UNIVERSIDADE FEDERAL DE SÃO CARLOS
CENTRO DE CIÊNCIAS EXATAS E DE TECNOLOGIA
PROGRAMA DE PÓS GRADUAÇÃO EM MATEMÁTICA
❘❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ♣❛r❛ ❞✉❛s ❝❧❛ss❡s
❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s✉❛✈❡s ❝♦♠♣❧❡①♦s
▲✉❝✐❡❧❡ ❘♦❞r✐❣✉❡s ◆✉♥❡s
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙ ●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❘❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ♣❛r❛ ❞✉❛s ❝❧❛ss❡s
❞❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s s✉❛✈❡s ❝♦♠♣❧❡①♦s
▲✉❝✐❡❧❡ ❘♦❞r✐❣✉❡s ◆✉♥❡s
❇♦❧s✐st❛ ❈❆P❊❙
❖r✐❡♥t❛❞♦r✿ Pr♦❢ ❉r✳ ❏♦sé ❘✉✐❞✐✈❛❧ ❙♦❛r❡s ❞♦s ❙❛♥t♦s ❋✐❧❤♦
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s ❝♦♠♦ ♣❛rt❡ ❞♦s r❡✲ q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉✲ t♦r ❡♠ ▼❛t❡♠át✐❝❛✱ ár❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❆♥á❧✐s❡✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária UFSCar Processamento Técnico
com os dados fornecidos pelo(a) autor(a)
N972r
Nunes, Luciele Rodrigues
Resolubilidade local para duas classes de campos de vetores suaves complexos / Luciele Rodrigues Nunes. -- São Carlos : UFSCar, 2016.
106 p.
Tese (Doutorado) -- Universidade Federal de São Carlos, 2016.
❆♦s ♠❡✉s ♣❛✐s ❆ss✐s & ▼❛r❧❡♥❡
❆❣r❛❞❡❝✐♠❡♥t♦
❆❣r❛❞❡ç♦✱
❛ ❉❡✉s✱ q✉❡ s❡♠♣r❡ ♠❡ ❞❡✉ ❢♦rç❛s ♣❛r❛ s❡❣✉✐r ♠❡✉ s♦♥❤♦ ❡ s❡♠♣r❡ ♠❡ ❣✉✐♦✉ à ❢❛③❡r ❛s ❡s❝♦❧❤❛s ❝❡rt❛s❀
❛♦s ♠❡✉s ♣❛✐s ❆ss✐s ❡ ▼❛r❧❡♥❡✱ ❛♦s q✉❛✐s ❞❡❞✐❝♦ ❡ ❞❡✈♦ t✉❞♦ q✉❡ ❝♦♥q✉✐st❡✐ ❛té ❤♦❥❡✳ ❖❜r✐❣❛❞❛ ♣♦r s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❡♠ ❡ ♠❡ ❛❥✉❞❛r❡♠ ❡♠ t✉❞♦ q✉❡ ♣r❡❝✐s❡✐✳ ❆s ❝♦♥s❡❧❤♦s ♣❡❧♦ t❡❧❡❢♦♥❡ ❞♦ ♠❡✉ P❛✐✱ ♦ q✉❛❧ s❡♠♣r❡ ❢♦✐ ❛ ♣❛❧❛✈r❛ ✜♥❛❧ ❡♠ t♦❞❛s ❞❡❝✐sõ❡s q✉❡ ♣r❡❝✐s❡✐ t♦♠❛r✳ P♦❞❡♠ t❡r ❝❡rt❡③❛ q✉❡ ❛ ♣❛rt❡ ♠❛✐s ❞✐❢í❝✐❧ ❞❡ss❛ ❡t❛♣❛ ❢♦✐ ❞❡✐①❛r ✈♦❝ês ❛ ♠❛✐s ❞❡ ✶✻✵✵❦♠ ❞❡ ♠✐♠✳✳✳ ▼❛s✱ ❡st♦✉ ✈♦❧t❛♥❞♦✦✦✦
❛♦ ❘❛❢❛❡❧✱ ♥♦✐✈♦✱ ❝♦♠♣❛♥❤❡✐r♦✱ ❛♠✐❣♦✳✳✳ ❆❝❤♦ q✉❡ ✐ss♦ ❞❡s❝r❡✈❡ t✉❞♦ q✉❡ ❡❧❡ r❡♣r❡✲ s❡♥t❛ ♣r❛ ♠✐♠❀
❛ ♠✐♥❤❛ ✐r♠ã ▲✉❛♥❛✱ q✉❡ ♠❡ ❞❡✉ ❛♦ ❧♦♥❣♦ ❞❡ss❡ ❞♦✉t♦r❛❞♦ ✉♠❛ s♦❜r✐♥❤❛ ✭❛✜❧❤❛❞❛✮ ❧✐♥❞❛✱ q✉❡ s❡♠♣r❡ ❛❧❡❣r♦✉ ♠❡✉s r❡t♦r♥♦s ♣❛r❛ ❝❛s❛❀
❛ ❞✉❛s ❝♦♠♣❛♥❤❡✐r✐♥❤❛s ❞❡ ❡st✉❞♦s✱ P❛ç♦❝❛ ❡ ▲✉♣✐t❛✱ q✉❡ s❡♠♣r❡ ❡st✐✈❡r❛♠ ❛♦ ❧❛❞♦ ❞♦s ❧✐✈r♦s ❡st✉❞❛♥❞♦ ❝♦♠ ❛ ♠❛♠ã❡ ❡ ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝❛♥s♦✱ ❞❡✐t❛❞❛s ❡♠ ♠✐♥❤❛ ❜❛rr✐❣❛❀ ❛♦ Pr♦❢❡ss♦r ❘✉✐❞✐✈❛❧ ❡ ❛♦ ♣r♦❢❡ss♦r ▼❛✉rí❝✐♦✱ ♣❡❧❛ s❛❜❡❞♦r✐❛ q✉❡ ♦r✐❡♥t❛r❛♠ ♠❡✉ ❞♦✉t♦r❛❞♦ ❡ ♠❡str❛❞♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀
❛♦s ❛♠✐❣♦s✱ ♦s q✉❡ ❣❛♥❤❡✐ ❡♠ ❙ã♦ ❈❛r❧♦s ♣❡❧❛s ❛❧❡❣r✐❛s✱ ❝♦♠❡♠♦r❛çõ❡s✱ ❛♣r❡♥❞✐ ♠✉✐t♦ ❝♦♠ ✈♦❝ês ❡ ♦s q✉❡ ❞❡✐①❡✐ ❡♠ ❘✐♦ ●r❛♥❞❡✱ ♦♥❞❡ s❡♠♣r❡ q✉❡ ✈♦❧t❛✈❛ ❡r❛ ❝♦♠♦ s❡ ♥✉♥❝❛ t✐✈❡ss❡ ✐❞♦ ❡♠❜♦r❛❀
❛ ❈❆P❊❙✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st✉❞❛ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❧✐♥❡❛r❡s ❝♦♠♣❧❡①♦s ❞❡ ♦r❞❡♠ ✉♠ s❡♠ t❡r♠♦s ❞❡ ♦r❞❡♠ ③❡r♦ ✭❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❝♦♠♣❧❡①♦s✮✳ Pr✐♠❡✐r❛♠❡♥t❡ ❛♣r❡s❡♥t❛♥❞♦ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s q✉❡ ❞❡✐①❛♠ ❞❡ s❡r ❡❧í♣t✐❝♦s ♣r❡❝✐s❛♠❡♥t❡ ❡♠ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ ✶✲❞✐♠❡♥s✐♦♥❛❧ ❡ ♣♦r ✜♠ ❝♦♥str✉✐♥❞♦ s♦❧✉çõ❡s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦
(P)❡ q✉❡ ❞❡✐①❛♠ ❞❡ s❡r ❡❧í♣t✐❝♦s ♣r❡❝✐s❛♠❡♥t❡ ❡♠ ✉♠ ♣♦♥t♦✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ st✉❞② ❝♦♠♣❧❡① ❧✐♥❡❛r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦❢ ♦r❞❡r ♦♥❡ ✇✐t❤♦✉t ❤❛✈✐♥❣ t❡r♠s ♦❢ ③❡r♦ ♦r❞❡r ✭❝♦♠♣❧❡① ✈❡❝t♦r ✜❡❧❞s✮✳ ❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❧♦❝❛❧ s♦❧✈❛❜✐❧✐t② ♦❢ ❛ ❝❧❛ss ♦❢ ♦♣❡r❛t♦rs ✇❤✐❝❤ ❛r❡ ♥♦t ❡❧❧✐♣t✐❝ ♣r❡❝✐s❡❧② ✐♥ ❛ ✶✲❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞ ❛♥❞ ✇❡ ❝♦♥str✉❝t s♦❧✉t✐♦♥s ❢♦r ❛ ❝❧❛ss ♦❢ ♦♣❡r❛t♦rs s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥ (P) ❛♥❞ ✇❤✐❝❤ ❛r❡ ♥♦t ❡❧❧✐♣t✐❝ ♣r❡❝✐s❡❧② ✐♥ ❛ ♣♦✐♥t✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✺
✶✳✶ ❚ó♣✐❝♦s ❞❡ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ ▼✉❞❛♥ç❛s ❞❡ ✈❛r✐á✈❡✐s ♥♦ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡ ✉♠ ♦♣❡r❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸ ❖♣❡r❛❞♦r❡s ♣❛r❝✐❛❧♠❡♥t❡ ❤✐♣♦❡❧í♣t✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s ❝♦♠♣❧❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✹✳✶ ❆ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡ X(Ω) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✶✳✹✳✷ ❊str✉t✉r❛s ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹✳✸ ❋♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✹✳✹ ❖ ❝♦♥❥✉♥t♦ ❝❛r❛❝t❡ríst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✹✳✺ ❊str✉t✉r❛s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹✳✻ ●❡r❛❞♦r❡s ❧♦❝❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✷ ❖♣❡r❛❞♦r❡s q✉❛s❡ ℓ✲▼✐③♦❤❛t❛ ✷✼
✷✳✶ ❯♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❋♦r♠❛ ♥♦r♠❛❧ ✲ P❛rt❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸ ❋♦r♠❛ ◆♦r♠❛❧ ✲ P❛rt❡ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸ ❈❛r❛❝t❡r✐③❛çã♦ ❞♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ✺✺
✸✳✶ ❊st❛❜❡❧❡❝✐♠❡♥t♦ ❞❛ ❝♦♥❞✐çã♦ ❝r✉❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
✸✳✷ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✸✳✷✳✶ ❆ ❝♦♥❞✐çã♦ é s✉✜❝✐❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✸✳✷✳✷ ❆ ❝♦♥❞✐çã♦ é ♥❡❝❡ssár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✹ ❉❡t❡r♠✐♥❛çã♦ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ❧♦❝❛❧♠❡♥t❡ r❡✲
s♦❧ú✈❡✐s ✽✶
✹✳✶ ❍✐♣ót❡s❡s s♦❜r❡ ♦ ♦♣❡r❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✷ ▲❡♠❛ té❝♥✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✸ ❖♣❡r❛❞♦r ✐♥t❡❣r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ✹✳✹ ❖♣❡r❛❞♦r❡s s❛t✐s❢❛③❡♥❞♦ ❛s ❤✐♣ót❡s❡s ✭✐✮✱ ✭✐✐✮ ❡ ✭✐✐✐✮ ❞❛ ❙❡çã♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✹✳✹✳✶ ❊①❡♠♣❧♦ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✹✳✹✳✷ ❊①❡♠♣❧♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶
■♥tr♦❞✉çã♦
❯♠ ❞♦s ♣r♦❜❧❡♠❛s ♠❛✐s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s ❧✐♥❡❛r❡s é ❞❡❝✐❞✐r s♦❜r❡ ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r✳ ◆❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞❛ ❞é❝❛❞❛ ❞❡ ✺✵ s✉r❣✐r❛♠ ❞♦✐s tr❛❜❛❧❤♦s ❝r✉❝✐❛✐s s♦❜r❡ ♦ t❡♠❛✳ ❖ ♣r✐♠❡✐r♦ ❞❡❧❡s ❛♣r❡s❡♥t❛ ♦ ❝é❧❡❜r❡ ❡①❡♠♣❧♦ ❞❡ ❍✳ ▲❡✇② ❬✶✹❪✳ ❊♠ ✶✾✺✻ ❡❧❡ ♠♦str♦✉ q✉❡ ♦ ♦♣❡r❛❞♦r ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠
L=− ∂
∂x1 −
i ∂ ∂x2
+ 2i(x1+ix2)
∂ ∂x3
♥ã♦ é r❡s♦❧ú✈❡❧ ❡♠ ♣♦♥t♦ ❛❧❣✉♠ ❞❡ R3✳ ◆❛ r❡❛❧✐❞❛❞❡ ♦ r❡s✉❧t❛❞♦ é ❛✐♥❞❛ ♠❛✐s ❢♦rt❡✿ ❡①✐st❡
f ∈C∞(R3) ♣❛r❛ ❛ q✉❛❧ ❛ ❡q✉❛çã♦ Lu=f ♥ã♦ t❡♠ s♦❧✉çã♦ ❡♠ ❛❜❡rt♦ ❛❧❣✉♠ ✭♥ã♦ ✈❛③✐♦✮
❞❡R3✳
❖ s❡❣✉♥❞♦ tr❛❜❛❧❤♦ ❝✐t❛❞♦ é ♦ ❞❡ ▲✳ ❍ör♠❛♥❞❡r ❬✶✵❪ ❡ ❛♣r❡s❡♥t❛ ✉♠❛ ❡①♣❧✐❝❛çã♦ ♣❛r❛ ♦ ❡①❡♠♣❧♦ ❞❡ ❍✳ ▲❡✇②✳ ◆❡st❡ tr❛❜❛❧❤♦ ❡❧❡ ❞❡♠♦♥str❛ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r P✱ ❡♠ ✉♠ ❞❛❞♦ ♣♦♥t♦ x0✱
❡♥✈♦❧✈❡♥❞♦ ♦ ❝♦♠✉t❛❞♦r ❡♥tr❡ P ❡ s❡✉ ❝♦♥❥✉❣❛❞♦ ❝♦♠♣❧❡①♦ P✿ C = [P,P¯] = ¯P P −PP .¯
❉❡✈❡✲s❡ ♦❜s❡r✈❛r q✉❡✱ s❡ P é ❞❡ ♦r❞❡♠ m✱ ❡♥tã♦ C é ❞❡ ♦r❞❡♠ ≤ 2m−1✳ ❆ ❝♦♥❞✐çã♦
♥❡❝❡ssár✐❛ ♠❡♥❝✐♦♥❛❞❛ é ❛ s❡❣✉✐♥t❡✿
❚❡♦r❡♠❛ ✶✳ ❙❡ P é r❡s♦❧ú✈❡❧ ❡♠ x0 ❡♥tã♦
σp(P)(x0, ξ) = 0, ξ ∈Rn ⇒ σp(C)(x0, ξ) = 0.
❆ ♥ã♦ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦r ❞❡ ❍✳ ▲❡✇② é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❡st❡ t❡♦r❡♠❛✳ ❯♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ✭❝♦♥❞✐çã♦ ✭P✮✮ ❢♦✐ ❡♥✉♥❝✐❛❞❛ ♣♦r ▲✳ ◆✐r❡♥❜❡r❣ ❡ ❋✳ ❚r❡✈❡s ❬✶✽❪ ♥♦ ❞❡❝♦rr❡r
❞❛ ❞é❝❛❞❛ s❡❣✉✐♥t❡ ❡ ♣♦❞❡✲s❡ ❛✜r♠❛r q✉❡ ❡❧❛ t❡♠ s✉❛ ♦r✐❣❡♠ ♥❛ ❛♥á❧✐s❡ ♣♦r ❡❧❡s ❢❡✐t❛✱ ❞♦s ❝❤❛♠❛❞♦s ♦♣❡r❛❞♦r❡s ❞❡ ▼✐③♦❤❛t❛✱ q✉❡ sã♦ ❞❛❞♦s ♣♦r
Mℓ =
∂
∂t +i(ℓ+ 1)t
ℓ ∂
∂x, ℓ∈N.
❘❡♣r❡s❡♥t❛♠♦s ❛q✉✐ ❛s ✈❛r✐á✈❡✐s ❡♠ R2 ♣♦r(x, t)✳ P❛r❛ ❡♥❢❛t✐③❛r ♦ í♥❞✐❝❡ ℓ ❝❤❛♠❛r❡♠♦s ♦
♦♣❡r❛❞♦r Mℓ ❞❡ ♦♣❡r❛❞♦r ℓ✲▼✐③♦❤❛t❛✳ ❖ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡ Mℓ é ❛ ❢✉♥çã♦ ❡♠ R4
σp(Mℓ)(x, t, ξ, τ) = τ+i(ℓ+ 1)tℓξ.
❖❜s❡r✈❡ q✉❡ s❡t 6= 0❡♥tã♦ Mℓ é ❡❧í♣t✐❝♦ ❡♠(x, t)✳ ❆ss✐♠✱ ♦s ú♥✐❝♦s ♣♦♥t♦s q✉❡ ♥❡❝❡ss✐t❛♠ ❞❡ ❛♥á❧✐s❡ q✉❛♥t♦ ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ sã♦ ♦s ❞❛ ❢♦r♠❛ (x,0)✳
❖❜s❡r✈❡ q✉❡✱ s❡ℓ= 1✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❚❡♦r❡♠❛ ✶ ❡ ❝♦♥❝❧✉✐r q✉❡M1 ♥ã♦ é r❡s♦❧ú✈❡❧
♥❛ ♦r✐❣❡♠✳ ❏á✱ s❡ ℓ ≥ 2 ❡ss❡ t❡♦r❡♠❛ ♥ã♦ ❢♦r♥❡❝❡ ♥❡♥❤✉♠❛ ✐♥❢♦r♠❛çã♦✳ ▼❛s ♣❛r❛ ❡ss❡s
♦♣❡r❛❞♦r❡s ♠♦str❛✲s❡ r❛③♦❛✈❡❧♠❡♥t❡ ❞❡ ♠❛♥❡✐r❛ ❡❧❡♠❡♥t❛r✱ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❚❡♦r❡♠❛ ✷✳ Mℓ é r❡s♦❧ú✈❡❧ ♥❛ ♦r✐❣❡♠ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ℓ é ♣❛r✳
◆♦ ❝♦♠❡ç♦ ❞❛ ❞é❝❛❞❛ ❞❡ ✽✵✱ ❋✳ ❚r❡✈❡s ❬✷✵❪ ❡ ❏✳ ❙❥östr❛♥❞ ❬✷✶❪ ❝♦♥s✐❞❡r❛r❛♠ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ❞♦ t✐♣♦ ▼✐③♦❤❛t❛ ❡♠ R2✱ q✉❡ ♣♦r ❞❡✜♥✐çã♦ sã♦ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s L
s✉❛✈❡s ❝♦♠♣❧❡①♦s ❞❡✜♥✐❞♦s ♣❡rt♦ ❞❛ ♦r✐❣❡♠(0,0)✱ t❛✐s q✉❡
✭✐✮ L(0,0), L(0,0) sã♦ ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✱ ❡
✭✐✐✮ L(0,0),(L(0,0), L(0,0) sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳
❆ss✉♠✐♥❞♦ q✉❡ ❡st❡ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡sL é ❞❛ ❢♦r♠❛ L= ∂
∂t+iλ(x, t) ∂ ∂x,
❝♦♠ λ❞❡✜♥✐❞❛ ♣❡rt♦ ❞❛ ♦r✐❣❡♠ ❞❡R2✱ r❡❛❧ ❡ s✉❛✈❡✳ ❖s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ❢♦r❛♠ ♦❜t✐❞♦s
♣♦r ❋✳ ❚r❡✈❡s ❡ ❏✳ ❙❥östr❛♥❞✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✿
❚❡♦r❡♠❛ ✸✳ ❆ss✉♠❛ q✉❡ L s❛t✐s❢❛ç❛ λ(0,0) = 0 ❡ ∂a
∂t(0,0)6= 0✳ L é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ♥❛ ♦r✐❣❡♠ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s t❛❧ q✉❡ L é ✉♠
♠ú❧t✐♣❧♦✱ ♣♦r ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ♥ã♦ ♥✉❧❛✱ ❞♦ ♦♣❡r❛❞♦r 1✲▼✐③♦❤❛t❛✳
❚❡♦r❡♠❛ ✹✿ ❆ss✉♠❛ q✉❡ L s❛t✐s❢❛ç❛ λ(0,0) = 0 ❡ ∂a
∂t(0,0) 6= 0✳ ❊♥tã♦ ❡①✐st❡♠ ❢✉♥çõ❡s s✉❛✈❡s u+ ❡ u− ❞❡✜♥✐❞❛s ❡♠ t ≥ 0 ❡ t ≤ 0✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❛✐s q✉❡ u±(x,0) sã♦ r❡❛✐s✱
∂u±
∂x (x,0) >0 ❡ Lu
± = 0✳ ❆❧é♠ ❞✐ss♦✱ L é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ ♥❛ ♦r✐❣❡♠ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛ ❢✉♥çã♦ (u+)−1◦u−(x,0)é ❛♥❛❧ít✐❝❛ r❡❛❧ ♥❛ ♦r✐❣❡♠✳
❖s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❛❝✐♠❛ ❛♣r❡s❡♥t❛♠ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ ♦ ♦♣❡r❛❞♦rLs❡r ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ ❏á ♥❛ s❡❣✉♥❞❛ ♠❡t❛❞❡ ❞❛ ❞é❝❛❞❛ ❞❡ ✽✵✱ ◆✳ ❍❛♥❣❡s ❡♠
❬✾❪ ❛♣r❡s❡♥t♦✉ ✉♠ r❡s✉❧t❛❞♦ s♦❜r❡ ❛ r❡s♦❧✉❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ss❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s✱ ❛ q✉❛❧ ❡❧❡ ❝❤❛♠♦✉ ❞❡ ♦♣❡r❛❞♦r❡s q✉❛s❡ ▼✐③♦❤❛t❛✳ ❊♠ ❬✾❪ ❡❧❡ ❛♣r❡s❡♥t♦✉ ✉♠❛ ❝♦♥❞✐çã♦ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠✱ ❡st❛r ♥❛ ✐♠❛❣❡♠ ❞❡ss❡s ♦♣❡r❛❞♦r❡s✳ ■st♦ é✱ ❞❛❞❛ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ f✱ ❡❧❡ ❡♥❝♦♥tr♦✉ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛
❡①✐st✐r ✉♠❛ s♦❧✉çã♦ ❞✐str✐❜✉✐❝✐♦♥❛❧ u t❛❧ q✉❡ Lu=f✳
❯♠❛ ❡①t❡♥sã♦ ♣❛r❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ♣♦r ❍✳ ◆✐♥♦♠✐②❛ ❡♠ ❬✶✼❪ ♥♦ ✜♥❛❧ ❞❛ ❞é❝❛❞❛ ❞❡ ✾✵✳ ◆❡st❡ tr❛❜❛❧❤♦ ❡❧❡ ❝♦♥s✐❞❡r♦✉ ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s q✉❡ ❝♦♥té♠ ♦ ❖♣❡r❛❞♦rℓ✲▼✐③♦❤❛t❛✱ ♥♦ ❝❛s♦ ❡♠ q✉❡ℓ é í♠♣❛r✱ q✉❡ ♣♦r ❞❡✜♥✐çã♦ sã♦ ♦s ❝❛♠♣♦s
❞❡ ✈❡t♦r❡sLs✉❛✈❡s✱ ❝♦♠♣❧❡①♦s✱ ❞❡✜♥✐❞♦s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠ ❞❡R2s❛t✐s❢❛③❡♥❞♦✿
✭✐✐✮ L(x,0) ❡Cℓ(x,0)sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱
♦♥❞❡ C0 = L, C1 = L, L, C2 = [L, C1], . . . , Cn = [L, Cn−1], n = 1,2,· · · , ℓ. ❊♠ ❬✶✼❪ ❢♦✐
❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❧♦❝❛❧ ❞❡ss❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s✳
◆❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♠♦s ✉♠❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s ❝♦♠♦ ❡♠ ❬✶✼❪✳ ❆♣r❡s❡♥t❛♠♦s ✉♠❛ ❡①t❡♥sã♦ ❞♦ tr❛❜❛❧❤♦ ❞❡ ◆✳ ❍❛♥❣❡s✳ ❈❤❛♠❛♠♦s ❡st❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s q✉❛s❡ ℓ✲▼✐③♦❤❛t❛ ❡♠ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡ Σ✳ ❈♦♥s✐❞❡r❛♠♦s ❛ s✉❜✈❛r✐❡❞❛❞❡ Σ ={t = 0} ❡ ♥♦s r❡str✐♥❣✐♠♦s ❛♦ ❝❛s♦ q✉❡ ℓ é í♠♣❛r✳
◆♦ ❈❛♣ít✉❧♦ ✷ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛ ♣❛r❛ ❡st❡ ♦♣❡r❛❞♦r r❡str✐t♦ ❛♦s s❡♠✐♣❧❛♥♦st ≥0❡t ≤0❡ ✉♠❛ ❢♦r♠❛ ♥♦r♠❛❧ ♣❛r❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ♦♣❡r❛❞♦r❡s✳ ❏á ♥♦ ❈❛♣ít✉❧♦
✸✱ ♦❜t❡♠♦s ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ f ♣❡rt❡♥❝❡r ❛
✐♠❛❣❡♠ ❞❡st❡ ♦♣❡r❛❞♦r✳
❖❜s❡r✈❛♠♦s q✉❡ ♣❛r❛ ❣❡♥❡r❛❧✐③❛r ♦ r❡s✉❧t❛❞♦ ❛❝✐♠❛ t❡r✐❛♠♦s q✉❡ ❧✐❞❛r ❝♦♠ ♦♣❡r❛❞♦r❡s ❞♦ t✐♣♦L= ∂t∂ +iλ(x, t)∂x∂✱ ❝♦♠λ(x, t) =x2−t2✱ q✉❡ ♥ã♦ é ✉♠ ❝❛♠♣♦ ❧♦❝❛❧♠❡♥t❡ r❡s♦❧ú✈❡❧
❡①❛t❛♠❡♥t❡ ♥❛s r❡t❛s t = ±x✳ ◆♦ ❡♥t❛♥t♦✱ ❡①❝❡t♦ ♣❡❧❛ ♦r✐❣❡♠✱ ♠❛s s♦❜r❡ ❡st❛s r❡t❛s ❡❧❡
é ✶✲▼✐③♦❤❛t❛✱ ❥á ♥❛ ♦r✐❣❡♠ ❡❧❡ é q✉❛s❡ 2✲▼✐③♦❤❛t❛✳ ◆❛ ❜✉s❝❛ ❞❡ ✉♠❛ té❝♥✐❝❛ ❛♣r♦♣r✐❛❞❛
♣❛r❛ ❛❜♦r❞❛r ❡ss❛ t❛❧ s✐t✉❛çã♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ❛rt✐❣♦ ❞❡ ❈✳ ❈❛♠♣❛♥❛✱ P✳ ❉❛tt♦r✐ ❡ ❆✳ ▼❡③✐❛♥✐ ✭✈❡r ❬✹❪✮✱ q✉❡ ❛♣❡s❛r ❞❡ ❧✐❞❛r ❝♦♠ ♦ ❝❛s♦ ❞❡ ♦♣❡r❛❞♦r❡s ❧♦❝❛❧♠❡♥t❡ r❡s♦❧ú✈❡✐s✱ ♥♦s ♣❛r❡❝❡✉ ❛♣r♦♣r✐❛❞♦ tr❛t❛r ♦ ❝❛s♦ ❡♠ q✉❡ λ s❡ ❛♥✉❧❛ ❡①❛t❛♠❡♥t❡ ♥✉♠ ♣♦♥t♦✳ ❆ s❛❜❡r ❡♠
❬✹❪ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ r❡♣r❡s❡♥t❛çã♦ ❡①♣❧í❝✐t❛ ❞❡ s♦❧✉çõ❡s ♥♦ ❝❛s♦ ❡♠ q✉❡ λ ≥ 0 ❡ s❡ ❛♥✉❧❛
✐❞❡♥t✐❝❛♠❡♥t❡ ❡♠ t = 0✳ ◆♦ss♦ s❡❣✉♥❞♦ r❡s✉❧t❛❞♦✱ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ✈❡♠ ❞❡
❡♥❝♦♥tr♦ ❡♠ ❡st❡♥❞❡r ♦ t❡♦r❡♠❛ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✹❪ q✉❛♥❞♦ λ s❡ ❛♥✉❧❛ ❛♣❡♥❛s ♥❛ ♦r✐❣❡♠✳
❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ L t❡♠ ❛ ❢♦r♠❛ L= ∂
∂t +i(x
2+t2)λ 0(x, t)
∂
∂x, ❝♦♠ λ0(0,0)6= 0. ✭✶✮
❈❛♣ít✉❧♦
1
Pr❡❧✐♠✐♥❛r❡s
❊st❡ ❝❛♣ít✉❧♦ s❡rá ❞❡st✐♥❛❞♦ ❛ ❛♣r❡s❡♥t❛r ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳
✶✳✶ ❚ó♣✐❝♦s ❞❡ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s ❞❛ t❡♦r✐❛ ❞❡ ❢✉♥çõ❡s ❞❡ ✈❛r✐á✈❡✐s ❝♦♠♣❧❡①❛s✱ ❛♣❡♥❛s ❛ tít✉❧♦ ❞❡ r❡❢❡rê♥❝✐❛✳ ❖s ♠❡s♠♦s s❡rã♦ ✉t✐❧✐③❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✸✳
❙❡❥❛U ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠ ❞❡R2✱ ❞❡✜♥✐♠♦s ♦s ❝♦♥❥✉♥t♦s
U− ={(x, t)∈U :t <0} ❡
U+={(x, t)∈U :t >0}.
❉❡✜♥✐çã♦ ✶✳✶✳✶ ✭❍♦❧♦♠♦r❢❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❧❡♥t♦✮✳ ❯♠❛ ❢✉♥çã♦ H é ❤♦❧♦♠♦r❢❛ ❞❡ ❝r❡s✲
❝✐♠❡♥t♦ ❧❡♥t♦ ♥♦ s❡♠✐♣❧❛♥♦ ✐♥❢❡r✐♦r s❡ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❯ ❞❛ ♦r✐❣❡♠ ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ ❝♦♥st❛♥t❡ C >0 ❡ N ∈N t❛❧ q✉❡ H é ❤♦❧♦♠♦r❢❛ ❡♠ U− ❡
|H(x+it)| ≤ C
|t|N, x+it∈U −.
❚❡♦r❡♠❛ ✶✳✶✳✷✳ ❙❡❥❛♠ H ✉♠❛ ❢✉♥çã♦ ❤♦❧♦♠♦r❢❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❧❡♥t♦ ♥♦ s❡♠✐♣❧❛♥♦ ✐♥✲
❢❡r✐♦r ❡ U =I1×I2 ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠ ❡♠R2 t❛❧ q✉❡ I1 ❡ I2 sã♦ ✐♥t❡r✈❛❧♦s ❛❜❡rt♦s
❡♠ R✳ ❊♥tã♦ H(·+it)t❡♠ ❧✐♠✐t❡ H0 ∈D′N+1(I
1)✭N ❝♦♠♦ ❞❛ ❞❡✜♥✐çã♦ ❛♥t❡r✐♦r✮ q✉❛♥❞♦
t→0✱ q✉❡ é
hH0, φi= lim
t→0−
Z
H(x+it)φ(x) dx, φ∈CN+1
c (I1). ✭❱❡r ❬✶✶❪✱ ♣á❣✐♥❛ ✻✸✮✳
❚❡♦r❡♠❛ ✶✳✶✳✸✳ ❙❡❥❛ H ✉♠❛ ❢✉♥çã♦ ❤♦❧♦♠♦r❢❛ ❡♠ U−✱ ♦♥❞❡ U é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠ ❝♦♠♦ ♥♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ ❙❡ lim
t→0H(·+it) ❡①✐st❡ ❡♠
D′N(I
1) ❡♥tã♦
|H(x+it)| ≤ C
|t|N+1, x+it ∈U
′
,
♦♥❞❡ U′ é ♦ ♣r♦❞✉t♦ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ I1′ ⊂⊂I1 ❡ ❞♦ ✐♥t❡r✈❛❧♦ (−δ2,0)✱ s❡♥❞♦ I2 = (−δ, δ)✳
❖✉ s❡❥❛✱ H é ✉♠❛ ❢✉♥çã♦ ❤♦❧♦♠♦r❢❛ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❧❡♥t♦ ♥♦ s❡♠✐♣❧❛♥♦ ✐♥❢❡r✐♦r✳
✭❱❡r ❬✶✶❪✱ ♣á❣✐♥❛ ✻✻✮✳
❖s ♣ró①✐♠♦s r❡s✉❧t❛❞♦s s❡rã♦ ✉t✐❧✐③❛❞♦s ♥❛ ❙❡çã♦ ✷ ❞♦ ❈❛♣ít✉❧♦ ✸✳
Pr♦♣♦s✐çã♦ ✶✳✶✳✹ ✭❉❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❈❛✉❝❤②✮✳ ❙❡❥❛ f ∈ C∞(I) ♣❛r❛ ❛❧❣✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ I✳ ❆ ❢✉♥çã♦ f é ❛♥❛❧ít✐❝❛ r❡❛❧ ❡♠ I s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ ❝❛❞❛ α ∈ I✱ ❡①✐st❡ ✉♠
✐♥t❡r✈❛❧♦ ❛❜❡rt♦ J✱ ❝♦♠ α ∈J ⊆I ❡ ❝♦♥st❛♥t❡s C > 0 ❡ R >0 t❛❧ q✉❡ ❛s ❞❡r✐✈❛❞❛s ❞❡ f
s❛t✐s❢❛③❡♠
|f(j)(x)
| ≤C j!
Rj, ∀x∈J.
Pr♦♣♦s✐çã♦ ✶✳✶✳✺✳ ❉❛❞♦ ❛❜❡rt♦ D ⊂ Cn✱ G(z, ξ) ❝♦♥tí♥✉❛ ❡♠ D×Rm✱ z 7−→ G(z, ξ) ❤♦❧♦♠♦r❢❛ ❡♠ D ❡ |G(z, ξ)| ≤h(ξ)∈L1✳ ❊♥tã♦✱
P(z) =
Z
G(z, ξ) dξ é ❤♦❧♦♠♦r❢❛ ❡♠ D.
❚❡♦r❡♠❛ ✶✳✶✳✻ ✭❚❡♦r❡♠❛ ❞❡ ▼♦♥t❡❧✮✳ ❙❡❥❛♠ D ⊆ C ❛❜❡rt♦ ❡ F ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❢✉♥çõ❡s
❤♦❧♦♠♦r❢❛s ❡♠ D✳ ❙✉♣♦♥❤❛ q✉❡ F é ❧♦❝❛❧♠❡♥t❡ ❧✐♠✐t❛❞❛✳ ❊♥tã♦ F é ♥♦r♠❛❧✱ ✐st♦ é✱
❝❛❞❛ s❡q✉ê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ F t❡♠ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣✐♥❞♦ ✉♥✐❢♦r♠❡♠❡♥t❡ ❡♠ ❝❛❞❛
s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❞❡ D✳
✶✳✷ ▼✉❞❛♥ç❛s ❞❡ ✈❛r✐á✈❡✐s ♥♦ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡ ✉♠
♦♣❡r❛❞♦r
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❛ s❡çã♦ é ♠♦str❛r ❝♦♠♦ ♦ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ✭❖❉P▲✮ s❡ tr❛♥s❢♦r♠❛ ❛tr❛✈és ❞❡ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s✳ ❊ss❡ r❡s✉❧t❛❞♦ s❡rá út✐❧ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ t❡①t♦ q✉❛♥❞♦ r❡❛❧✐③❛r♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♥♦ ♦♣❡r❛❞♦r✳
❈♦♥s✐❞❡r❛r❡♠♦s ♥❡st❛ s❡çã♦ ❝♦♠♦ ✉♠ ❖❉P▲ ❞❡ ♦r❞❡♠k ∈N❡♠ ✉♠ ❛❜❡rt♦U ⊂Rn ✉♠❛ ❛♣❧✐❝❛çã♦ P :C∞(U)→C∞(U)✱ ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
P(x, D) = P = X
|α|≤k
aα(x)Dα, s❡♥❞♦ Dα = (
−i)|α|∂α ❡a
α ∈C∞(U),|α| ≤k✳ ❆ss♦❝✐❛♠♦s ❛P ♦s s❡❣✉✐♥t❡s ♣♦❧✐♥ô♠✐♦s ❡♠
ξ✿
p(x, ξ) = X
|α|≤k
aα(x)ξα, x∈U, ξ ∈Rn ❡
pk(x, ξ) =
X
|α|=k
aα(x)ξα, x∈U, ξ ∈Rn. ❉❡✜♥✐♠♦s ♦ sí♠❜♦❧♦ ❞❡ P ❡ ♦ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡ P ♣♦r
σ(P)(x, ξ) = p(x, ξ)
❡
σp(P)(x, ξ) = pk(x, ξ),
r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚❡♠♦s ♦s s❡❣✉✐♥t❡s r❡s✉❧t❛❞♦s ❛ r❡s♣❡✐t♦ ❞♦ sí♠❜♦❧♦ ♣r✐♥❝✐♣❛❧ ❞❡ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ P✳
▲❡♠❛ ✶✳✷✳✶✳ ❙❡❥❛♠ P ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❞❡✜♥✐❞♦ ❡♠ U✱ x ∈ U ❡ ξ ∈ Rn✳ ❙❡
f, ϕ∈C∞(U) sã♦ t❛✐s q✉❡ f(x) = 1 ❡ dϕ(x) = ξ✱ ❡♥tã♦
σp(P)(x, ξ) = lim t→∞t
−k
e−itϕ(x)P(eitϕf)(x).
❉❡♠♦♥str❛çã♦✳ ❙❡❣✉❡ ❞❛ ❛♣❧✐❝❛çã♦ ❞❛ ❢ór♠✉❧❛ ❞❡ ▲❡✐❜♥✐③✳
❈♦♥s✐❞❡r❛r❡♠♦s ❛❣♦r❛ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦h :U →U′✱ ♦♥❞❡ U′ é ♦✉tr♦ ❛❜❡rt♦ ❞❡ Rn✳
❖ ❞✐❢❡♦♠♦r✜s♠♦ h ✐♥❞✉③ ❛s s❡❣✉✐♥t❡s ❛♣❧✐❝❛çõ❡s✿ h∗ :C∞(U′
)→C∞(U)
f 7→f ◦h;
h∗ :End(C∞(U))→End(C∞(U
′ ))
Q7→(h∗)−1◦Q◦h∗;
T∗h:U′
×Rn →U×Rn
(y, η)7→(h−1(y), dh(h−1(y))·η)
❡ T∗h ✐♥❞✉③ ❛ ❛♣❧✐❝❛çã♦
(T∗h)∗ :C∞(U ×Rn)→C∞(U′
×Rn)
σ 7→σ◦(T∗h). ❈♦♠ ✐ss♦ t❡♠♦s ❛ s❡❣✉✐♥t❡
Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ❆ ❛♣❧✐❝❛çã♦ h∗ ❛♣❧✐❝❛ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❞❡✜♥✐❞♦ ❡♠U ❜✐❥❡t✐✈❛✲ ♠❡♥t❡ ❡♠ ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❞❡✜♥✐❞♦ ❡♠ U′✳ ❆❧é♠ ❞✐ss♦✱
σp(h∗(P)) = (T∗h)∗(σp(P)), ❡♠ U
′
×Rn.
❉❡♠♦♥str❛çã♦✳ ❆ ♣r♦✈❛ ❞❛ ♣r✐♠❡✐r❛ ❛♣❧✐❝❛çã♦ s❡❣✉❡ ❞❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛ ❡ ❞❛ ❢ór♠✉❧❛ ❞❡ ▲❡✐❜♥✐③✳ ❏á ♣❛r❛ ❛ s❡❣✉♥❞❛ ❛✜r♠❛çã♦✱ ✜①❛♠♦s (x, ξ)∈U ×Rn ❡ t♦♠❛♠♦s
(y, η) = (h(x), ξ·(dh(x))−1)
❡ f, ϕ∈ C∞(U′
) t❛✐s q✉❡ f(y) = 1 ❡ dϕ(y) =η✳ ❈♦♠ ✐ss♦ ❛♣❧✐❝❛♥❞♦ ♦ ❧❡♠❛ ❛♥t❡r✐♦r ❞✉❛s
✈❡③❡s✱ ❝♦♥❝❧✉í♠♦s ♦ r❡s✉❧t❛❞♦✳
❊①❡♠♣❧♦ ✶✳✷✳✸✳ ❙❡❥❛P ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♦r❞❡♠ ✉♠ ❞❡✜♥✐❞♦ ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦
❛❜❡rt♦ U ❞❡ R2✳ ❉✐❣❛♠♦s q✉❡ P t❡♠ ❛ ❢♦r♠❛
P =A(x, t) ∂
∂x +B(x, t) ∂
∂t. ✭✶✳✶✮
❊♥tã♦ ❛♣ós ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ϕ : U → U′✱ ϕ(x, t) = (y(x, t), s(x, t)) t❡♠♦s q✉❡ ♦
♦♣❡r❛❞♦r P t❡♠ ❛ ❢♦r♠❛
˜
P =ϕ∗(P) = P(y)(x, t)
∂
∂y +P(s)(x, t) ∂
∂s. ✭✶✳✷✮
❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡
σp(P)(x, t, ξ, η) =A(x, t)ξ+B(x, t)η, (x, t)∈U,(ξ, η)∈R2.
❆ss✐♠✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r t❡♠♦s
σp( ˜P)(ϕ(x, t), ξ, η) = (T∗ϕ)∗(σp(P))(ϕ(x, t), ξ, η)
= σp(P)◦(T∗h)∗(ϕ(x, t), ξ, η)
= σp(P)(x, t, dϕ(x, t)·(ξ, η))
= σp(P)(x, t, ξyx+ηsx, ξyt+ηst)
= A(x, t)(ξyx+ηsx) +B(x, t)(ξyt+ηst)
= (A(x, t)yx+B(x, t)yt)ξ+ (B(x, t)sx+B(x, t)st)η
= P(y)(x, t)ξ+P(s)(x, t)η.
▲♦❣♦ P˜ t❡♠ ❛ ❢♦r♠❛ ✭✶✳✷✮✳
❊①❡♠♣❧♦ ✶✳✷✳✹✳ ❙❡❥❛ L ❝♦♠♦ ❡♠ ✭✶✳✶✮✱ ❝♦♠ A ♦✉ B ♥ã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦s✳ ❊♥tã♦ L
♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠ ♠ú❧t✐♣❧♦ ♥ã♦ ♥✉❧♦ ❞❡
L= ∂
∂t+iλ(x, t) ∂
∂x, ✭✶✳✸✮
❝♦♠ λ r❡❛❧✱ s✉❛✈❡ ❡ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠✳
❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❛♣ós tr❛♥s❧❛çã♦ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ L ❡stá ❞❡✜♥✐❞♦ ❡♠ ✉♠❛
✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠✳ P♦❞❡♠♦s ❛ss✉♠✐r t❛♠❜é♠ q✉❡ B(0,0) 6= 0✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠❛
✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠ t❛❧ q✉❡ B(x, t)6= 0✳ ▲♦❣♦✱ t♦♠❛♥❞♦ C= A
B✱ t❡r❡♠♦s
L=B
∂ ∂t +
A B
∂ ∂x
=B
∂ ∂t+C
∂ ∂x
.
❆ss✐♠✱ ✈❛♠♦s s✉♣♦r q✉❡ Lt❡♠ ❛ ❢♦r♠❛ L= ∂
∂t+C(x, t) ∂ ∂x,
♦♥❞❡✱ C(x, t) =C1(x, t) +iC2(x, t)✳
▲♦❣♦ q✉❡r❡♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❞❛ ❢♦r♠❛x=x(y, s) ❡t =s t❛❧ q✉❡ ∂
∂s = ∂
∂t+C1(x, t) ∂ ∂x.
❆ss✐♠✱ t♦♠❛♠♦sy(x, t)s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡ P✳❱✳■
∂y
∂t +C1(x, t) ∂y ∂x = 0 y|t=0 =x,
♦ q✉❛❧ ♣♦ss✉✐ ú♥✐❝❛ s♦❧✉çã♦✳
◆♦t❡ q✉❡(x, t)7−→(y, s)é ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❛ ♦r✐❣❡♠✳
❆ss✐♠✱ ♣❡❧♦ ❊①❡♠♣❧♦ ✶✳✷✳✸ t❡♠♦s q✉❡ L s❡ tr❛♥s❢♦r♠❛ ♣♦r ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ❡♠ ✉♠
♦♣❡r❛❞♦r ❞❛ ❢♦r♠❛
˜
L =
∂y
∂t +C(x, t) ∂y ∂x
∂ ∂y +
∂ ∂s
=
∂y
∂t +C1(x, t) ∂y ∂x
∂
∂y +iC2(x, t) ∂y ∂x
∂ ∂y +
∂ ∂s
= ∂
∂s+iC2(x, t) ∂y ∂x
∂ ∂y
= ∂
∂s+iλ(y, s) ∂ ∂y.
❝♦♠ λ r❡❛❧ ❡ s✉❛✈❡✳
❘❡♥♦♠❡❛♥❞♦ ❛s ✈❛r✐á✈❡✐s t❡♠♦s q✉❡L t❡♠ ❛ ❢♦r♠❛ ✭✶✳✸✮✳
✶✳✸ ❖♣❡r❛❞♦r❡s ♣❛r❝✐❛❧♠❡♥t❡ ❤✐♣♦❡❧í♣t✐❝♦s
❊♠ ✉♠ tr❛❜❛❧❤♦ ❞❡ ▲✳ ●❛r❞✐♥❣ ❡ ❇✳ ▼❛❧❣r❛♥❣❡ ❡♠ ✶✾✻✶ ✭✈❡r ❬✼❪✮ ❢♦✐ ❛♣r❡s❡♥t❛❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❛ ❞❡✜♥✐çã♦ ❞❡ ♦♣❡r❛❞♦r ♣❛r❝✐❛❧♠❡♥t❡ ❤✐♣♦❡❧í♣t✐❝♦✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s
❝♦♥s✐❞❡r❛r❛♠ ♦♣❡r❛❞♦r❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✳ ◆♦ ❛♥♦ s❡❣✉✐♥t❡✱ ❙✳ ▼✐③♦❤❛t❛ ✭✈❡r ❬✶✻❪✮ ♣✉❜❧✐❝♦✉ ✉♠ tr❛❜❛❧❤♦ ♦♥❞❡ ❡❧❡ ♠♦str❛ ❝♦♠♦ ❧✐❞❛♠♦s ❝♦♠ ❡ss❡s r❡s✉❧t❛❞♦s ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❝♦❡✜❝✐❡♥t❡s ✈❛r✐á✈❡✐s✳
❈♦♥s✐❞❡r❛r❡♠♦s ♥❡ss❛ s❡çã♦✱ P(D) ❡ P(x, D) ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐á✈❡✐s ❝♦♠ ❝♦❡✜❝✐✲
❡♥t❡s ❝♦♥st❛♥t❡s ❡ ❝♦❡✜❝✐❡♥t❡s ✈❛r✐á✈❡✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❉❡✜♥✐çã♦ ✶✳✸✳✶ ✭❖♣❡r❛❞♦r ❤✐♣♦❡❧í♣t✐❝♦✮✳ ❯♠ ♦♣❡r❛❞♦r P(x, D)é ❞✐t♦ s❡r ❤✐♣♦❡❧í♣t✐❝♦ s❡✱
❡ s♦♠❡♥t❡ s❡✱ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ Ω⊂Rn ❡ u∈D′(Ω) t❡♠♦s
SS(P(x, D)u) = SS(u).
❉❡✜♥✐çã♦ ✶✳✸✳✷ ✭❉✐str✐❜✉✐çã♦ ♣❛r❝✐❛❧♠❡♥t❡ r❡❣✉❧❛r✮✳ ❙❡❥❛Ω✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡Rm×Rn
❡ u(x, t) ∈ D′(Ω)✳ ❉✐③❡♠♦s q✉❡ u é r❡❣✉❧❛r ❡♠ t s❡✱ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ❛❜❡rt♦s V ⊂ Rm✱
W ⊂Rn✱ V
×W ⊂Ω ❡ t♦❞❛ ϕ ∈C∞
c (V)✱ ❛ ❞✐str✐❜✉✐çã♦ uϕ ❡♠ W ❞❡✜♥✐❞❛ ♣♦r
uϕ :φ(t)7−→ hu(x, t), φ(t)ϕ(x)i é ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ❞❡ t✳
◆♦t❛çã♦✿ u∈C∞(W,D′(V))✳
❉❡✜♥✐çã♦ ✶✳✸✳✸ ✭❖♣❡r❛❞♦r ♣❛r❝✐❛❧♠❡♥t❡ ❤✐♣♦❡❧í♣t✐❝♦✮✳ ❙❡❥❛ L=L(x, t, Dx, Dt) ✉♠ ♦♣❡✲ r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧✳ L é ❞✐t♦ s❡r ❤✐♣♦❡❧í♣t✐❝♦ ❡♠ t s❡✱ ♣❛r❛ ❝❛❞❛ f r❡❣✉❧❛r ❡♠ t t❡♠✲s❡ q✉❡ ❛
❞✐str✐❜✉✐çã♦ u t❛❧ q✉❡ Lu=f é r❡❣✉❧❛r ❡♠ t✳
P❛r❛ ✉♠ ♦♣❡r❛❞♦r ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡✱ P(Dx, Dt)✱ ♣♦❞❡♠♦s ❞❛r ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦ ❡q✉✐✈❛❧❡♥t❡ ♣❛r❛ ❤✐♣♦❡❧✐♣t✐❝✐❞❛❞❡ ♣❛r❝✐❛❧✳ ▼❛s ❛♥t❡s ♣r❡❝✐s❛r❡♠♦s ❞❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✿
❉❡✜♥✐çã♦ ✶✳✸✳✹ ✭❖♣❡r❛❞♦r ❡str✐t❛♠❡♥t❡ ♠❡♥♦s ❢♦rt❡✮✳ ❙❡❥❛ P ✉♠ ♦♣❡r❛❞♦r ❤✐♣♦❡❧í♣t✐❝♦✳
❉✐③❡♠♦s q✉❡ ✉♠ ♦♣❡r❛❞♦r Q é ❡str✐t❛♠❡♥t❡ ♠❡♥♦s ❢♦rt❡ q✉❡ P ✭❡s❝r❡✈❡♠♦s Q << P✮✱s❡ σ(Q)(ξ)
σ(P)(ξ) −→0 q✉❛♥❞♦ ξ → ∞.
❚❡♦r❡♠❛ ✶✳✸✳✺ ✭❬✼❪✱ ♣á❣✐♥❛ ✾✮✳ P❛r❛ q✉❡ P = P(Dx, Dt) s❡❥❛ ♣❛r❝✐❛❧♠❡♥t❡ ❤✐♣♦❡❧í♣t✐❝♦ ❡♠ t é ♥❡❝❡ssár✐♦ ❡ s✉✜❝✐❡♥t❡ q✉❡ ❛s s❡❣✉✐♥t❡ ❝♦♥❞✐çõ❡s s❡❥❛♠ ❡q✉✐✈❛❧❡♥t❡s✿
✶✳ σ(P)(ξ, η) = 0✱ Reξ ❡ η ❧✐♠✐t❛❞♦s ⇒ Imξ ❧✐♠✐t❛❞♦❀
✷✳ P ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❛ ❢♦r♠❛ σ(P)(ξ, η) = σ(P0)(η) +
X
σ(Pj)(η)σ(Qj)(ξ), j >0, ♦♥❞❡ P0 é ❤✐♣♦❡❧í♣t✐❝♦ ❡ Pj << P0✳
❏á ♣❛r❛ ✉♠ ♦♣❡r❛❞♦r ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✈❛r✐á✈❡✐s✱ ❙✳ ▼✐③♦❤❛t❛✱ ♠♦str♦✉ ✉♠ r❡s✉❧t❛❞♦ ❡q✉✐✈❛❧❡♥t❡✳ P❛r❛ ✐ss♦✱ ❝♦♥s✐❞❡r❛♠♦s P(x, t, Dt) ✉♠ ♦♣❡r❛❞♦r ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ✈❛r✐á✈❡✐s ❡ t❛❧ q✉❡✱ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ (x0, t0)✱ P s❡ ❡①♣r❡ss❛ ♣♦r
P(x, t, Dt) =
X
aj(x, t)Mj(Dt) ✭✶✳✹✮ ♦♥❞❡✱
✶♦✮ ♦s ❝♦❡✜❝✐❡♥t❡s a′
js sã♦ s✉❛✈❡s❀ ✷♦✮ M(D
t) = Paj(x0, t0)Mj(Dt) é ❤✐♣♦❡❧í♣t✐❝♦❀ ✸♦✮ ♦s ♦♣❡r❛❞♦r❡s M′
js sã♦ ❤✐♣♦❡❧í♣t✐❝♦s ❡ ❡q✉✐✈❛❧❡♥t❡s à M✳
❚❡♦r❡♠❛ ✶✳✸✳✻ ✭❬✶✻❪✱ ♣á❣✐♥❛ ✹✷✸✮✳ ❙❡❥❛L(x, t, Dx, Dt)✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❧✐♥❡❛r ❝♦♠ ❝♦❡✜❝✐❡♥t❡s s✉❛✈❡s✳ ❙✉♣♦♥❤❛ q✉❡
L=P(x, t, Dt) +
X
Pj(x, t, Dt)Qj(Dx) ✭✶✳✺✮ ✈❡r✐✜❝❛ ❛s ❝♦♥❞✐çõ❡s✿
✭✐✮ ♦ ♦♣❡r❛❞♦r P s❛t✐s❢❛③ ✭✶✳✹✮❀
✭✐✐✮ ❝❛❞❛ ♦♣❡r❛❞♦r Pj é ❡str✐t❛♠❡♥t❡ ♠❡♥♦s ❢♦rt❡ q✉❡ ♦ ♦♣❡r❛❞♦r M❀ ✭✐✐✐✮ ♦s Q′
js sã♦ ♦♣❡r❛❞♦r❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡s✳ ❊♥tã♦✱ L é ❤✐♣♦❡❧í♣t✐❝♦ ❡♠ t✳
❊①❡♠♣❧♦ ✶✳✸✳✼✳ ❈♦♥s✐❞❡r❛♠♦s ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♦r❞❡♠ ✉♠ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛
L=Dt+λ(x, t)Dx,
♦♥❞❡ é λ s✉❛✈❡ ❡ ❞❡✜♥✐❞♦ ♥♦ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ Ω ❞❡ R2✳ ❊♥tã♦ L é ❤✐♣♦❡❧í♣t✐❝♦ ❡♠ t✳
❉❡♠♦♥str❛çã♦✳ P♦❞❡♠♦s ❡s❝r❡✈❡r L ❞❛ ❢♦r♠❛
L=P(Dt) +P0(Dt)Q0(Dx),
♦♥❞❡ P(Dt) =Dt✱ P0(Dt) =λ(x, t) ✭♦♣❡r❛❞♦r ❞❡ ♦r❞❡♠ ③❡r♦✮ ❡ Q0(Dx) =Dx✳ ❚❡♠♦s q✉❡
L ✈❡r✐✜❝❛ ❛s ❝♦♥❞✐çõ❡s ✭✐✮✲✭✐✐✐✮✳ ❉❡ ❢❛t♦✱P s❛t✐s❢❛③ ✭✶✳✹✮✱ ♣♦✐s P =Dt é ❤✐♣♦❡❧í♣t✐❝♦✱ Q0 é
✉♠ ♦♣❡r❛❞♦r ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❝♦♥st❛♥t❡✳ ❆❧é♠ ❞✐ss♦✱
σ(P0)(η)
σ(P)(η) =
λ(x, t)
η →0q✉❛♥❞♦ η →0,
♦✉ s❡❥❛✱ P0 é ❡str✐t❛♠❡♥t❡ ♠❡♥♦s ❢♦rt❡ q✉❡ P✳ P♦rt❛♥t♦✱ L é ❤✐♣♦❡❧í♣t✐❝♦ ✭♣❛r❝✐❛❧♠❡♥t❡
❤✐♣♦❡❧í♣t✐❝♦✮ ❡♠ t✳
✶✳✹ ❈❛♠♣♦s ❞❡ ✈❡t♦r❡s ❝♦♠♣❧❡①♦s
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛r❡♠♦s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s q✉❡ ❛♣❛r❡❝❡r❛♠ ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❖ ❝♦♥t❡ú❞♦ ❞❡ss❛ s❡çã♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ ❈❛♣ít✉❧♦ ✶ ❞❡ ❬✸❪✳
❆♦ ❧♦♥❣♦ ❞❡ t♦❞❛ ❡st❛ s❡çã♦✱ ❝♦♥s✐❞❡r❛r❡♠♦s Ω ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ ❞❡ ❞✐✲
♠❡♥sã♦ n✳
❉❡✜♥✐çã♦ ✶✳✹✳✶✳ ❯♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s ❝♦♠♣❧❡①♦ ✭s✉❛✈❡✮ s♦❜r❡Ωé ✉♠❛ ❛♣❧✐❝❛çã♦C✲❧✐♥❡❛r
L:C∞(Ω) →C∞(Ω)
q✉❡ s❛t✐s❢❛③ ❛ r❡❣r❛ ❞❡ ▲❡✐❜♥✐③
L(f g) =f(Lg) +g(Lf), f, g ∈C∞(Ω).
❱❛♠♦s ❞❡♥♦t❛r ♣♦rX(Ω) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❝♦♠♣❧❡①♦s s♦❜r❡
Ω✳
❊①❡♠♣❧♦ ✶✳✹✳✷✳ ❯♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ❞❡ ♦r❞❡♠ ✉♠ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠
C∞ s❡♠ t❡r♠♦ ❞❡ ♦r❞❡♠ ③❡r♦ é ✉♠ ❡①❡♠♣❧♦ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✸✳ ❙❡ L∈X(Ω) ❡ s❡ f é ❝♦♥st❛♥t❡ ❡♥tã♦ Lf = 0✳ ❆❧é♠ ❞✐ss♦✱
S(Lf)⊂S(f), ∀f ∈C∞(Ω), L∈X(Ω).
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ r❡s✉❧t❛❞♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ r❡str✐çã♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦L∈
X(Ω) ❛ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ W ❞❡ Ω✱ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✱ s❡ p ∈ W ❡ f ∈ C∞(W)✱
❞❡✜♥✐♠♦s ❛ ❛♣❧✐❝❛çã♦
LW(f)(p) = L(fe)(p),
♦♥❞❡feé q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❡♠C∞(Ω) q✉❡ ❝♦✐♥❝✐❞❡ ❝♦♠f ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡p✳ ❈❛❞❛
LW ❞❡✜♥❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ X(W)✳ ❯s✉❛❧♠❡♥t❡ ✈❛♠♦s ❡s❝r❡✈❡r L ❛♦ ✐♥✈és ❞❡ LW✳
✶✳✹✳✶ ❆ ❡str✉t✉r❛ ❛❧❣é❜r✐❝❛ ❞❡
X
(Ω)
❉❛❞♦g ∈C∞(Ω) ❡L∈X(Ω) ♣♦❞❡♠♦s ❞❡✜♥✐r gL∈X(Ω) ♣♦r
(gL)(f) =g·L(f), f ∈C∞(Ω).
❚❛❧ ♠✉❧t✐♣❧✐❝❛çã♦ ❡①t❡r♥❛ ❞á ❛X(Ω) ✉♠❛ ❡str✉t✉r❛ ❞❡ C∞✲♠ó❞✉❧♦✳
❯♠❛ ♦♣❡r❛çã♦ ✭✐♥t❡r♥❛✮ ✐♠♣♦rt❛♥t❡ ❡♠ X(Ω) é ❝❤❛♠❛❞❛ ❞❡ ❝♦♠✉t❛❞♦r ✭♦✉ ❝♦❧❝❤❡t❡
❞❡ ▲✐❡✮ ❡♥tr❡ ❞♦✐s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s✳ ❉❛❞♦s L✱M ∈X(Ω) ❞❡✜♥✐♠♦s
[L, M] (f) = L(M(f))−M(L(f)), f ∈C∞(Ω).
❱❡r✐✜❝❛✲s❡ q✉❡ [L, M] ∈ X(Ω)✳ ❊st❛ ♦♣❡r❛çã♦ tr❛♥s❢♦r♠❛ X(Ω) ❡♠ ✉♠❛ á❧❣❡❜r❛ ❞❡ ▲✐❡
s♦❜r❡ C✳
❙❡❥❛ (U,①) ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❡♠ Ω ❡ s❡❥❛ t❛♠❜é♠ L ∈ X(U)✳ ❋✐①❡♠♦s p ∈ U ❡
❡s❝r❡✈❡♠♦s
①(q) = (x1(q), x2(q), . . . , xn(q)), q ∈U.
❆❣♦r❛✱ t♦♠❛♠♦s V ⊂U ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ t❛❧ q✉❡ ①(V) é ✉♠❛ ❜♦❧❛ ❛❜❡rt❛ ❝❡♥tr❛❞❛ ❡♠
①(p) = a = (a1, . . . , an)✳ ❉❛❞❛ f ∈C∞(U)✱ ❡s❝r❡✈❡ f∗ =f ◦①−1✳ ❙❡ (x1, . . . , xn) ∈ ①(V)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞♦ ❈á❧❝✉❧♦ ❛♣❧✐❝❛❞♦ ❛ ❢✉♥çã♦
s❡❣✉❡ q✉❡
f∗(x1, . . . , xn) = f∗(a1, . . . , an) + n
X
j=1
hj(x1, . . . , xn)(xj−aj), ♦♥❞❡ hj ∈C∞(①(V))❡ hj(a) =
∂f∗ ∂xj
(a)✳ ❙❡ ❞❡✜♥✐r♠♦s gj =hj ◦①∈C∞(U)✱ ♦❜t❡♠♦s
f(q) =f(p) +
n
X
j=1
gj(q)(xj(q)−xj(p)), q ∈V ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♣❡❧❛ ❘❡❣r❛ ❞❡ ▲❡✐❜♥✐③
L(f)(p) =
n
X
j=1
gj(p)(Lxj)(p). ✭✶✳✻✮ ❉❡✜♥✐çã♦ ✶✳✹✳✹✳ ❆ ❛♣❧✐❝❛çã♦ C−❧✐♥❡❛r C∞(U)→C∞(U) ❞❛❞❛ ♣♦r
f 7→ ∂f
∗
∂xj ◦ ①
❞❡✜♥❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ X(U)✱ q✉❡ é ❞❡♥♦t❛❞♦ ♣♦r ∂
∂xj✳
❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r
gj(p) =hj(①(p)) =
∂f∗
∂xj
(①(p)) =
∂ ∂xj
(f)(p). ✭✶✳✼✮
❆❣♦r❛✱ s✉❜st✐t✉✐♥❞♦ ✭✶✳✼✮ ❡♠ ✭✶✳✻✮ ♦❜t❡♠♦s
L(f)(p) =
n
X
j=1
(Lxj)(p)
∂ ∂xj
(f)(p).
❈♦♠♦p∈U é ❛r❜✐trár✐♦✱ ♦❜t❡♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡L♥❛s ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s(x1, . . . , xn) ♣♦r
L=
n
X
j=1
(Lxj)
∂ ∂xj
.
❖❜s❡r✈❛çã♦ ✶✳✹✳✺✳ ❊ss❛ r❡♣r❡s❡♥t❛çã♦ ♠♦str❛ q✉❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦r é ✉♠ ♦♣❡r❛❞♦r ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❞❡ ♦r❞❡♠ ✉♠ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ C∞✱ s❡♠ t❡r♠♦ ❞❡ ♦r❞❡♠ ③❡r♦✳
❆❧é♠ ❞✐ss♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❡ss❛ r❡♣r❡s❡♥t❛çã♦ ♠♦str❛ q✉❡n ∂ ∂x1, . . . ,
∂ ∂xn
o
é ✉♠❛ ❜❛s❡ ❞❡ X(U)✳ ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ s❡M ∈ X(U) ❡♥tã♦ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ [L, M] ❡♠ ❝♦♦r❞❡✲
♥❛❞❛s ❧♦❝❛✐s (x1, . . . , xn) é ❞❛❞❛ ♣♦r ❞❛❞❛ ♣♦r
[L, M] =
n
X
j=1
{L(M xj)−M(Lxj)}
∂ ∂xj
.
✶✳✹✳✷ ❊str✉t✉r❛s ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s
❉❡♥♦t❡ ♣♦rBp ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣❛r❡s(V, f)✱ ♦♥❞❡V é ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡p ❡ f ∈C∞(V)✳ ❊♠ B
p ✐♥tr♦❞✉③✐♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✿
• (V1, f1)∼(V2, f2) s❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ V ❞❡p✱ V ⊂V1∩V2✱ t❛❧ q✉❡ f1 ❡
f2 ❝♦✐♥❝✐❞❡♠ ❡♠ V✳
❯♠ ❣❡r♠❡ ❞❡ ✉♠❛ ❢✉♥çã♦ C∞✭s✉❛✈❡✮ ❡♠p é ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❡s♣❛ç♦ q✉♦❝✐❡♥t❡
C∞(p)=. Bp/∼.
❖❜s❡r✈❛♠♦s q✉❡ C∞(p)é t❛♠❜é♠ ✉♠❛ C✲á❧❣❡❜r❛✳ ❉❛❞❛ ✉♠❛ ❢✉♥çã♦ f s✉❛✈❡ ❞❡✜♥✐❞❛ ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ ❞❡ p✱ ♦ ❣❡r♠❡ ❡♠ p ❞❡✜♥✐❞♦ ♣♦r f s❡rá ❞❡♥♦t❛❞♦ ♣♦rf✳ ◆♦t❡ q✉❡
❡①✐st❡ ✉♠ ❤♦♠❡♦r♠♦r✜s♠♦ ♥❛t✉r❛❧ C∞(p)→C ❞❡✜♥✐❞♦ ♣♦r f →f(p)✳
❉❡✜♥✐çã♦ ✶✳✹✳✻✳ ❯♠ ✈❡t♦r t❛♥❣❡♥t❡ ❝♦♠♣❧❡①♦ ✭❡♠ Ω✮ ❡♠ p é ✉♠❛ ❛♣❧✐❝❛çã♦ C✲❧✐♥❡❛r
v :C∞(p)→C
s❛t✐s❢❛③❡♥❞♦
v(f g) =f(p)v(g) +g(p)v(f).
❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ✈❡t♦r❡s t❛♥❣❡♥t❡s ❝♦♠♣❧❡①♦s ❡♠ p s❡rá ❞❡♥♦t❛❞♦ ♣♦r CTpΩ✱
t❡♥❞♦ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ C❡ s❡rá ❝❤❛♠❛❞♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❝♦♠♣❧❡①♦
❞❡ Ω ❡♠ p✳
❙❡L∈X(Ω) ❡♥tã♦ Lp :C∞(p)→C ❞❡✜♥✐❞♦ ♣♦r
Lp(f) =L(f)(p)
♣❡rt❡♥❝❡ à CTpΩ✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❛❞❛ p∈Ω ✉♠ ❡❧❡♠❡♥t♦vp ∈CTpΩ
é ❞❛❞♦ t❛❧ q✉❡
p7→vp(f)∈C∞(Ω), ∀f ∈C∞(Ω). ❊♥tã♦ ❡①✐st❡ L∈X(Ω) t❛❧ q✉❡ Lp =vp ♣❛r❛ t♦❞♦p∈Ω✳
❖ ✜❜r❛❞♦ t❛♥❣❡♥t❡ ❝♦♠♣❧❡①✐✜❝❛❞♦ ❞❡Ω é ❞❡✜♥✐❞♦ ❝♦♠♦ ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛
CTΩ = [
p∈Ω
CTpΩ.
❱❛♠♦s ♣r❡❝✐s❛r t❛♠❜é♠ ❞❛ ♥♦t❛çã♦ ❞❡ ✉♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ CTΩ ❞❡ ♣♦st♦
n✳ P♦r ❡ss❡✱ ♥♦s r❡❢❡r✐♠♦s ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛
V = [
p∈Ω
Vp ⊂CTΩ s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
✭❛✮ ♣❛r❛ ❝❛❞❛ p∈Ω✱Vp é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ CTpΩ ❞❡ ❞✐♠❡♥sã♦n❀
X(U0)t❛✐s q✉❡ L1p, . . . , Lnp ❣❡r❛♠Vp ♣❛r❛ ❝❛❞❛ p∈U0✳
❖ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ Vp é ❝❤❛♠❛❞♦ ❞❡ ✜❜r❛ ❞❡V ❡♠ p✳
❉❛❞♦s ✉♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ V ❞❡ CTΩ ❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ W ❞❡
Ω✱ ✉♠❛ s❡çã♦ ❞❡ V s♦❜r❡W é ✉♠ ❡❧❡♠❡♥t♦ L ❞❡X(W) t❛❧ q✉❡Lp ∈ Vp ♣❛r❛ ❝❛❞❛ p∈W✳
❆❣♦r❛ ❡st❛♠♦s ♣r♦♥t♦s ♣❛r❛ ❞❡✜♥✐r ♦ q✉❡ ✈❡♠ ❛ s❡r ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡ss❛ t❡♦r✐❛✳
❉❡✜♥✐çã♦ ✶✳✹✳✼✳ ❯♠❛ ❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s♦❜r❡ Ω é ✉♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧
❝♦♠♣❧❡①♦ V ❞❡ CTΩ s❛t✐s❢❛③❡♥❞♦ ❛ ❝♦♥❞✐çã♦ ✐♥✈♦❧✉t✐✈❛✿
• ❙❡ W ⊂Ωé ❛❜❡rt♦ ❡ L, M ∈X(W) sã♦ s❡çõ❡s ❞❡ V s♦❜r❡W ❡♥tã♦ [L, M] é t❛♠❜é♠
✉♠❛ s❡çã♦ ❞❡ V s♦❜r❡ W✳
✶✳✹✳✸ ❋♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s
❱❛♠♦s ❞❡♥♦t❛r ♣♦rN(Ω)♦ ❞✉❛❧ ❞❡ X(Ω)❡ ✈❛♠♦s ♥♦s r❡❢❡r✐r ❛♦s s❡✉s ❡❧❡♠❡♥t♦s ❝♦♠♦
❝♦♠♦ ❢♦r♠❛s ❞✐❢❡r❡♥❝✐❛✐s s♦❜r❡ Ω ❞❡ ❣r❛✉ ✉♠ ✭♦✉ ✶✲❢♦r♠❛s✮✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ✉♠❛
✶✲❢♦r♠❛ ❡♠ Ω é ✉♠❛ ❛♣❧✐❝❛çã♦C∞(Ω)✲❧✐♥❡❛r
ω :X(Ω)→C∞(Ω).
❙❡ ❞❡✜♥✐r♠♦s
CTp∗Ω=. ❞✉❛❧ ❞❡CTpΩ,
♣❛r❛ ❝❛❞❛ w∈N(Ω) ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ❡❧❡♠❡♥t♦ ωp ∈CTp∗Ω♣❡❧❛ ❢ór♠✉❧❛
ωp(v) =ω(L)(p), ♦♥❞❡ L∈X(Ω) é t❛❧ q✉❡ Lp =v✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✽✳ CT∗
pΩ = {ωp :ω ∈N(Ω)}✳
❉❡✜♥✐çã♦ ✶✳✹✳✾✳ ❉❛❞❛ f ∈C∞(Ω) ❞❡✜♥✐♠♦sdf ∈N(Ω) ♣❡❧❛ ❢ór♠✉❧❛
df(L) =L(f), L ∈X(Ω).
P♦❞❡♠♦s ♦❜t❡r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❧♦❝❛✐s ❞❛❞❛ ♣♦r
df =
N
X
j=1
df
∂ ∂xj
dxj = N
X
j=1
∂f ∂xj
dxj.
❆❣♦r❛ ✐♥tr♦❞✉③✐♠♦s ♦ ✜❜r❛❞♦ ❝♦t❛♥❣❡♥t❡ ❝♦♠♣❧❡①✐✜❝❛❞♦ ❞❡ Ω ❝♦♠♦ s❡♥❞♦ ❛ ✉♥✐ã♦
❞✐s❥✉♥t❛
CT∗Ω=. [
p∈Ω
CTp∗Ω.
❈♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ♥♦çã♦ ❞❡ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡CT∗Ω❞❡
♣♦st♦m ❝♦♠♦ s❡♥❞♦ ❛ ✉♥✐ã♦ ❞✐s❥✉♥t❛
W = [
p∈Ω
Wp,
♦♥❞❡ ❝❛❞❛ Wp é ✉♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ CTp∗Ω ❞❡ ❞✐♠❡♥sã♦ m✱ s❛t✐s❢❛③❡♥❞♦ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✿
• ❉❛❞♦ p0 ∈ Ω ❡①✐st❡♠ ✉♠ ❝♦♥❥✉♥t♦ ❛❜❡rt♦ U0 ❝♦♥t❡♥❞♦ p0 ❡ ✶✲❢♦r♠❛s ω1, . . . , ωm ∈
N(U0)t❛✐s q✉❡ ω1p, . . . , ωmp ❣❡r❛♠ Wp ♣❛r❛ ❝❛❞❛ p∈U0✳
❈♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡ ✈❛♠♦s ♥♦s r❡❢❡r✐r ❛♦ ❡s♣❛ç♦ Wp ❝♦♠♦ ✉♠❛ ✜❜r❛ ❞❡ W ♥♦ ♣♦♥t♦ p✳ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✵✳ ❙❡❥❛V = S
p∈ΩV
p ✉♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡CTΩ❡ s❡❥❛✱ ♣❛r❛ ❝❛❞❛ p∈Ω
Vp⊥
.
=λ∈CTp∗Ω :λ= 0 ❡♠ Vp .
❊♥tã♦ V⊥ =. S p∈ΩV
⊥
p é ✉♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡ CT∗Ω✳
◗✉❛♥❞♦ V é ✉♠❛ ❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s♦❜r❡ Ω ❞❡ ❞✐♠❡♥sã♦ N ✈❛♠♦s
❞❡♥♦t❛r ♦ s✉❜✜❜r❛❞♦ V⊥ ♣♦r T′
✳ ❱❛♠♦s t❛♠❜é♠ ❞❡♥♦t❛r ♣♦r n ♦ ♣♦st♦ ❞❡ V ❡ ♣♦r m ♦
♣♦st♦ ❞❡ T′✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ n+m =N✳
❱❛♠♦s ✉s❛r t❛♠❜é♠ ❛s s❡❣✉✐♥t❡s ♥♦t❛çõ❡s✿
TpΩ=. {v ∈CTpΩ :v é r❡❛❧};
Tp∗Ω=. ξ∈CT∗
pΩ :ξ é r❡❛❧ ;
TΩ=. [
p∈Ω
TpΩ;
T∗Ω=. [
p∈Ω
Tp∗Ω.
❉❛❞♦ L∈ X(Ω) s❡✉ ❝♦♥❥✉❣❛❞♦ ✭❝♦♠♣❧❡①♦✮ é ♦ ❝❛♠♣♦ ❞❡ ✈❡t♦r❡s L ∈X(Ω) ❞❡✜♥✐❞♦
♣♦r
L(f) = L(f), f ∈C∞(Ω).
❊♠ ♣❛rt✐❝✉❧❛r✱ ❞✐③❡♠♦s q✉❡L é ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ r❡❛❧ s❡ L=L✳
❊①❡♠♣❧♦ ✶✳✹✳✶✶✳ ❙❡ L=A(x, t)∂
∂x+B(x, t) ∂
∂t✱ ♦♥❞❡ A, B sã♦ ❢✉♥çõ❡s s✉❛✈❡s ❝♦♠♣❧❡①❛s ❞❡✜♥✐❞❛s ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ W ❞❡ R2 ❡♥tã♦
L(f)L(f) = A∂f ∂x +B
∂f ∂t.
❉♦ ♠❡s♠♦ ♠♦❞♦ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♥❥✉❣❛❞♦ ✭❝♦♠♣❧❡①♦✮ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠CTpΩ✳
❉❛❞♦ ✉♠ s✉❜❡s♣❛ç♦Vp ⊂CTpΩ❞❡✜♥✐♠♦s
➱ ❝❧❛r♦ ❞❛ ❞❡✜♥✐çã♦ q✉❡ s❡ V é ✉♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ ❞❡CTΩ❡♥tã♦ ♦ ♠❡s♠♦ é
✈❡r❞❛❞❡ ♣❛r❛V =. S
p∈ΩV
p✳ ❱❛♠♦s ♥♦ r❡❢❡r✐r ❛V ❝♦♠♦ ♦ ❝♦♥❥✉❣❛❞♦ ✭❝♦♠♣❧❡①♦✮ ❞♦ s✉❜✜❜r❛❞♦ V✳ ❆♥❛❧♦❣❛♠❡♥t❡ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ✐♥tr♦❞✉③✐❞♦s ❡ ♦❜t✐❞♦s ♣❛r❛ CT∗Ω ❡
s✉❛s ✜❜r❛s CT∗
pΩ✳ ➱ ✐♠♣♦rt❛♥t❡ t❛♠❜é♠ ♠❡♥❝✐♦♥❛r ❛ ✐❣✉❛❧❞❛❞❡
V⊥ =V⊥,
❛ q✉❛❧ é ✈á❧✐❞❛ ♣❛r❛ ❝❛❞❛ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ V ❞❡ CTΩ✳
✶✳✹✳✹ ❖ ❝♦♥❥✉♥t♦ ❝❛r❛❝t❡ríst✐❝♦
❙❡❥❛ V ⊂ CTΩ ✉♠❛ ❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s♦❜r❡ Ω✳ ❖ ❝♦♥❥✉♥t♦ ❝❛r❛❝t❡✲
ríst✐❝♦ ❞❡ V é ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ T∗Ω ❞❡✜♥✐❞♦ ♣♦r
T0 =. T′ ∩T∗Ω.
❉❛❞♦ p ∈ Ω✱ ✈❛♠♦s t❛♠❜é♠ ❡s❝r❡✈❡r T0
p = T
′
p ∩Tp∗Ω✳ ❘❡❝♦r❞❛♠♦s q✉❡ ♦ sí♠❜♦❧♦ ❞❡ ✉♠ ❝❛♠♣♦ ❞❡ ✈❡t♦rL∈X(Ω) é ✉♠❛ ❢✉♥çã♦ σ(L) :T∗Ω→C❞❛❞❛ ♣♦r
σ(L)(ξ) = ξ(Lp), s❡ξ ∈Tp∗Ω. ❊♥tã♦ ✈❡♠♦s q✉❡ ξ∈T0
p s❡✱ ❡ s♦♠❡♥t❡ s❡ σ(L)(ξ) = 0✱ ♣❛r❛ ❝❛❞❛ s❡çã♦ L ❞❡V✳
❙❡❥❛ (U,①)✱ ① = (x1, . . . , xN) ✉♠❛ ❝❛rt❛ ❧♦❝❛❧ ❞❡ Ω✳ ❚♦♠❡ p ∈ U ❡ ξ ∈ Tp∗Ω✳ ❙❡ ❡s❝r❡✈❡r♠♦s ξ=
N
P
j=1
ξjdxjp (ξj ∈R)❡ L= N
P
j=1
aj
∂ ∂xj
❡♥tã♦
σ(L)(ξ) =
N
X
j=1
aj(p)ξj.
❆ss✐♠✱ s❡Lj = N
P
k=1
ajk
∂ ∂xk
sã♦ns❡çõ❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡V s♦❜r❡U ♣♦❞❡♠♦s
❞❡s❝r❡✈❡r T0∩T∗U ♣❡❧♦ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s N
X
k=1
ajk(p)ξk= 0, p∈U, ξk ∈R, j = 1, . . . , n.
✶✳✹✳✺ ❊str✉t✉r❛s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡✐s
❯♠ s✉❜✜❜r❛❞♦ ✈❡t♦r✐❛❧ ❝♦♠♣❧❡①♦ V ❞❡CTΩ✱ ❞❡ ♣♦st♦ n✱ ❞❡✜♥❡ ✉♠❛ ❡str✉t✉r❛ ❧♦❝❛❧✲
♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡ ❞❛❞♦ ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ p0 ∈ Ω✱ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛ U0
❞❡p0 ❡ ❢✉♥çõ❡s Z1, . . . , Zm ∈C∞(U0)✱ ❝♦♠m =N −n✱ t❛✐s q✉❡
spam{dZ1p, . . . ,dZmp}=Vp⊥, ∀p∈U0.
❙❡ ♦❜s❡r✈❛r♠♦s q✉❡ ❛ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡g é ✉♠❛ s❡çã♦ ❞❡ V⊥ s❡✱ ❡ s♦♠❡♥t❡ s❡✱Lg = 0♣❛r❛ ❝❛❞❛ s❡çã♦ ❞❡V✱ s❡❣✉❡ ❝❧❛r❛♠❡♥t❡ q✉❡ t♦❞❛ ❡str✉t✉r❛ ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧
❞❡✜♥❡ ✉♠❛ ❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ ❚❡♠♦s✿
• ❆ ❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧V é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❞❛❞♦s
p0 ∈Ω❡ ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s L1, . . . , Ln✱ ♦s q✉❛✐s ❣❡r❛♠V ❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛
U0❞❡p0✱ ❡①✐st❡♠ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❛❜❡rt❛V0 ⊂U0 ❞❡p0 ❡ ❢✉♥çõ❡s s✉❛✈❡sZ1, . . . , Zm ∈
C∞(V
0) t❛✐s q✉❡✿
dZ1∧. . .dZm 6= 0 ❡♠ V0;
LjZk= 0, j = 1, . . . , n, k= 1, . . . , m. ❆s ❢✉♥çõ❡sZ1, . . . , Zm sã♦ ❝❤❛♠❛❞❛s ❞❡ ✐♥t❡❣r❛✐s ♣r✐♠❡✐r❛s ❞❡ V✳
❉❡✜♥✐çã♦ ✶✳✹✳✶✷✳ ❙❡❥❛ V ✉♠❛ ❡str✉t✉r❛ ❢♦r♠❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ s♦❜r❡ Ω✳ ❉✐③❡♠♦s q✉❡ V
