∈ Ω✱ ❡♥tã♦ ❡①✐st❡♠ ❢✉♥çõ❡s λl
jk ∈ C
∞
(Ω)✱ j, k, l = 1, . . . , n✱
t❛✐s q✉❡✱ ❡♠ t♦r♥♦ ❞❡ p✱ ✈❛❧❡
[Lj, Lk] = n X
l=1
λljkLl , j, k=1, . . . , n
❉❡♠♦♥str❛çã♦✿ P♦r ✐♥✈♦❧✉t✐✈✐❞❛❞❡✱ ❝♦♠♦ ♣❡rt♦ ❞❡p❛ ❡str✉t✉r❛ é ❣❡r❛❞❛ ♣❡❧♦s Lj✱ t❡♠♦s
q✉❡✱ ♣❛r❛ ❝❛❞❛ q s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦ ❛ p✱ ❡①✐st❡♠ λν
jk(q) ∈ C✱ j, k, ν= 1, . . . , n✱ t❛✐s
q✉❡
[Lj, Lk]q = n X
ν=1
λνjk(q)(Lν)q
❖ tr❛❜❛❧❤♦ ❛❣♦r❛ é ♣r♦✈❛r q✉❡ ❛s ❢✉♥çõ❡s λν
jk ❛ss✐♠ ❞❡✜♥✐❞❛s sã♦ s✉❛✈❡s✱ ♣♦rq✉❡ s❡ ✐st♦
❡st✐✈❡r ♣r♦✈❛❞♦ ❡♥tã♦ t❡r❡♠♦s✱ ♣❛r❛ t♦❞♦ q♣ró①✐♠♦ ❛ p ❡ t♦❞❛ f∈C∞
(Ω)✱
[Lj, Lk](f)(q) = [Lj, Lk]q(f) = n X
ν=1
λνjk(q)(Lν)q(f) = n X
ν=1
λνjk(q)Lν(f)(q)
❙❡ ❧❡♠❜r❛r♠♦s q✉❡ ❛s ❢✉♥çõ❡s λν
jk ♣♦❞❡♠ s❡r ❡st❡♥❞✐❞❛s ❛ Ω ❝♦♠ ❢✉♥çõ❡s ❞❡ ❝♦rt❡✱ ♦
t❡♦r❡♠❛ ❡st❛rá ♣r♦✈❛❞♦✳
❉✐♠✐♥✉✐♥❞♦ s❡ ♥❡❝❡ssár✐♦ ❛ ✈✐③✐♥❤❛♥ç❛ ❞❡p❝♦♥s✐❞❡r❛❞❛✱ t♦♠❡✲❛ ❝♦♠♦ ✉♠❛ ❝❛rt❛ ❧♦❝❛❧
U✳ ❊①✐st❡♠ ❝❛♠♣♦s ∂x1, . . . , ∂xN q✉❡ ❣❡r❛♠ V s♦❜r❡ U✱ ❡ ❡♥tã♦ ❡①✐st❡♠ ajk ∈ C ∞
✽ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
j=1 . . . , n✱ k=1 . . . , Nt❛✐s q✉❡
Lj = n X
k=1
ajk
∂ ∂xk
, j=1, . . . , n
◆♦t❡ q✉❡ ❛s ❢✉♥çõ❡s ajk sã♦ ❞❛❞❛s ♣♦r
ajk(q) = N X
m=1
ajm(q)
∂xk
∂xm
=Lj(xk)(q) = (Lj)q(xk)
❈♦♠♦ ♦ ♣♦st♦ ❞❛ ❡str✉t✉r❛ é n✱ ♦ ♣♦st♦ ❞❛ ♠❛tr✐③ (ajk(p) ; 1 ≤ j ≤ n, 1 ≤ k ≤ N) é n✳
P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ r❡♦r❞❡♥❡ ❛s ❝♦♦r❞❡♥❛❞❛s xk ❞❡ ❢♦r♠❛ q✉❡ ❛ ♠❛tr✐③ (ajk(p) ; 1 ≤
j≤n, 1≤k≤n) s❡❥❛ ✐♥✈❡rtí✈❡❧✳ ❊♥tã♦
A(q) = (ajk(q) ; 1≤j≤n, 1≤k≤n)
♣❡r♠❛♥❡❝❡ ✐♥✈❡rtí✈❡❧✱ ♣♦r ❝♦♥t✐♥✉✐❞❛❞❡✱ ♣❛r❛ q ❡♠ ✉♠ ❛❜❡rt♦ V ⊆ U ❡♠ t♦r♥♦ ❞❡ p✱ ❡
♦s ❡❧❡♠❡♥t♦s ❞❡ A−1(q) sã♦ ❢✉♥çõ❡s s✉❛✈❡s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ✐♥✈❡rsã♦ ❞❡
♠❛tr✐③❡s✳
P❛r❛ ❝❛❞❛ q∈V ❡ ❝❛❞❛j, k=1, . . . , n✱ ❞❡✜♥❛ ♦ ✈❡t♦r
Vjk(q) = (λ1jk(q), . . . , λ n jk(q))
P♦rt❛♥t♦✱
[Lj, Lk](xl)(q) = [Lj, Lk]q(xl) = n X
ν=1
λνjk(q)(Lν)q(xl) = n X
ν=1
λνjk(q)aνl(q)
■ss♦ ♣r♦✈❛ q✉❡ ♦ ✈❡t♦r Vjk(q)A(q) é ✐❣✉❛❧ ❛
([Lj, Lk](x1)(q), . . . ,[Lj, Lk](xn)(q))
❊♠ ♣❛rt✐❝✉❧❛r s❡✉s ❡❧❡♠❡♥t♦s sã♦ ❢✉♥çõ❡s s✉❛✈❡s ❞❡ q✳ ▲♦❣♦✱ ♦s ❡❧❡♠❡♥t♦s ❞♦ ✈❡t♦r
(λ1jk(q), . . . , λnjk(q)) =Vjk(q) = (Vjk(q)A(q))A−1(q)
sã♦ ❢✉♥çõ❡s s✉❛✈❡s ❞❡ q✱ ❝♦♠♦ q✉❡rí❛♠♦s ♣r♦✈❛r✳
✶✳✸ ❋♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s
❖ q✉❡ ♥♦s s❡rá út✐❧ é ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♦ ❝♦♥❝❡✐t♦ ❣❡♦♠étr✐❝♦ ❞❡1✲❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s✿
❉❡✜♥✐çã♦ ✶✳✶✹✳ ❯♠❛ ✭1✲✮❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧ ❡♠ Ω é ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❞✉❛❧ Λ1(Ω) ❞♦
C∞
(Ω)✲♠ó❞✉❧♦ X(Ω)✱ ✐st♦ é✱ ✉♠❛ ❛♣❧✐❝❛çã♦ C∞
(Ω)✲❧✐♥❡❛r
ω:X(Ω)→C∞
✶✳✸ ❋♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ✾
Pr♦♣♦s✐çã♦ ✶✳✶✺✳ ❙❡ L∈X(Ω) é ♦ ❝❛♠♣♦ ♥✉❧♦ ❡♠ ✉♠ ❛❜❡rt♦ U⊆Ω✱ ❡♥tã♦ ω(L) =0
❡♠ U ♣❛r❛ t♦❞❛ ω∈Λ1(Ω)✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ p ∈ U✳ ❖❜t❡♥❤❛ g ∈ C∞
(Ω) q✉❡ ✈❛❧❡ 1 ❡♠ p ❡ 0 ❢♦r❛ ❞❡ U❀ ❧♦❣♦ L= (1−g)L ❡♠ Ω✱ ❡ t❡♠✲s❡
ω(L) =ω((1−g)L) = (1−g)ω(L)
q✉❡ ✈❛❧❡ 0 ❡♠ p✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ ♣r♦✈❛✲s❡ ❛ ❜♦❛✲❞❡✜♥✐çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ r❡str✐çã♦ ❞❡ ✉♠❛ ❢♦r♠❛ ❛
✉♠ ❛❜❡rt♦✿ ❞❛❞♦s ✉♠❛ ❢♦r♠❛ω∈Λ1(Ω)❡ ✉♠ ❛❜❡rt♦U⊆Ω✱ ♣♦❞❡✲s❡ ❞❡✜♥✐rω|
U ∈Λ1(U)
❞❡ ❢♦r♠❛ q✉❡ ♦ ❞✐❛❣r❛♠❛ ❛ s❡❣✉✐r ❝♦♠✉t❡ ✭❛s s❡t❛s ✈❡rt✐❝❛✐s ✐♥❞✐❝❛♠ ❛ r❡str✐çã♦ ❞❡ ❝❛♠♣♦s ❡ ❞❡ ❢✉♥çõ❡s s✉❛✈❡s✮✿
X(Ω) ω //
C∞
(Ω)
X(U)
ω|U//C
∞
(U)
❉❡ ❢❛t♦✱ ❞❛❞♦sM∈X(U) ❡p∈U✱ q✉❡r❡♠♦s ❞❡✜♥✐rωU(M)(p)✳ ❚♦♠❡g ∈C∞
c (Ω)t❛❧
q✉❡ g≡1 ♣❡rt♦ ❞❡p✱ ❡ ❞❡✜♥❛ ♦ ❝❛♠♣♦ ✄M∈X(Ω) ♣♦r
✄
M(f) =M(gf) , f∈C∞
(Ω)
➱ s✐♠♣❧❡s ♠♦str❛r q✉❡ M é ❞❡ ❢❛t♦ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ✲ ✐st♦ é ✈❡r❞❛❞❡ ♣❛r❛ q✉❛❧q✉❡r
g ∈ C∞
✳ ❆❧é♠ ❞✐ss♦✱ M = M✄ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ p✳ P♦rt❛♥t♦✱ ❛ ❢♦r♠❛
❞✐❢❡r❡♥❝✐❛❧
ωU(M)(p) =ω(M✄)(p)
❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ❡ s❡ L∈X(Ω)✱ ❡♥tã♦
ωU LU(p) =ω (LU)∼(p) =ω(L)(p)
■st♦ é✱ ♦ ❞✐❛❣r❛♠❛ ❛❝✐♠❛ ❝♦♠✉t❛✳
Pr♦♣♦s✐çã♦ ✶✳✶✻✳ ❙❡ L∈X(Ω) é t❛❧ q✉❡ Lp =0 ♣❛r❛ ❛❧❣✉♠ p∈Ω✱ ❡♥tã♦ ω(L)(p) =0
♣❛r❛ t♦❞❛ ω∈Λ1(Ω)✳
❉❡♠♦♥str❛çã♦✿ ❊s❝♦❧❤❛ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐sx1, . . . , xN✳ ❚❡♠♦sL= N X
j=1
Lxj
∂
∂xj✱ ❡ ❛♣❧✐❝❛♥❞♦✲
s❡ ω✿
ω(L) =
N X
j=1
(Lxj)ω
∂ ∂xj
✶✵ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
▲♦❣♦
ω(L)(p) =
N X
j=1
Lxj(p)ω
∂ ∂xj
(p) =
N X
j=1
Lp(xj)ω
∂ ∂xj
(p) =0
❙❡❥❛ (U, x)✉♠❛ ❝❛rt❛ ❧♦❝❛❧✳ ❈♦♠♦ {∂x1, . . . , ∂xN} é ✉♠❛ ❜❛s❡ ♣❛r❛ X(U)s♦❜r❡ C∞(U) ❡
❝♦♠♦ ❛s ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s sã♦C∞✲❧✐♥❡❛r❡s✱ ✉♠❛ ❢♦r♠❛ ❡stá t♦t❛❧♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛ ❡♠
U
♣❡❧♦s s❡✉s ✈❛❧♦r❡s ❡♠ ❝❛❞❛ ❝❛♠♣♦ ∂xj✳ ❉❡ ✐♠♣♦rtâ♥❝✐❛ ❡s♣❡❝✐❛❧ sã♦ ❛s ❢♦r♠❛s dx1, . . . , dxN
❞❡✜♥✐❞❛s ♣♦r dxj ∂ ∂xk
=δjk , j, k=1, . . . , N
✐st♦ é✱ {dx1, . . . , dxN} é ❛ ❜❛s❡ ❞✉❛❧ ❞❡ {∂x1, . . . , ∂xN}✳ ❙❡ ω∈Λ1(U) ❡ L∈X(U)✱ t❡♠♦s ❞❛
r❡♣r❡s❡♥t❛çã♦ ❧♦❝❛❧ ♣❛r❛ L q✉❡ dxj(L) =Lxj❀ ❧♦❣♦
ω(L) =
N X
j=1
Lxjω
∂ ∂xj = N X
j=1
dxj(L)ω
∂ ∂xj ❡ ❛ss✐♠ ω= N X
j=1
ω
∂ ∂xj
dxj ✭✶✳✶✼✮
▼❛✐s ❣❡r❛❧♠❡♥t❡✱ s❡ f∈C∞
(Ω)✱ ♣♦❞❡✲s❡ ❞❡✜♥✐r df ∈Λ1(Ω) ♣♦r
df(L) =Lf , L∈X(Ω)
P❛r❛ ❡st❛ ❢♦r♠❛ ❞✐❢❡r❡♥❝✐❛❧✱ ❛ ❡q✉❛çã♦ ✭✶✳✶✼✮ t♦♠❛ ❛ ❢♦r♠❛ ❝♦♥❤❡❝✐❞❛
df=
N X
j=1
df
∂ ∂xj
dxj= N X
j=1
∂f ∂xj
dxj
❖ ❡s♣❛ç♦ ❝♦t❛♥❣❡♥t❡ ✭❝♦♠♣❧❡①✐✜❝❛❞♦✮ ❛ Ω ❡♠ p✱ ❞❡♥♦t❛❞♦ ♣♦r CTp∗Ω✱ é ❞❡✜♥✐❞♦
❝♦♠♦ ♦ ❞✉❛❧ ❞♦ C✲❡s♣❛ç♦ ✈❡t♦r✐❛❧ CTpΩ✱ ❡ s❡✉s ❡❧❡♠❡♥t♦s sã♦ ❞✐t♦s ✈❡t♦r❡s ❝♦t❛♥❣❡♥t❡s
✭❝♦♠♣❧❡①♦s✮ ❛ Ω ❡♠ p✳ P♦rt❛♥t♦ ✉♠ ✈❡t♦r ❝♦t❛♥❣❡♥t❡ ❡♠ p é ✉♠❛ ❛♣❧✐❝❛çã♦ C✲❧✐♥❡❛r
λ:CTpΩ→C
❆ ❝❛❞❛ ❢♦r♠❛ω∈Λ1(Ω)❡stá ❛ss♦❝✐❛❞♦ ✉♠ ✈❡t♦r ❝♦t❛♥❣❡♥t❡ωp❞❡CTp∗Ω✱ s✉❛ ❧♦❝❛❧✐③❛çã♦
❡♠ p✱ ❞❛❞♦ ♣♦r
ωp(ν) =ω(L)(p) , ν∈CTpΩ
♦♥❞❡ L ∈ X(Ω) é q✉❛❧q✉❡r ❝❛♠♣♦ t❛❧ q✉❡ Lp = ν✳ ❊st❡ ❝♦♥❝❡✐t♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❡♠
✈✐st❛ ❞❛s ♣r♦♣♦s✐çõ❡s ✶✳✶✵ ❡ ✶✳✶✻✳ ❚❛♠❜é♠ ❡stá ❝❧❛r❛ ❛C∞
(Ω)✲❧✐♥❡❛r✐❞❛❞❡ ❞❛ ♦♣❡r❛çã♦ ❞❡
❧♦❝❛❧✐③❛çã♦✱ ✐st♦ é✱
(ω1+fω2)p = (ω1)p+f(ω2)p , ω1, ω2∈Λ1(Ω), f ∈C
∞
✶✳✸ ❋♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ✶✶
❖❜s❡r✈❡ q✉❡✱ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s✱ ❛ ❜❛s❡ ❞✉❛❧ ❞❡ {(∂x1)p, . . . ,(∂xN)p} ⊆ CTpΩ é {(dx1)p, . . . ,(dxN)p} ⊆ CTp∗Ω✳ ❉❡ ❢❛t♦✱ (dxj)p((∂xk)p) é ❞❡✜♥✐❞♦ ❝♦♠♦ dxj(L)(p) ♦♥❞❡ L é q✉❛❧q✉❡r ❝❛♠♣♦ t❛❧ q✉❡Lp= (∂xk)p✳ P♦❞❡♠♦s ❡♥tã♦ t♦♠❛rL=∂xk ❡ ♦❜t❡r(dxj)p((∂xk)p) =
dxj(∂xk)(p) = δjk✱ ❝♦♠♦ q✉❡rí❛♠♦s✳ ▼❛✐s ❣❡r❛❧♠❡♥t❡✱ s❡ L ∈ X(Ω) ❡ ω ∈ Λ1(Ω)✱ t❡♠♦s
♣♦r ❞❡✜♥✐çã♦ ωp(Lp) =ω(L)(p) ♣❛r❛ t♦❞♦ p∈Ω✳
P❛rt✐♥❞♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❝❛♥ô♥✐❝❛ ❧♦❝❛❧ ❞❡ ✉♠❛ ❢♦r♠❛ω∈Λ1(Ω)✱
ω =
N X
k=1
ω
∂ ∂xk
dxk
❡♠ t♦r♥♦ ❞❡ ✉♠ ❞❛❞♦ p∈Ω✱ ♦❜t❡♠♦s ❛ ❢ór♠✉❧❛ ❞❡ s✉❛ ❧♦❝❛❧✐③❛çã♦ ❡♠ p✿
ωp= N X
k=1
ω
∂ ∂xk
(p)(dxk)p ✭✶✳✶✽✮
❚❛♠❜é♠ ✈❛❧❡ ♣❛r❛ ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ✉♠ r❡s✉❧t❛❞♦ ❛♥á❧♦❣♦ à ♣r♦♣♦s✐çã♦ ✶✳✶✵✿ Pr♦♣♦s✐çã♦ ✶✳✶✾✳ CTp∗Ω={ωp ; ω∈Λ1(Ω)} ♣❛r❛ t♦❞♦ p∈Ω✳
❉❡♠♦♥str❛çã♦✿ ❉❡✈✐❞♦ à ❡q✉❛çã♦ ✭✶✳✶✽✮ ❡ ❛♦ ❢❛t♦ q✉❡ {(dxk)p}k=1,...,N é ❜❛s❡ ♣❛r❛ CTp∗Ω✱
♦ q✉❡ ♣r❡❝✐s❛♠♦s ♠♦str❛r é q✉❡✱ ❞❛❞❛ ✉♠❛ ❝❛rt❛ U ∋ p ❡ ❞❛❞♦s a1, . . . , aN ∈ C✱ ❡①✐st❡
✉♠❛ ❢♦r♠❛ ω ∈Λ1(Ω) t❛❧ q✉❡
ωU
∂ ∂xk
(p) =ak , k=1, . . . , N
P❛r❛ t❛♥t♦✱ ❞❡✜♥❛ ω0 ∈ Λ1(W) ❡♠ ❛❧❣✉♠ ❛❜❡rt♦ W ⊂⊂ U ❝♦♥t❡♥❞♦ p ❝♦♠♦ ω0 =
PN
k=1akdxk✳ ❙❡ g é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝♦rt❡ q✉❡ ✈❛❧❡ 1 ❡♠ W ❡ 0 ❢♦r❛ ❞❡ U✱ ❛ ❢♦r♠❛ q✉❡
♣r♦❝✉r❛♠♦s é ω=gω0✳
❉❡✜♥✐çã♦ ✶✳✷✵✳ ❖ ✜❜r❛❞♦ ❝♦t❛♥❣❡♥t❡ ✭❝♦♠♣❧❡①✐✜❝❛❞♦✮ ❞❡ Ω é ❛ r❡✉♥✐ã♦ ❞✐s❥✉♥t❛
CT∗Ω ❞♦s CTp∗Ωs♦❜r❡ t♦❞♦s p∈Ω✳ ❯♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ✭❝♦♠♣❧❡①♦✮ ❞❡ CT∗Ω ❞❡
♣♦st♦ ♠ ❡ ❝♦♣♦st♦ ◆−♠ é ✉♠❛ r❡✉♥✐ã♦ ❞✐s❥✉♥t❛ W=∪p∈ΩWp✱ ♦♥❞❡ ❝❛❞❛ Wp é ✉♠
s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ CT∗
pΩ ❞❡ ❞✐♠❡♥sã♦ m✱ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ tr✐✈✐❛❧✐❞❛❞❡
❧♦❝❛❧✿ ♣❛r❛ t♦❞♦ p ∈ Ω✱ ❡①✐st❡♠ ✉♠ ❛❜❡rt♦ U ∋ p ❡ ❢♦r♠❛s ω1, . . . , ωm ∈ Λ1(U) ❞❡
♠♦❞♦ q✉❡✱ ♣❛r❛ ❝❛❞❛ q ∈ U✱ Wq é ❣❡r❛❞♦ ♣♦r (ω1)q, . . . ,(ωm)q✳ ❈♦♠♦ ❛♥t❡s✱ ❞✐③✲s❡
♥❡st❡ ❝❛s♦ q✉❡ ω1, . . . , ωm ❣❡r❛♠ W ❡♠ t♦r♥♦ ❞❡ ♣✳ ❆✐♥❞❛✱ Wq é ❞✐t❛ ✜❜r❛ ❞❡ W
s♦❜r❡ q✱ ❡ ♣♦❞❡♠♦s ❝❤❛♠❛r W ❞❡ s✉❜✜❜r❛❞♦ ❝♦t❛♥❣❡♥t❡✳
Pr♦♣♦s✐çã♦ ✶✳✷✶✳ ❙❡ ♦s ❝❛♠♣♦s L1, . . . , Ln ∈X(Ω) sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠
✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ U ❞❡ ✉♠ ♣♦♥t♦ p ∈ Ω✱ ❡♥tã♦ ❡①✐st❡♠ ✉♠ ❛❜❡rt♦ V✱ ❝♦♠
p∈V ⊆U✱ ❡ ❢♦r♠❛s ω1, . . . , ωN−n∈Λ1(V) ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s t❛✐s q✉❡
✶✷ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
❉❡♠♦♥str❛çã♦✿ ❉✐♠✐♥✉❛U❛té q✉❡ ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ ✉♠❛ ❝❛rt❛ ❡♠ t♦r♥♦ ❞❡p✱ ❡ ❡s❝r❡✈❛
♦s ❝❛♠♣♦s Lj ❡♠ U ❝♦♠♦
Lj = N X
k=1
ajk
∂ ∂xk
, j=1, . . . , n
♣❛r❛ ❢✉♥çõ❡s ajk∈C
∞
(U)✳ ❉❡✜♥❛ ❛ ♠❛tr✐③ ❞❡ ❢✉♥çõ❡s s✉❛✈❡s
A(q) =
a11(q) . . . a1N(q)
✳✳✳ ✳✳✳ ✳✳✳
an1(q) . . . anN(q)
n×N
=
=
a11(q) . . . a1n(q) a1,n+1(q) . . . a1N(q)
✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳
an1(q) . . . ann(q) an,n+1(q) . . . anN(q)
n×N
=
= C(q)n×n D(q)(N−n)×n
n×N
❊♥tã♦ t❡♠♦s
(L1, . . . , Ln)T =A(∂x1, . . . , ∂xN)T
❡ ❝♦♠♦ ♦s Lj sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❛ ♠❛tr✐③ A(q)t❡♠ ♣♦st♦ n ♣❛r❛ t♦❞♦q∈U✳
P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♥♦t❛çã♦✱ s✉♣♦♥❤❛ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡C(p)é ✐♥✈❡rtí✈❡❧✱ ❡ t♦♠❡
V ⊆U t❛❧ q✉❡ C(q) é ✐♥✈❡rtí✈❡❧ ♣❛r❛ t♦❞♦ q∈V✳ ❉❡✜♥❛ ❛s ♠❛tr✐③❡s
E(q) = −C−1(q)D(q) , B(q) = E(q)n×(N−n)
IN−n
!
N×(N−n)
♦♥❞❡ IN−n é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❝♦♠ N−n ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s✳ ◆♦t❡ q✉❡ AB=CE+DI=
−D+D=0✳ P♦rt❛♥t♦✱ ❜❛st❛ ❞❡✜♥✐r♠♦s ❢♦r♠❛s ω1, . . . , ωN−n ∈Λ1(V)♣❡❧❛ ❡q✉❛çã♦
(ω1, . . . , ωN−n)T =B(dx1, . . . , dxN)T
■st♦ é✱ s❡ B= (bjk)✱ ❡♥tã♦ ωi =PNp=1bipdxp✱ ❡ ❛ss✐♠
ωi(Lj) = N X
p=1 N X
k=1
ajkbpidxp
∂ ∂xk
=
N X
k=1
ajkbki
❊st❡ é ♦ ❡❧❡♠❡♥t♦ (j, i) ❞❛ ♠❛tr✐③ AB✱ q✉❡ é ③❡r♦✳
P♦❞❡✲s❡ ♦❜t❡r ✉♠ s✉❜✜❜r❛❞♦ ❞❡ CT∗Ω ❞❡ ♣♦st♦ N−n❛ ♣❛rt✐r ❞❡ ✉♠ s✉❜✜❜r❛❞♦ V ❞❡
CTΩ ❞❡ ♣♦st♦ n✱ ❞✐t♦ ❝♦♠♣❧❡♠❡♥t♦ ♦rt♦❣♦♥❛❧ ♦✉ ❛♥✉❧❛❞♦r ❞❡ V✱ ❡ ❞❡♥♦t❛❞♦ ♣♦r V⊥✱
❞❡✜♥✐♥❞♦ s✉❛s ✜❜r❛s ❝♦♠♦ s❡♥❞♦ ♦s s✉❜❡s♣❛ç♦s
✶✳✹ ❈❛♠♣♦s ❡ ❢♦r♠❛s r❡❛✐s ✶✸
➱ ♥❡❝❡ssár✐♦ ✈❡r✐✜❝❛r q✉❡ s❡ tr❛t❛ ❞❡ ✉♠ s✉❜✜❜r❛❞♦✳ ◗✉❡ ❝❛❞❛ ✜❜r❛ é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧
(N−n)✲❞✐♠❡♥s✐♦♥❛❧ ❡stá ❝❧❛r♦❀ ♣r♦✈❡♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ tr✐✈✐❛❧✐❞❛❞❡ ❧♦❝❛❧✳ ❙❡❥❛ p ∈ Ω❀
♣r❡❝✐s❛♠♦s ❡♥❝♦♥tr❛r ✉♠ ❛❜❡rt♦ V ∋ p ❡ ❢♦r♠❛s ω1, . . . , ωN−n ∈ Λ1(V) t❛✐s q✉❡✱ ♣❛r❛
❝❛❞❛ q ∈ V✱ ❛ ✜❜r❛ V⊥
q s❡❥❛ ❣❡r❛❞❛ ♣♦r (ω1)q, . . . ,(ωN−n)q✳ ▼❛s s❛❜❡♠♦s✱ s❡♥❞♦ V ✉♠
s✉❜✜❜r❛❞♦ ❞❡ CTΩ✱ q✉❡ ❡①✐st❡♠ ✉♠ ❛❜❡rt♦ U ∋ p ❡ ❝❛♠♣♦s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s
L1, . . . , Ln ∈ X(U) t❛✐s q✉❡✱ ♣❛r❛ ❝❛❞❛ q ∈ U✱ ❛ ✜❜r❛ Vq é ❣❡r❛❞❛ ♣♦r (L1)q, . . . ,(Ln)q✳
❊♥tã♦ ❞❡✜♥❛ V ❡ ωj t❛✐s ❝♦♠♦ ❞❛❞♦s ♣❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ❋✐①❡ q ∈ V✳ ❈♦♠♦ ❛s
❢♦r♠❛s ωj sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ t❛♠❜é♠ ♦s ✈❡t♦r❡s ❝♦t❛♥❣❡♥t❡s (ωj)q ♦ sã♦✱
♣❛r❛ j = 1, . . . , N −n✳ ❘❡st❛ ❛ss✐♠ ♠♦str❛r q✉❡ ♣❡rt❡♥❝❡♠ ❛ V⊥q❀ ♠❛s ❞❡ ❢❛t♦✱ ♣❛r❛
q✉❛✐sq✉❡r j=1, . . . , N−n ❡ k=1, . . . , n✱ t❡♠♦s
(ωj)q((Lk)q) =ωj(Lk)(q) =0
❧♦❣♦ t❛♠❜é♠ (ωj)q(ν) =0 ♣❛r❛ q✉❛❧q✉❡r ν ∈Vq =s♣❛♥{(L1)q, . . . ,(Ln)q}✱ ✐st♦ é✱ (ωj)q ∈
V⊥
q✳
✶✳✹ ❈❛♠♣♦s ❡ ❢♦r♠❛s r❡❛✐s
❋❛③✲s❡ ♥❡❝❡ssár✐♦ ❛♦ ❧♦♥❣♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❛ t❡♦r✐❛ ❡s♣❡❝✐✜❝❛r q✉❛♥❞♦ ❝❛♠♣♦s✱ ✈❡t♦r❡s✱ s✉❜✜❜r❛❞♦s ❡t❝✳ sã♦ ❞✐t♦s r❡❛✐s✳ P❛r❛ t❛♥t♦✱ é s✉✜❝✐❡♥t❡ ❡st❡♥❞❡r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♥❥✉❣❛çã♦ ❝♦♠♣❧❡①❛ ❛ ❡ss❛s ❡str✉t✉r❛s✳ ❆ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ ❡♥✈♦❧✈❡ ❝á❧❝✉❧♦s tr❛❜❛❧❤♦s♦s ♣♦ré♠ tr✐✈✐❛✐s✱ ♥❡❝❡ssár✐♦s ♣❛r❛ ♣r♦✈❛r q✉❡ ♦s ♦❜❥❡t♦s ❞❡✜♥✐❞♦s sã♦ ❞❡ ❢❛t♦ ❝❛♠♣♦s✱ ❢♦r♠❛s ♦✉ ✈❡t♦r❡s ✭❝♦✲✮t❛♥❣❡♥t❡s✿
❉❡✜♥✐çã♦ ✶✳✷✷✳ ❙❡❥❛♠ L∈ X(Ω)✱ ω∈ Λ1(Ω)✱ ν ∈CTpΩ✱ λ∈ CTp∗Ω✳ ❉❡✜♥✐♠♦s s❡✉s
❝♦♥❥✉❣❛❞♦s ✠L∈X(Ω)✱ ✠ω∈Λ1(Ω)✱ ✠ν∈CTpΩ ❡ ✠λ∈CTp∗Ω ♣♦r
✠
Lf=L(f✠), f ∈C∞
(Ω) , ωL✠ =ω(L✠), L∈X(Ω)
✠
ν(f) =ν(f✠), f ∈C∞
(p) , ✠λ(ν) =λ(ν✠), ν∈CTpΩ
❉✐③❡♠♦s q✉❡ L✱ ω✱ ν ❡ λ sã♦ r❡❛✐s q✉❛♥❞♦ L =✠L✱ ✠ω=ω✱ ν=ν✠ ❡ λ=✠λ✱ r❡s♣❡❝t✐✈❛✲
♠❡♥t❡✳
❙❡❥❛♠ ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿
C∞
(Ω;R) ={f∈C∞
(Ω) ; f ❛ ✈❛❧♦r❡s r❡❛✐s}
TpΩ={ν∈CTpΩ; νr❡❛❧} Tp∗Ω={λ∈CTp∗Ω; λr❡❛❧}
TΩ= [
p∈Ω
TpΩ T∗Ω=
[
p∈Ω
✶✹ ❈❛♣ít✉❧♦ ✶ ✖ ❙✐st❡♠❛s ■♥✈♦❧✉t✐✈♦s
➱ s✐♠♣❧❡s ✈❡r q✉❡ ✉♠ ❝❛♠♣♦Lé r❡❛❧ s❡ ❡ só s❡L(C∞
(Ω;R))⊆C∞
(Ω;R)✳ ❆✐♥❞❛✱ ♦✉tr❛
❝♦♥❞✐çã♦ ❡q✉✐✈❛❧❡♥t❡ é q✉❡ s✉❛s ❝♦♦r❞❡♥❛❞❛s ♥✉♠ s✐st❡♠❛ ❧♦❝❛❧ ♥❛ ❜❛s❡{∂x1, . . . , ∂xN}s❡❥❛♠
❡❧❡♠❡♥t♦s ❞❡ C∞
(Ω;R)✱ ❡ ♦ ♠❡s♠♦ ✈❛❧❡ ♣❛r❛ ❢♦r♠❛s✳ ❆ ♣r♦♣r✐❡❞❛❞❡ ❛♥á❧♦❣❛ t❛♠❜é♠ ✈❛❧❡
♣❛r❛ ✈❡t♦r❡s ✭❝♦✲✮t❛♥❣❡♥t❡s✱ ❞❛❞♦s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❝♦♠♣❧❡①❛s ❧♦❝❛✐s ❝♦♠ r❡❧❛çã♦ às ❜❛s❡s
{(∂x1)p, . . . ,(∂xN)p} ♦✉ {(dx1)p, . . . ,(dxN)p}✿ ❡ss❛s ❝♦♦r❞❡♥❛❞❛s ❞❡✈❡♠ s❡r ♥ú♠❡r♦s r❡❛✐s✳
❙❡❥❛ V ✉♠ s✉❜✜❜r❛❞♦ ❞❡ CTΩ✳ ❊♥tã♦ ❛ r❡✉♥✐ã♦ ❞✐s❥✉♥t❛
V= [
p∈Ω
Vp
♦♥❞❡ ♣♦r ❞❡✜♥✐çã♦ Vp = {ν✠ ; ν ∈ Vp}✱ ❝♦♥st✐t✉✐ ✉♠ s✉❜✜❜r❛❞♦ ❞❡ CTΩ ❞❡ ♠❡s♠♦ ♣♦st♦✳
❆ ❥✉st✐✜❝❛t✐✈❛ é q✉❡✱ s❡ L1, . . . , Ln ❣❡r❛♠ V❡♠ t♦r♥♦ ❞❡ ✉♠ ♣♦♥t♦✱ ❡♥tã♦ L1, . . . , Ln ❣❡r❛♠
V ❡♠ t♦r♥♦ ❞♦ ♠❡s♠♦ ♣♦♥t♦✱ ❡ ✐ss♦ s❡ ❞❡✈❡ ❛♦ ❢❛t♦ q✉❡✱ ♣❛r❛ q✉❛✐sq✉❡r cj ∈C✱
X
cj(Lj)q(f) = X
cj(Lj)q(f✠) = X
cj(Lj)q(f✠) = X
cj(Lj)q(f)
✐st♦ é X
cj(Lj)q=cj(Lj)q
❆ ❞❡✜♥✐çã♦ ❛♥á❧♦❣❛ ✈❛❧❡ t❛♠❜é♠ ❝❛s♦ V s❡❥❛ s✉❜✜❜r❛❞♦ ❞❡ CT∗Ω✳
Pr♦♣♦s✐çã♦ ✶✳✷✸✳ ❙❡❥❛ V s✉❜✜❜r❛❞♦ ❞❡ CTΩ✳ ❊♥tã♦
V⊥ =V⊥
❉❡♠♦♥str❛çã♦✿ ❇❛st❛ ♠♦str❛r♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❛s ✜❜r❛s✳ ❋✐①❡ p∈Ω✳ ❚❡♠♦s
λ∈Vp ⊥
⇐⇒ λ(ν✠) =0 ♣❛r❛ t♦❞♦ ν∈Vp
λ∈V⊥
p ⇐⇒ ✠λ(ν) =0 ♣❛r❛ t♦❞♦ ν∈Vp
P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✠λ✱ ❡ss❛s ❞✉❛s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✳
✶✳✺ ❚❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s
❖ ❚❡♦r❡♠❛ ❞❡ ❋r♦❜❡♥✐✉s ♣❡r♠✐t❡ ❡♥❝♦♥tr❛r✱ s♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s
♥❛s q✉❛✐s ✉♠ s✉❜✜❜r❛❞♦ ✜❝❛ ❣❡r❛❞♦ ♣❡❧♦s ❝❛♠♣♦s ❞❛ ❜❛s❡ ❝❛♥ô♥✐❝❛ ❧♦❝❛❧ ❞❡ X(Ω)✳ ➱
❝♦♥st❛♥t❡♠❡♥t❡ ✉t✐❧✐③❛❞♦ ❡♠ tr❛❜❛❧❤♦s q✉❡ ❜✉s❝❛♠ r❡s✉❧t❛❞♦s ❞❡ ♥❛t✉r❡③❛ ❧♦❝❛❧✱ ✉♠❛ ✈❡③ q✉❡ s✐♠♣❧✐✜❝❛ ❛s ❡str✉t✉r❛s ❡st✉❞❛❞❛s s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✳ ❖ ♣ró①✐♠♦ ❧❡♠❛ é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ❡♠ s✉❛ ❞❡♠♦♥str❛çã♦✳
▲❡♠❛ ✶✳✷✹✳ ❙❡❥❛ L∈X(Ω) ✉♠ ❝❛♠♣♦ r❡❛❧ q✉❡ ♥✉♥❝❛ s❡ ❛♥✉❧❛✳ ❊♥tã♦✱ ❞❛❞♦ p∈Ω✱
❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ p ❡ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s y1, . . . , yN ♥❡st❛ ✈✐③✐✲
