❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❚❡♦r✐❛ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❡ ♦ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②
♣❛r❛ ❛ ❊q✉❛çã♦ ❞❛ ❖♥❞❛ ❈ú❜✐❝❛
❆❧❞♦ ❱✐❡✐r❛ P✐♥t♦
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙➹❖ ❈❆❘▲❖❙
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❉❊ ❚❊❈◆❖▲❖●■❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ▼❆❚❊▼➪❚■❈❆
❚❡♦r✐❛ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❡ ♦ Pr♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤②
♣❛r❛ ❛ ❊q✉❛çã♦ ❞❛ ❖♥❞❛ ❈ú❜✐❝❛
❆❧❞♦ ❱✐❡✐r❛ P✐♥t♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❘✉✐❞✐✈❛❧ ❞♦s ❙❛♥t♦s ❋✐❧❤♦
❉✐ss❡rt❛çã♦ ❛♣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♦ ❞♦ ❚ít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária da UFSCar
P659tl
Pinto, Aldo Vieira.
Teoria de Littlewood-Paley e o problema de Cauchy para a equação da onda cúbica / Aldo Vieira Pinto. -- São Carlos : UFSCar, 2010.
82 f.
Dissertação (Mestrado) -- Universidade Federal de São Carlos, 2010.
1. Análise. 2. Equações diferenciais parciais. 3. Equação da onda. 4. Estimativas de Strichartz. 5. Bony,
Decomposição. I. Título.
Banca Examinadora:
afael Augusto dos Santos Kapp
DM- UFSCar
~~~.b!~
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ♣♦r ❙❡✉ ✐♥✜♥✐t♦ ❛♠♦r✱ ❜♦♥❞❛❞❡ ❡ s❛❜❡❞♦r✐❛❀ ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ♣❡❧♦ ❞✐r❡❝✐♦♥❛✲ ♠❡♥t♦ ❛♦s ❝❛♠✐♥❤♦s ❛ s❡r❡♠ tr✐❧❤❛❞♦s❀ ♣❡❧❛s ♦♣♦rt✉♥✐❞❛❞❡s ❡ ♣♦r ♠❡ ❞❛r ❢♦rç❛ ♣❛r❛ ❡♥❢r❡♥t❛r ♦s ♠♦♠❡♥t♦s ❞❡ ❞✐✜❝✉❧❞❛❞❡ ✭❡ ❛♣r❡♥❞❡r ❝♦♠ ❡❧❡s✮✳
➚ ♠✐♥❤❛ ♠ã❡ ❈é❧✐❛✱ ♣❡❧♦ ❛♣♦✐♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❡ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♠ ❡ ♥♦s ♠❡✉s s♦♥❤♦s✱ ♣♦r s❡r ♣r❡s❡♥ç❛ ❝♦♥st❛♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❛♣❡s❛r ❞❛ ❞✐stâ♥❝✐❛✳ ❆♦ ♠❡✉ ✐r♠ã♦ ❆♥❞❡rs♦♥✱ ❛♦ ♠❡✉ ♣❛✐ ❈❡❧s♦ ❡ ❛♦s ♠❡✉s t✐♦s ❡ ❛✈ós✱ ♣♦r ❡st❛r❡♠ s❡♠♣r❡ ❛ ♠✐♥❤❛ ❡s♣❡r❛ ❝♦♠ ✉♠ ❛❜r❛ç♦ s✐♥❝❡r♦❀ ❡♠ ❡s♣❡❝✐❛❧ ❛ t✐❛ ▲❡❝✐✱ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ ♣❡rs❡✈❡r❛♥ç❛ ❡ ♣♦r ♣❛rt✐❝✐♣❛r ❞✐r❡t❛♠❡♥t❡ ❞❡ ❝❛❞❛ ❡t❛♣❛ ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳ ➚ t♦❞❛ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣❡❧♦ ❛♠♦r✱ ✐♥❝❡♥t✐✈♦ ❡ ♣r♦t❡çã♦✿ ♠✐♥❤❛ ❜❛s❡ ❡ r❡❢❡rê♥❝✐❛✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❏♦sé ❘✉✐❞✐✈❛❧ ❞♦s ❙❛♥t♦s ❋✐❧❤♦✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ❞❡❞✐❝❛❞❛ ❡ ❡①♣❡r✐ê♥❝✐❛s ❣❡♥❡r♦s❛♠❡♥t❡ ❝♦♠♣❛rt✐❧❤❛❞❛s❀ ♣❡❧❛ ❛♠✐③❛❞❡✱ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✱ ♣❛❝✐ê♥❝✐❛ ❡ r❡s♣❡✐t♦✳
➚ ♠✐♥❤❛ ♠❛❞r✐♥❤❛ ❱❡r❛ ▲ú❝✐❛✱ ♣♦r ♠❡ ❞❡s♣❡rt❛r ♦ ❣♦st♦ ♣❡❧♦ ✧❛♣r❡♥❞❡r✧❀ à ♣r♦❢❡ss♦r❛ ❘✐t❛ ❩❛♥❝❤❡t ❈♦✉t♦✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❝♦♥st❛♥t❡✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❝✉rs♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯◆■P▲❆❈✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ❆✐❧t♦♥ ❉✉r✐❣♦♥✱ ❛♦ ❈❛r❧♦s ▲❡ã♦ ❡ ❛♦ ❱❛❧❞❡❝✐ ❈♦st❛✱ ♣❡❧♦s ❝♦♥s❡❧❤♦s ❡ ❞✐r❡❝✐♦♥❛♠❡♥t♦✳ ❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉▼✲❯❋❙❈❛r✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s ❡ r❡❝❡♣t✐✈✐❞❛❞❡✳ ❆ t♦❞♦s ♦s ❡❞✉❝❛❞♦r❡s q✉❡ ♣❛ss❛r❛♠ ❡♠ ♠✐♥❤❛ ✈✐❞❛ ❡ ❞❡✐①❛r❛♠ ♥ã♦ só ❛ s❡♠❡♥t❡ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ♠❛s t❛♠❜é♠ ❧✐çõ❡s ❞❡ ✈✐❞❛ ❡ ❡①❡♠♣❧♦s ❞❡ ❝❛rát❡r✳
❆♦ ❛♠✐❣♦ ❙✐❧✈❡str❡❀ ❛♦ ❏✉❧✐❛♥♦✱ ❛♦ ❘♦❞r✐❣♦ ❡ à ❊❧❛✐♥❡✱ ♣❡❧❛ ❝♦♥✈✐✈ê♥❝✐❛✱ ❡ ❛♦s ❛♠✐❣♦s ❞♦ ❉▼❀ ❡♠ ♣❛rt✐❝✉❧❛r ❛ t✉r♠❛ ❞❡ ♠❡str❛❞♦ ❞❡ ✷✵✵✽✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❛♠✐③❛❞❡ ❡ r❡❝✐♣r♦❝✐❞❛❞❡✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛♠♦s ♦ r❡s✉❧t❛❞♦ ❞❡ ❜♦❛✲❝♦❧♦❝❛çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❝ú❜✐❝❛ u+u3 = 0❡♠R3✱ ❞❡✈✐❞♦ ❛ ❍✳ ❇❛❤♦✉r✐ ❡ ❏✳✲❨✳ ❈❤❡♠✐♥✱ ♥♦ q✉❛❧ ♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② ❡stã♦
♥♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❤♦♠♦❣ê♥❡♦H˙3/4(R3)×H˙−1/4(R3)✳ ❆ ♣r♦✈❛ ✉t✐❧✐③❛ ✉♠ ♠ét♦❞♦ ❞❡
✐♥t❡r♣♦❧❛çã♦ ♥ã♦✲❧✐♥❡❛r✱ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥② ❡ ❞❡s✐❣✉❛❧❞❛❞❡ ❧♦❣❛rít♠✐❝❛ ❞❡ ❙tr✐❝❤❛rt③✱ t♦❞❛s ❢♦r♠✉❧❛❞❛s ♥❛ ❚❡♦r✐❛ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡②✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ st✉❞② t❤❡ r❡s✉❧t ♦❢ ✇❡❧❧✲♣♦s❡❞♥❡ss ❢♦r t❤❡ ❝✉❜✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥ u+u3 = 0 ✐♥ R3✱ ❞✉❡ t♦ ❍✳ ❇❛❤♦✉r✐ ❡ ❏✳✲❨✳ ❈❤❡♠✐♥✱ ✇❤❡r❡ t❤❡ ❈❛✉❝❤② ❞❛t❛ ✐s ✐♥
t❤❡ ❍♦♠♦❣❡♥❡♦✉s ❙♦❜♦❧❡✈ s♣❛❝❡H˙3/4(R3)×H˙−1/4(R3)✳ ❚❤❡ ♣r♦♦❢ r❡❧✐❡s ♦♥ ♥♦♥❧✐♥❡❛r ✐♥✲
t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞✱ t❤❡ ❇♦♥②✬s ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠✐❝ ❙tr✐❝❤❛rt③ ❡st✐♠❛t❡s✱ ❛s ❢♦r♠✉❧❛t❡❞ ✐♥ t❤❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❚❤❡♦r②✳
✐
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✸
✶✳✶ ❚❡r♠✐♥♦❧♦❣✐❛s ❡ ◆♦t❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❊s♣❛ç♦s Lp ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✸ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✸✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❉✐str✐❜✉✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹ ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s ❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ P❛❧❡②✲❲✐❡♥❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✺ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✺✳✶ ▼❡r❣✉❧❤♦s ❞❡ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✺✳✷ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❍♦♠♦❣ê♥❡♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷ ❚❡♦r✐❛ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ✷✾
✷✳✶ ❉❡s✐❣✉❛❧❞❛❞❡s ❞❡ ❇❡r♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸ ❊s♣❛ç♦s ❞❡ ❇❡s♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✹ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ❇♦♥② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✺ ❚❡♦r✐❛ ❍♦♠♦❣ê♥❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✸ ❆ ❊q✉❛çã♦ ❞❛ ❖♥❞❛ ❈ú❜✐❝❛ ♥♦ R3 ✺✹
❙✉♠ár✐♦ ✐✐
✸✳✹ Pr♦✈❛ ❞❛ Pr♦♣♦s✐çã♦ ✸✳✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
✶
■♥tr♦❞✉çã♦
❈♦♥s✐❞❡r❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② s❡♠✐❧✐♥❡❛r
u+u3 = 0 ❡♠ R×R3
u(0, x) =u0(x)
∂tu(0, x) = u1(x)
✭✶✮
♦♥❞❡=∂2
t −∆ é ♦ ♦♣❡r❛❞♦r ❞❛ ♦♥❞❛✳
❆❧❣✉♠❛ ❤✐♣ót❡s❡ ❞❡ ♣❡q✉❡♥❡③ ❡✴♦✉ ❞❡❝❛✐♠❡♥t♦ ❞❡✈❡♠ s❡r ✐♠♣♦st❛s ♥♦s ❞❛❞♦s ❞❡ ❈❛✉❝❤② ♣❛r❛ q✉❡ ❡①✐st❛♠ s♦❧✉çõ❡s ❣❧♦❜❛✐s✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡♠♦s ❛ ❊❉❖
u′′(t) +u3(t) = 0
u(0) = 0 ❡ u′(0) = √1 2
♦✉ s❡❥❛✱ ♣r♦❝✉r❛♠♦s ♣❛r❛ ✭✶✮ s♦❧✉çõ❡s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❡x❝♦♠ ✈❛❧♦r❡s ✐♥✐❝✐❛✐s ❝♦♥st❛♥t❡s✳
▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❊❉❖ ♣♦ru′(t)t❡r❡♠♦s
d dt
u′2(t)
2 +
u4(t) 4
= 0 ✭✷✮
❡✱ ♣♦r ✐♥t❡❣r❛çã♦✱
F(u)❞❡❢=
Z u
0
dx
√
1−x4 =
t
√
2.
❖❜s❡r✈❡♠♦s q✉❡ F é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✱ ❡ ❛ ♣r✐♥❝✐♣✐♦ ❞❡✜♥✐❞❛ ♣❛r❛ u < 1✳ ❈♦♠♦
T = R01 √dx
1−x4 é ✜♥✐t♦✱ ❛ ❢✉♥çã♦ u(t) = F−
1(√t
2) é s♦❧✉çã♦ ✈á❧✐❞❛ ♣❛r❛ t < T✳ ❚❛♠❜é♠✱
♦ ❧✐♠✐t❡ ❞❡ u(t) ✈❛❧❡ 1 q✉❛♥❞♦ t → T✳ P♦r ❝♦♥s❡❣✉✐♥t❡✱ ❛ ❞❡r✐✈❛❞❛ ❞❡ u ❡♠ t =T s❡r✐❛
✐❣✉❛❧ ❛ ③❡r♦ ♣❡❧❛ ❝♦♥s❡r✈❛çã♦ ♣♦♥t✉❛❧ ❞❡ ❡♥❡r❣✐❛ ❡♠ ✭✷✮✳ ❆ss✐♠✱ ❞❡✈❡♠♦s t❡ru(t) = 1 s❡
t > T✳
▼❛s s❡ u ❢♦ss❡ ❞❡ ❝❧❛ss❡ C2 ❡ s♦❧✉çã♦ ❞❛ ❊❉❖ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❛ ❝♦♥❝❛✈✐❞❛❞❡ s❡r✐❛
✈♦❧t❛❞❛ ♣❛r❛ ❜❛✐①♦✱ ✐♠♣♦ss✐❜✐❧✐t❛♥❞♦ ❞❡ s❡r ❝♦♥st❛♥t❡ ♣❛r❛ t > T✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ u
■♥tr♦❞✉çã♦ ✷
◆♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛ ❡st❛s q✉❡stõ❡s✱ ❡①✐st❡ ✉♠❛ ✈❛st❛ ❜✐❜❧✐♦❣r❛✜❛ ❞✐r❡❝✐♦♥❛❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ s✐♥❣✉❧❛r✐❞❛❞❡s ♣❛r❛ ❡q✉❛çã♦ ❤✐♣❡r❜ó❧✐❝❛s ♥ã♦✲❧✐♥❡❛r❡s✱ r❡✈✐t❛❧✐③❛❞❛s ♥♦s tr❛❜❛❧❤♦s ❞❡ ❋r✐t③ ❏♦❤♥ ✭✈❡r ❬✶✻❪✮✳
❆❧❣✉♥s r❡s✉❧t❛❞♦s ❡①✐st❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛ ♠♦str❛♠ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ✭✶✮ é ❣❧♦✲ ❜❛❧♠❡♥t❡ ❜❡♠✲♣♦st♦ q✉❛♥❞♦ (u0, u1) ∈ H˙s(R3)×H˙s−1(R3)✱ ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦s ✈❛❧♦r❡s
❞❡ s✳ P♦r ❡①❡♠♣❧♦✱ ❍✳ ▲✐♥❞❜❧❛❞ ❡ ❈✳ ❉✳ ❙♦❣❣❡✱ ❡♠ ❬✶✾❪✱ ♣r♦✈❛r❛♠ q✉❡ ❛ ❡q✉❛çã♦ ✭✶✮
❛❞♠✐t❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çã♦ ♣❛r❛s = 1/2❝♦♠ ❞❛❞♦s ✐♥✐❝✐❛✐s s✉✜❝✐❡♥t❡♠❡♥t❡
♣❡q✉❡♥♦s✳ ❏á ❈✳ ❊✳ ❑❡♥✐❣✱ ●✳ P♦♥❝❡ ❡ ▲✳ ❱❡❣❛✱ ❡♠ ❬✶✼❪✱ ♦❜t✐✈❡r❛♠ ♦ r❡s✉❧t❛❞♦ q✉❛♥❞♦
s >3/4✱ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❛❞✐❝✐♦♥❛❧ ❞❡ q✉❡u0 ∈L4✱ ♠❛s s❡♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ♣❡q✉❡♥❡③✳
❖ ♣ró①✐♠♦ ✭❡ ♥❛t✉r❛❧✮ ♣❛ss♦ é ✐♥✈❡st✐❣❛r s❡ ♦ ♠❡s♠♦ ♦❝♦rr❡ q✉❛♥❞♦ s = 3/4 ✭♣♦✐s ♥❡st❡
❝❛s♦✱ u0 ∈ L4✱ ♣❡❧♦s t❡♦r❡♠❛s ❞❡ ♠❡r❣✉❧❤♦✮✳ ❖ r❡s✉❧t❛❞♦ ❛♣r❡s❡♥t❛❞♦ ♣♦r ❍✳ ❇❛❤♦✉r✐ ❡
❏✳✲❨✳ ❈❤❡♠✐♥ ❡♠ ❬✷❪✱ ♦ q✉❛❧ ❡st✉❞❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦✱ r❡s♣♦♥❞❡ ❛ ♣❡r❣✉♥t❛ ❝♦♠♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ s❡❣✉✐♥t❡
❚❡♦r❡♠❛ ✶ ❙✉♣♦♥❤❛♠♦s q✉❡(u0, u1)∈B˙23,/44×B˙− 1/4
2,4 t❛❧ q✉❡ S˙0u0 ♣❡rt❡♥❝❡ ❛ L2✳ ❊♥tã♦
❡①✐st❡ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞❡ ✭✶✮ ♥♦ ❡s♣❛ç♦ L4
❧♦❝(R;L4)✳
❆ ♣r♦✈❛ ✉t✐❧✐③❛ ✉♠ ♠ét♦❞♦ ❞❡ ✐♥t❡r♣♦❧❛çã♦ ♥ã♦✲❧✐♥❡❛r✱ ♣♦r ♠❡✐♦ ❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ▲✐tt❧❡✇♦♦❞✲P❛❧❡②✳ ❆❧é♠ ❞✐ss♦✱ r❡q✉❡r ♦ ✉s♦ ❞❛s ❡st✐♠❛t✐✈❛s ❞❡ ❙tr✐❝❤❛rt③✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❧♦❣❛rít♠✐❝❛✱ q✉❡ r❡❧❛❝✐♦♥❛ ♥♦r♠❛s ❞❡ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❛ ♦♥❞❛ ❧✐✈r❡ ❝✉❥♦ s✉♣♦rt❡ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❡stá s✉♣♦rt❛❞❛ ❡♠ ❝♦♥❥✉♥t♦s ❞✐á❞✐❝♦s✳
❖ ♣r❡s❡♥t❡ t❡①t♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❡①♣♦r ❛❧❣✉♠❛s ❞❛s ❢❡rr❛♠❡♥t❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ ❞❡♠♦♥str❛çã♦ ❞❡st❡ r❡s✉❧t❛❞♦✳ ◆♦ ❈❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s tó♣✐❝♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ❚❡♦r✐❛ ❞❛s ❉✐str✐❜✉✐çõ❡s ❡✱ ❛tr❛✈és ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❞❡✜♥✐♠♦s ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈✳
✸
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
❊st❡ ❝❛♣ít✉❧♦ ❞❡st✐♥❛✲s❡ ❛ ❡st❛❜❡❧❡❝❡r ❛ ❧✐♥❣✉❛❣❡♠ ❡ ♦s ❢❛t♦s ❜ás✐❝♦s ♥❡❝❡ssár✐♦s ❛ ❝♦♠✲ ♣r❡❡♥sã♦ ❞♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✳ ❊♥✉♥❝✐❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❛ r❡s♣❡✐t♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❞❡✜♥✐♠♦s ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❡ ❡①♣❧♦r❛♠♦s ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ❆s r❡❢❡rê♥❝✐❛s ❜ás✐❝❛s sã♦ ❬✶✷❪✱ ❬✶✹❪ ❡ ❛s ♥♦t❛s ❞❡ ❝✉rs♦ ❬✺❪✳
✶✳✶ ❚❡r♠✐♥♦❧♦❣✐❛s ❡ ◆♦t❛çõ❡s
❈♦♥s✐❞❡r❡♠♦s ♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦Rd ♠✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❛♥ô♥✐❝♦
x.y =
d X
i=1
xiyi,
♦♥❞❡x= (x1,· · · , xd) ❡y= (y1,· · · , yd) sã♦ ❡❧❡♠❡♥t♦s ❞❡ Rd✳
❆ss✐♠✱ s❡ | · | é ❛ ♥♦r♠❛ ✉s✉❛❧ ❡♠ Rd ♣r♦✈❡♥✐❡♥t❡ ❞❡st❡ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❡ r
0, r1 > 0✱
✐♥❞✐❝❛♠♦s ♣♦r
B(x0, r0) =
x∈Rd;
|x−x0|< r0
❛ ❜♦❧❛ ❝❡♥tr❛❞❛ ❡♠ x0 ❡ r❛✐♦r0 ❡
C(x0, r0, r1) =
x∈Rd;r
0 <|x−x0|< r1
❛ ❝♦r♦❛ ❝❡♥tr❛❞❛ ❡♠x0✱ ❝♦♠ r❛✐♦ ♠❡♥♦r r0 ❡ r❛✐♦ ♠❛✐♦r r1✳
❙❡Ωé ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❞❡ Rd ❡k ✉♠ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦✱ ❛ ❝❧❛ss❡Ck(Ω)❝♦♥s✐st❡
❞❡ t♦❞❛s ❛s ❢✉♥çõ❡su: Ω→Rq✉❡ t❡♠ ❞❡r✐✈❛❞❛s ❝♦♥tí♥✉❛s ❞❡ ♦r❞❡♠ ♠❡♥♦r ♦✉ ✐❣✉❛❧ q✉❡
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✹
❯s✉❛❧♠❡♥t❡ ❡s❝r❡✈❡♠♦s ❛s ❞❡r✐✈❛❞❛s ❞❡ ✉♠❛ ❢✉♥çã♦f ∈C∞(Ω) ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❝♦♥❝✐s❛
♣♦r ♠❡✐♦ ❞❛ ♥♦t❛çã♦ ❞♦s ♠✉❧t✐✲í♥❞✐❝❡s✳ ❯♠ ♠✉❧t✐✲í♥❞✐❝❡ é ✉♠❛ ❞✲✉♣❧❛ ❞❡ ✐♥t❡✐r♦s ♥ã♦✲ ♥❡❣❛t✐✈♦s✳ ❆ss✐♠✱ ❞❛❞♦(α1,· · · , αd)∈ Nd ✉♠ ♠✉❧t✐✲í♥❞✐❝❡✱ s✉❛ ♦r❞❡♠ é ❞❛❞❛ ♣♦r |α| =
α1+· · ·+αd ❡ ❞✐③❡♠♦s q✉❡ α≤β s❡ αi ≤β1,∀i= 1,· · · , d✳
❚❛♠❜é♠✱ ♣❛r❛x= (x1,· · · , xd)∈Rd✱
xα =xα1
1 xα22· · ·x
αd
d
❆ss✐♠✱
∂αf = ∂|
α|f
∂α1
x1 · · ·∂
αd
xd
=∂α1
x1 · · ·∂
αd
xdf,
♦✉ s❡❥❛✱ ∂αi
xi r❡♣r❡s❡♥t❛ ❛ ❞❡r✐✈❛❞❛ ♣❛r❝✐❛❧ s✉❝❡ss✐✈❛ ✭αi✲✈❡③❡s✮ ❝♦♠ r❡s♣❡✐t♦ ❛ ✈❛r✐á✈❡❧xi✳
✶✳✷ ❊s♣❛ç♦s
L
p❋✐①❡♠♦s ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛(X,M, µ)✱ ♦✉ s❡❥❛✱ X é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦✱Mé ✉♠❛ σ−á❧❣❡❜r❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s ❞❡ X ❡ µ : M → [0,∞] ✉♠❛ ♠❡❞✐❞❛✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦
♠❡♥s✉rá✈❡❧ s♦❜r❡ ❳ ❡1≤p < ∞✱ ❞❡✜♥✐♠♦s
kfkLp =
Z
X|
f|pdµ
1/p
❚❛♠❜é♠✱ ♣❛r❛p=∞✱
kfkL∞ = inf{a≥0 :µ({x:|f(x)|> a}) = 0}
❝♦♠ ❛ ❝♦♥✈❡♥çã♦ ❞❡ q✉❡inf∅=∞✳ ❊♠ ❛❧❣✉♥s ❝❛s♦s✱kfkL∞ é ❝❤❛♠❛❞♦ s✉♣r❡♠♦ ❡ss❡♥❝✐❛❧
❞❡f ❡ ❡s❝r❡✈❡♠♦s
kfkL∞ =s✉♣❡ssx
∈X|f(x)|.
❉❡✜♥✐♠♦s
Lp(X,M, µ) = {f :X →C; ❢ é ♠❡♥s✉rá✈❡❧ ❡ kfkLp <∞}.
❉✐③❡♠♦s q✉❡ ❞✉❛s ❢✉♥çõ❡s ❞❡✜♥❡♠ ♦ ♠❡s♠♦ ❡❧❡♠❡♥t♦ ❞❡ Lp q✉❛♥❞♦ ❡❧❛s s❛♦ ✐❣✉❛✐s ❡♠
q✉❛s❡ t♦❞❛ ❛ ♣❛rt❡✳ ▼❡❞✐❛♥t❡ ❡st❛ ✐❞❡♥t✐✜❝❛çã♦✱ t❡♠♦s q✉❡ ♦ ❡s♣❛ç♦Lp(X,M, µ)✱ ♠✉♥✐❞♦
❞❛ ♥♦r♠❛ k · kLp✱ é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ✭♣❛r❛ ❛ ♣r♦✈❛✱ ✈❡r ❬✷✶❪✱ ♣♦r ❡①❡♠♣❧♦✮✳
❘❡❝♦r❞❡♠♦s q✉❡ ❞♦✐s ♥ú♠❡r♦s r❡❛✐sp❡p′ s❛t✐s❢❛③❡♥❞♦ 1
p+
1
p′ = 1✱ ❝♦♠p, p′ >1sã♦ ❞✐t♦s
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✺
Pr♦♣♦s✐çã♦ ✶✳✷✳✶ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✮ ❙❡❥❛♠1≤p, p′ ≤ ∞ ❡①♣♦❡♥t❡s ❝♦♥❥✉✲
❣❛❞♦s✳ ❙❡ ❢ ❡ ❣ sã♦ ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s s♦❜r❡ X✱
kf gkL1 ≤ kfkLpkgkLp′ ✭✶✳✶✮
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✾❪✱ ♣á❣✳ ✶✼✹✳
❙❡p=p′ = 2✱ ✭✶✳✶✮ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛rt③✳
◆♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ µ é ❛ ♠❡❞✐❞❛ ❞❛ ❝♦♥t❛❣❡♠ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ A✱ é ❝♦st✉♠❡
❞❡♥♦t❛r ♦ ❡s♣❛ç♦ Lp ❝♦rr❡s♣♦♥❞❡♥t❡ ♣♦r ℓp(A)✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ℓp✱ s❡ A é ❡♥✉♠❡rá✈❡❧✳
◆❡st❡ ❝❛s♦✱ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ℓp ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠❛ s❡q✉ê♥❝✐❛x= (x
j)✱j ∈A t❛❧ q✉❡ X
j∈A
|xj|p <∞,
s❡p < ∞✱ ♦✉
sup
j∈A |
xj|<∞,
❝❛s♦ p=∞✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✷ ❙❡ Aé ✉♠ ❝♦♥❥✉♥t♦ ❡ 0< p < q ≤ ∞✱ ❡♥tã♦ℓp(A)֒→ℓq(A)✱ ♦✉ s❡❥❛✱
❛ ❛♣❧✐❝❛çã♦ ✐♥❝❧✉sã♦ é ❝♦♥tí♥✉❛✱ ❡ ✈❛❧❡ kfkℓq ≤ kfkℓp✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✸ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❤❡❜②s❤❡✈✮ ❙❡ f ∈ Lp (1 ≤ p < ∞)✱ ❡♥tã♦
♣❛r❛ t♦❞♦λ >0✱
µ({x:|f(x)|> λ})≤
kfkLp
λ
p
❆ ♣r♦✈❛ ❞❡st❡s ❞♦✐s r❡s✉❧t❛❞♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✾❪✱ ♥❛s ♣á❣✐♥❛s ✶✼✽ ❡ ✶✽✺✱ r❡s✲ ♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥✉♥❝✐❛♠♦s ❛ s❡❣✉✐r ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❛ ❛ ♥♦r♠❛ ❡♠ Lp ❛tr❛✈és ❞❡
✐♥t❡❣r❛✐s s♦❜r❡[0,∞)✿
❚❡♦r❡♠❛ ✶✳✷✳✶ ❙❡p∈[1,∞)✱ ❡♥tã♦ ♣❛r❛ t♦❞❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ ❢ s♦❜r❡ (X,M, µ)✱
kfkpLp =p
Z ∞
0
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✻
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✾❪✱ ♣á❣✳ ✶✾✶✳
❙❡rá út✐❧✱ ❡♠ ❛❧❣✉♠❛s ❡st✐♠❛t✐✈❛s✱ ❛ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦ ❡ s❡✉ r❡s♣❡❝t✐✈♦ ❝♦r♦❧ár✐♦✱ q✉❡ ❣❡♥❡r❛❧✐③❛ ❛ ✐❞é✐❛ ❞❛ ✐♥t❡❣r❛çã♦ ❝♦♠ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s ♣❛r❛ Rd✳ ◆♦✈❛♠❡♥t❡✱ ❛ ♣r♦✈❛
♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❬✾❪✱ ♥❛ ♣á❣✐♥❛ ✼✺✳
Pr♦♣♦s✐çã♦ ✶✳✷✳✹ ❙❡❥❛ ❢ é ✉♠❛ ❢✉♥çã♦ ♠❡♥s✉rá✈❡❧ s♦❜r❡ Rd✱ ♥ã♦✲♥❡❣❛t✐✈❛ ♦✉ ✐♥t❡❣rá✈❡❧
t❛❧ q✉❡ f(x) = g(|x|)✱ ♣❛r❛ ❛❧❣✉♠❛ ❢✉♥çã♦ g ❡♠ (0,∞)✳ ❊♥tã♦
Z
Rd
f(x)dx=σ(Sd−1)
Z ∞
0
g(r)rd−1dr,
♦♥❞❡ σ(Sd−1) ❡①♣r❡ss❛ ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ Sd−1✳
❈♦r♦❧ár✐♦ ✶✳✷✳✶ ❙❡❥❛s ∈R✳ ❙❡ s > d/2✱ ❡♥tã♦
Z
Rd
dξ
(1 +|ξ|2)s <∞
◆❛ ♣ró①✐♠❛ s❡çã♦✱ ♣❛r❛ ♦❜t❡r♠♦s ❡①❡♠♣❧♦s ❞❡ ❞✐str✐❜✉✐çõ❡s✱ r❡❝♦r❞❡♠♦s ❛ s❡❣✉✐♥t❡ ❉❡✜♥✐çã♦ ✶✳✷✳✶ ❙❡ ❢ é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛✱ ▲❡❜❡s❣✉❡ ♠❡♥s✉rá✈❡❧ ❞❡✜♥✐❞❛ ♥♦ ❛❜❡rt♦
Ω⊂Rd t❛❧ q✉❡✱ ♣❛r❛ ❝❛❞❛ ❝♦♠♣❛❝t♦ K ⊂Ω Z
K|
f|dx <∞,
❞✐③❡♠♦s q✉❡ ❢ é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡♠♦sf ∈L1
❧♦❝(Ω)✳
P♦❞❡♠♦s ❛✐♥❞❛ ❡st❡♥❞❡r ❡st❛ ❞❡✜♥✐çã♦ ❞❡✜♥✐♥❞♦ Lp❧♦❝(Ω) ♣❛r❛ p ≥ 1✱ q✉❡ s❡rá ♦ ❡s♣❛ç♦
❞❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s ❝✉❥❛ ♣✲és✐♠❛ ♣♦tê♥❝✐❛ é ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✿
❉❡✜♥✐çã♦ ✶✳✷✳✷ ❙❡❥❛ ❢ é ✉♠❛ ❢✉♥çã♦ ❝♦♠♣❧❡①❛✱ ▲❡❜❡s❣✉❡ ♠❡♥s✉rá✈❡❧ ❞❡✜♥✐❞❛ ♥♦ ❛❜❡rt♦
Ω⊂Rd✳ ❉✐③❡♠♦s q✉❡ f ∈Lp
❧♦❝(Ω) s❡✱ ♣❛r❛ ❝❛❞❛ ❝♦♠♣❛❝t♦ K ⊂Ω✱ f ∈Lp(K)✳
❆ s✉❛ t♦♣♦❧♦❣✐❛ ❡stá ❞❡s❝r✐t❛ ✈✐❛ ❛s s❡♠✐✲♥♦r♠❛s
f 7−→
Z
K|
f|pdx
♦♥❞❡K é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ q✉❛❧q✉❡r ❞❡Ω✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✼
❉❡✜♥✐çã♦ ✶✳✷✳✸ ❙❡❥❛ (X,k · k) ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤ ❡ J ✉♠ s✉❜❝♦♥❥✉♥t♦ ▲❡❜❡s❣✉❡✲
♠❡♥s✉rá✈❡❧ ❞❡ R✳
✭✐✮ ❖ ❡s♣❛ç♦ Lp(J, X) ❝♦♥s✐st❡ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐s u:J →X ❝♦♠
kukLp(J,X) ❞❡❢=
Z
Jk
u(t)kp dt 1/p
<∞
♣❛r❛ 1≤p <∞✱ ❡
kukL∞(J,X) ❞❡❢= s✉♣❡sst∈Jku(t)k<∞.
✭✐✐✮ ❖ ❡s♣❛ç♦ C(J, X) ❝♦♠♣r❡❡♥❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s u:J →X ❝♦♠
kukC(J,X) ❞❡❢= sup
t∈J k
u(t)k<∞
❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱Lp❧♦❝(J, X)❝♦♥s✐st❡ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ♠❡♥s✉rá✈❡✐su:J →X t❛❧ q✉❡✱
♣❛r❛ q✉❛❧q✉❡r ❝♦♠♣❛❝t♦ K ⊂J✱ u∈Lp(K, X)✳
◗✉❛♥❞♦ J ❢♦r ♦ ✐♥t❡r✈❛❧♦ [0, T]✱ ♣❛r❛T >0✱ ✉t✐❧✐③❛♠♦s ❛ ♥♦t❛çã♦ Lp([0, T], X) = Lp T(X)✳
✶✳✸ ❉✐str✐❜✉✐çõ❡s
❙❡❥❛Ω⊂Rd ❛❜❡rt♦ ❡ φ : Ω→C ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛✳ ❉❡✜♥✐♠♦s ♦ s✉♣♦rt❡ ❞❡φ✱ ♦ q✉❛❧
❞❡♥♦t❛r❡♠♦s ♣♦r S(φ)✱ ❝♦♠♦ s❡♥❞♦ ♦ ❢❡❝❤♦ ❡♠ Ω❞♦ ❝♦♥❥✉♥t♦ {x∈Ω; φ(x)6= 0}✳
❉❡✜♥✐çã♦ ✶✳✸✳✶ ❙❡❥❛ Ω ✉♠ ❛❜❡rt♦ ❞❡ Rn✳ ❖ ❝♦♥❥✉♥t♦
Cc∞(Ω) ={φ: Ω→C; u∈C∞ ❡ S(φ) é ❝♦♠♣❛❝t♦}
é ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s t❡st❡s✳
P❛r❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❢✉♥çõ❡s t❡st❡s ♥ã♦✲♥✉❧❛s✱ s❡rá út✐❧ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✱ ❝✉❥❛ ❞❡✲ ♠♦♥str❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ♥❛ ♣á❣✐♥❛ ✼ ❞❡ ❬✶✷❪✳
Pr♦♣♦s✐çã♦ ✶✳✸✳✶ ❙❡❥❛ K ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❞❡ ✉♠ ❛❜❡rt♦ Ω⊂ Rd✳ ❊①✐st❡ ψ ∈
C∞
c (Ω) t❛❧ q✉❡ 0≤ψ ≤1 ❡ ψ = 1 ♥✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ K✳
❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ r❡❧❛❝✐♦♥❛ ♦ ❝♦♠♣♦rt❛♠❡♥t♦✱ ❝♦♠ r❡s♣❡✐t♦ ❛ ♥♦r♠❛ ❡♠L2✱ ❞❡ ❢✉♥çõ❡s
t❡st❡ ❡♠ C∞
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✽
▲❡♠❛ ✶✳✸✳✶ ✭▲❡♠❛ ❞❡ P♦✐♥❝❛ré✮ ❙❡❥❛ Ω ✉♠ s✉❜❝♦♥❥✉♥t♦ ❛❜❡rt♦ ❡♠ Rd✱ ❧✐♠✐t❛❞♦ ❡♠
❛❧❣✉♠❛ ❞✐r❡çã♦✱ ❞✐❣❛♠♦s Ω⊂(−R, R)×Rd−1✳ ❊♥tã♦✱ ♣❛r❛ q✉❛❧q✉❡r ϕ∈C∞ c (Ω)✱
kϕk2
L2 ≤4R2k∂x1ϕk
2
L2.
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✺❪✱ ♣á❣✳ ✷✷✳
❚❡♠♦s q✉❡ C∞
c (Ω) é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡♥s♦ ❡♠ Lp(Ω)✱ ❝♦♠ 1 ≤ p < ∞✱ ♣♦r ❬✷✶❪✳
❈♦♥❢♦r♠❡ ❬✸❪✱ é ♣♦ssí✈❡❧ ❡q✉✐♣á✲❧♦ ❝♦♠ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t♦♣♦❧ó❣✐❝♦✱ ♥ã♦✲♠❡tr✐③á✈❡❧✱ ❞❡ ♠♦❞♦ q✉❡ C∞
c (Ω) t♦r♥❡✲s❡ ✉♠ ❡s♣❛ç♦ ❝♦♠♣❧❡t♦✳ ❈♦♠ ❡st❛ ❡str✉t✉r❛
t♦♣♦❧ó❣✐❝❛✱ t❡r❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛(φj)j∈N❞❡ ❢✉♥çõ❡s t❡st❡ ❝♦♥✈❡r❣❡ ❛ ③❡r♦ ❡♠Cc∞(Ω)
s❡ ❡①✐st❡ ✉♠ ❝♦♠♣❛❝t♦ K ⊂ Ω t❛❧ q✉❡ S(φj) ⊂ Ω, ∀j ∈ N ❡✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦
m✱ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠ m ❝♦♥✈❡r❣❡ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ ③❡r♦ q✉❛♥❞♦ j → ∞✳
❉❡✜♥✐çã♦ ✶✳✸✳✷ ❙❡❥❛ Ω⊆Rd ❛❜❡rt♦✳ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦ u:C∞
c (Ω) →C é
❞✐t♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω✳ ❖ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s ❡♠ Ω s❡ ❞❡♥♦t❛ ❝♦♠ D′(Ω)✳
■♥❞✐❝❛♠♦s ♦ ✈❛❧♦r ❞❛ ❞✐str✐❜✉✐çã♦u ♥❛ ❢✉♥çã♦ t❡st❡φ ♣♦rhu, φi✳
❊①❡♠♣❧♦ ✶✳✸✳✶ ❙❡❥❛ f ∈L1
❧♦❝(Ω) ❡ ❞❡✜♥❛♠♦s ♦ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r
hTf, φi= Z
Ω
f(x)φ(x)dx, φ∈Cc∞(Ω) ✭✶✳✷✮
❙❡(φj)j∈N é ✉♠❛ s❡qüê♥❝✐❛ ❝♦♥✈❡r❣✐♥❞♦ ❛ ③❡r♦ ❡♠Cc∞(Ω)✱ s❡❥❛K ⊂Ω❝♦♠♣❛❝t♦ t❛❧ q✉❡
S(φj)⊂K✳ ❊♥tã♦
|hTf, φji| ≤ Z
K|
f(x)||φj(x)|dx
≤ sup
x∈K|
φj(x)| Z
K|
f(x)|dxj−→→∞ 0
P♦rt❛♥t♦✱ ❛ ❡①♣r❡ssã♦ ✭✶✳✷✮ ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠ Ω ✳ ❚❛♠❜é♠✱ s❡ f, g ∈ L1
❧♦❝(Ω) ❡
hTf, φi=hTg, φi✱ ♣❛r❛ t♦❞❛φ∈Cc∞(Ω)✱ ❡♥tã♦f =g q✳t✳♣ ✭✈❡r ♣r♦✈❛ ♥❛ ♣á❣✳ 11❞❡ ❬✶✷❪✮✳
❉❡st❡ ♠♦❞♦✱ ❛ ❛♣❧✐❝❛çã♦ ✐♥❥❡t✐✈❛f 7→Tf ♥♦s ♣❡r♠✐t❡ ❝♦♥s✐❞❡r❛r ✈ár✐♦s ❡s♣❛ç♦s ❞❡ ❢✉♥çõ❡s
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✾
❉✐③❡♠♦s q✉❡ ✉♠❛ ❞✐str✐❜✉✐çã♦u∈ D′(Ω) é ✐❣✉❛❧ ❛ ③❡r♦ ♥✉♠ ❛❜❡rt♦ U ⊂Ω s❡hu, φi= 0✱
♣❛r❛ t♦❞❛ φ ∈ C∞
c (U)✳ ❉❡✜♥✐♠♦s ❡♥tã♦ ♦ s✉♣♦rt❡ ❞❡ u✱ ❡ ❞❡♥♦t❡♠♦s ♣♦r S(u)✱ ❝♦♠♦ ❛
✐♥t❡rs❡çã♦ ❞❡ t♦❞♦s ♦s ❢❡❝❤❛❞♦s ❞❡ Ω ❢♦r❛ ❞♦s q✉❛✐s u é ♥✉❧❛✳ ❉❡♥♦t❛♠♦s ❝♦♠ E′(Ω) ♦
s✉❜❡s♣❛ç♦ ❞❡ D′(Ω) ❞❛s ❞✐str✐❜✉✐çõ❡s ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳
❚❡♦r❡♠❛ ✶✳✸✳✶ ❙❡❥❛ u ∈ D′(Ω)✳ ❊♥tã♦ S(u) é ❝♦♠♣❛❝t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠
❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥t✐♥✉♦ v ❡♠ C∞(Ω) ❝✉❥❛ r❡str✐çã♦ ❛ C∞
c (Ω) é ✐❣✉❛❧ ❛ u✳
❆ ♣r♦✈❛ ❞❡st❡ ❚❡♦r❡♠❛ s❡ ❡♥❝♦♥tr❛ ♥❛ ♣á❣✐♥❛ ✹✶ ❞❡ ❬✶✷❪✳ ❆q✉✐✱ ❛ ♥♦çã♦ ❞❡ s❡q✉❡♥❝✐❛❧♠❡♥t❡ ❝♦♥tí♥✉❛ é ❛ s❡❣✉✐♥t❡✿ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❢✉♥çõ❡sC∞(Ω) ❝♦♥✈❡r❣❡ ♣❛r❛ ③❡r♦ s❡✱ ♣❛r❛ t♦❞♦
❝♦♠♣❛❝t♦K ❡ t♦❞♦ ✐♥t❡✐r♦m✱ ❛s ❞❡r✐✈❛❞❛s ❞❡ ♦r❞❡♠m ❝♦♥✈❡r❣❡♠ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ ③❡r♦
❡♠ K q✉❛♥❞♦ j → ∞✳
❉❡✜♥✐çã♦ ✶✳✸✳✸ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ uj ∈ D′(Ω), j ∈ N✱ ❝♦♥✈❡r❣❡ ❛ u∈ D′(Ω)
s❡ huj, φi ❝♦♥✈❡r❣❡ ❛ hu, φi✱ ♣❛r❛ t♦❞❛ φ∈Cc∞(Ω)✳
❙✉♣♦♥❤❛♠♦s q✉❡un✱n = 1,2, ...é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❡♠D′(Ω) t❛❧ q✉❡un(φ)
é ❝♦♥✈❡r❣❡♥t❡ ♣❛r❛ ❝❛❞❛φ ∈C∞
c (Ω)✳ ❙❡ ❞❡✜♥✐r♠♦s u(φ) = lim
n→∞un(φ)✱ t❡♠♦s q✉❡ u é ✉♠
❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r✳ ▼❛✐s ❛✐♥❞❛✱u r❡s✉❧t❛ ❝♦♥tí♥✉♦ ❡♠ C∞ c (Ω)✳
❚❡♦r❡♠❛ ✶✳✸✳✷ ❙❡❥❛ (un)n∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❞✐str✐❜✉✐çõ❡s ❡♠ Ω✱ ❡ s✉♣♦♥❤❛♠♦s q✉❡✱
♣❛r❛ t♦❞❛ φ ∈ C∞
c (Ω)✱ hun, φi é ✉♠❛ s❡q✉ê♥❝✐❛ ♥✉♠ér✐❝❛ ❞❡ ❈❛✉❝❤②✳ ❊♥tã♦ (un)n∈N é
❝♦♥✈❡r❣❡♥t❡ ❡♠ D′(Ω)✳
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✶✷❪✱ ♣á❣✐♥❛ ✺✻✳
✶✳✸✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❉✐str✐❜✉✐çõ❡s
❙❡❥❛u∈C∞
c (Ω)✳ ❈♦♠♦ué ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧✱ ❛ ❡①♣r❡ssã♦ ✭✶✳✷✮ ♥♦s ♣❡r♠✐t❡ ❝♦♥s✐❞❡rá✲
❧❛ ❝♦♠♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❡♠Ω✳ P♦r ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s✱ t❡♠♦s
Z ∂u
∂xj
(x)φ(x)dx=−
Z
u(x)∂φ
∂xj
(x)dx,
♣❛r❛ t♦❞❛ ❢✉♥çã♦ t❡st❡φ✳ ❉❡st❛ ♠❛♥❡✐r❛✱ ♣♦r ❞✉❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❞✐str✐❜✉✐çã♦
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✵
♣❛r❛ t♦❞❛u∈ D′(Ω)✳ P♦r ✐♥❞✉çã♦ ❡♠ |α|✱
h∂αu, φi= (−1)|α|hu, ∂αφi
P❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦✱ ❞❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠❛ ❢✉♥çã♦ f ∈ C∞(Ω) ❝♦♠♦
s❡♥❞♦ ❛ ❞✐str✐❜✉✐çã♦
hf u, φi=hu, f φi
❆❣♦r❛✱ s❡❥❛♠ ❢✱ ❣ ❞✉❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❡♠Rdt❛❧ q✉❡ ✉♠❛ ❞❡❧❛s t❡♥❤❛ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳
❊♥tã♦ ❛ ❝♦♥✈♦❧✉çã♦ ❞❡ ❢ ❡ ❣ s❡ ❞❡✜♥❡ ❝♦♠♦
f∗g(x) =
Z
f(x−y)g(y)dy =
Z
f(y)g(x−y)dy
❆ ✜♠ ❞❡ ❡st❡♥❞❡r ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♣❛r❛ ♦ ❝♦♥t❡①t♦ ❞❛s ❞✐str✐❜✉✐çõ❡s✱ ❝♦♥s✐❞❡r❡♠♦s ❛
❉❡✜♥✐çã♦ ✶✳✸✳✹ ❙❡❥❛ u∈ D′(Ω) ❡ φ ∈C∞
c (Ω) (♦✉ u ∈ E′(Ω) ❡ φ ∈C∞(Ω))✳ ❉❡✜♥✐♠♦s
❛ ❝♦♥✈♦❧✉çã♦ ❞❡ u ❡ φ✱ ❞❡♥♦t❛❞❛ ♣♦r u∗φ✱ ❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♣♦r u∗φ(x) = hu,φ˘xi,
♦♥❞❡ φ˘x(y) =φ(x−y)✳
❱❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✭✐✮ u∗φ∈C∞(Ω) ❡ s✉❛s ❞❡r✐✈❛❞❛s sã♦ ❞❛❞❛s ♣♦r
∂α(u∗φ) =∂αu∗φ =u∗∂αφ
✭✐✐✮ S(u∗φ)⊂S(u) +S(φ)
❚❡♦r❡♠❛ ✶✳✸✳✸ C∞
c (Ω) é ❞❡♥s♦ ❡♠ D′(Ω)✳
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✶
✶✳✹ ❉✐str✐❜✉✐çõ❡s ❚❡♠♣❡r❛❞❛s ❡ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r
❙❡f ∈L1(Rd)✱ ❞❡✜♥✐♠♦s ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ ❢ ♣♦r
Ff(ξ) = fb(ξ) =
Z
e−ix.ξf(x) dx, ξ ∈Rd
♦♥❞❡i é ❛ ✉♥✐❞❛❞❡ ✐♠❛❣✐♥ár✐❛ ❡ x.ξ é ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❝❛♥ô♥✐❝♦✳
❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡ ❛ ❛♣❧✐❝❛çã♦ f 7→ fb❞❡✜♥❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r
❞❡L1(Rd) ❡♠ L∞(Rd) s❛t✐s❢❛③❡♥❞♦
kfbkL∞ ≤ kfkL1. ✭✶✳✸✮
▼❛✐s ❛✐♥❞❛✱ ❡stá é ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ ❧❡✈❛ L1 ♥♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s q✉❡ s❡
❛♥✉❧❛♠ ♥♦ ✐♥✜♥✐t♦✳
▲❡♠❛ ✶✳✹✳✶ ✭❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡✮ ❙❡❥❛ f ∈ L1(Rd)✳ ❊♥tã♦ fbé ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛
s❛t✐s❢❛③❡♥❞♦ fb(ξ)→0 q✉❛♥❞♦ |ξ| → ∞✳
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✶✷❪✱ ♣á❣✳ ✼✺✳
❙❡φ é ✉♠❛ ❢✉♥çã♦ t❡st❡✱ ♣r♦✈❛✲s❡ q✉❡ s✉❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ❛♥❛❧ít✐❝❛ ❝♦♠♣❧❡①❛
❡♠ Cd✳ ❆ss✐♠✱ φb ♥ã♦ t❡rá s✉♣♦rt❡ ❝♦♠♣❛❝t♦✱ ❛ ♠❡♥♦s q✉❡ φ s❡❥❛ ♥✉❧❛✱ ✉♠❛ ✈❡③ q✉❡ ♦
❝♦♥❥✉♥t♦ ❞♦s ③❡r♦s ❞❡ ✉♠❛ ❢✉♥çã♦ ❛♥❛❧ít✐❝❛ ❝♦♠♣❧❡①❛ ♥ã♦✲♥✉❧❛ ❡♠ Cd t❡♠ ✐♥t❡r✐♦r ✈❛③✐♦✳
❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦ ✉♠ ❡s♣❛ç♦ q✉❡ ❝♦♥té♠ ❛s ❢✉♥çõ❡s ❞❡ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ ❡ q✉❡ s❡❥❛ ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳
❉❡✜♥✐çã♦ ✶✳✹✳✶ ✭❊s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③✮ ❉❡♥♦t❛♠♦s ♣♦rS ♦ s✉❜❡s♣❛ç♦ ❞❡C∞(Rd)❞❛s
❢✉♥çõ❡s φ t❛✐s q✉❡
kφkN,α = sup x∈Rd
(1 +|x|)N|∂αφ|<∞ ✭✶✳✹✮
♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦N ❡ ♣❛r❛ t♦❞♦ α∈Nd✳
❚❛♥t♦ ❛s ❢✉♥çõ❡s ❞❡ S q✉❛♥t♦ ❛s s✉❛s ❞❡r✐✈❛❞❛s ❞❡❝r❡s❝❡♠ ♥♦ ✐♥✜♥✐t♦ ♠❛✐s r❛♣✐❞❛♠❡♥t❡
❞♦ q✉❡ q✉❛❧q✉❡r ♣♦tê♥❝✐❛ ♥❡❣❛t✐✈❛ ❞❡ |x|✳ ▼✉♥✐r❡♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③ S ❝♦♠ ❛
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✷
❊①❡♠♣❧♦ ✶✳✹✳✶ ❖ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s✲t❡st❡s ❡stá ❝♦♥t✐❞♦ ❞❡♥s❛♠❡♥t❡ ❡♠ S✱ ♠❛s ♣❛r❛
x∈Rd✱φ(x) =e−|x|2
♣❡rt❡♥❝❡ ❛S✱ ♣♦ré♠ ♥ã♦ ♣♦ss✉✐ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳ ❆ss✐♠✱ C∞ c S✳
❚❡♥❞♦ ❡♠ ✈✐st❛ ❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✹✱ s❡❣✉❡ q✉❡ s❡ φ∈ S✱ k∂αφkLp =
Z
|∂αφ(x)|p(1 +|x|)
n+1 (1 +|x|)n+1dx
1/p
≤ C sup
x∈Rd
(1 +|x|)n+1p |∂α(x)φ| ≤Ckφk
N,α, ✭✶✳✺✮
♦♥❞❡ ◆ é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♠❛✐♦r q✉❡ n+1
p ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ S ֒→L p✳
❚❡♦r❡♠❛ ✶✳✹✳✶ ❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡rF :S → S é ✉♠ ♦♣❡r❛❞♦r ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦✱
❝♦♥t✐♥✉❛♠❡♥t❡ ✐♥✈❡rsí✈❡❧✱ ❝✉❥❛ tr❛♥s❢♦r♠❛❞❛ ✐♥✈❡rs❛ é ❞❛❞❛ ♣♦r
F−1φ(x) = ˇφ(x) = 1 (2π)n
Z
eix.ξφ(ξ)dξ, φ∈ S
❆ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛✲s❡ ♥❛ ♣á❣✐♥❛ ✼✼ ❞❡ ❬✶✷❪✳ ❙❡ φ✱ ψ ∈ S✱ ✉t✐❧✐③❛♥❞♦ ♦
t❡♦r❡♠❛ ❛♥t❡r✐♦r ❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❝♦♥✈♦❧✉çã♦✱ ♣r♦✈❛♠✲s❡ ❛s s❡❣✉✐♥t❡s ✐❣✉❛❧❞❛❞❡s✿
✭✐✮ ∂dαφ(ξ) = (iξ)αφb(ξ)
✭✐✐✮ F(xαφ(x))(ξ) =i|α|∂αφb(ξ)
✭✐✐✐✮ Z φψ dxb =Z φψ dxb
✭✐✈✮ φ[∗ψ =φbψb
✭✈✮ φψc = (2π)−dφb∗ψb
✭✈✐✮ ubb= (2π)−du˘✱ ❝♦♠ u˘(ξ) = u(−ξ)✳
❚❛♠❜é♠✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱ ❣❛r❛♥t✐❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✳✶✱ ♦✉tr❛s ❢❛♠í❧✐❛s ❞❡ s❡♠✐✲♥♦r♠❛s q✉❡ ♣♦❞❡♠ s❡r ✉s❛❞❛s ♣❛r❛ ❞❡✜♥✐r ❛ t♦♣♦❧♦❣✐❛ ❡♠ S sã♦
❞❛❞❛s ♣♦r
kfkk,S = sup
|α|≤k x∈Rd
(1 +|ξ|)k |∂αf(x)|, k ∈N
kfbkk = sup |α|≤k ξ∈Rd
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✸
❉❡✜♥✐çã♦ ✶✳✹✳✷ ❯♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❡ ❝♦♥t✐♥✉♦ ❡♠Sé ❞✐t♦ ✉♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳
❖ ❡s♣❛ç♦ ❞❛s ❞✐str✐❜✉✐çõ❡s t❡♠♣❡r❛❞❛s s❡ ❞❡♥♦t❛ ❝♦♠ S′✳
❊①❡♠♣❧♦ ✶✳✹✳✷ ❯♠❛ ✈❡③ q✉❡ t♦❞❛ ❞✐str✐❜✉✐çã♦ ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦ s❡ ❡st❡♥❞❡ ❝♦♥✲ t✐♥✉❛♠❡♥t❡ ❛ C∞(Rd)✱ ✈❛❧❡ ❛ ✐♥❝❧✉sã♦ E′ ⊂ S′✳ P♦r r❡str✐çã♦ ❛ C∞
c (Rd)✱ ❡ ❝♦♠♦ ❡st❡ é
❞❡♥s♦ ❡♠S✱ t❡♠♦s q✉❡ S′ ⊂ D′✳ ❙❡ f ∈Lp✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝á✲❧❛ ❝♦♠♦ ✉♠❛ ❞✐str✐❜✉✐çã♦
t❡♠♣❡r❛❞❛ ❞❡✜♥✐♥❞♦✱ ♣❛r❛ ❝❛❞❛ φ∈ S✱
hTf, φi= Z
f(x)φ(x)dx.
❆ ❧✐♥❡❛r✐❞❛❞❡ é ✐♠❡❞✐❛t❛ ❡ ❛ ✐♥t❡❣r❛❧ ❛❝✐♠❛ é ✜♥✐t❛✱ ♣❡❧❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❍ö❧❞❡r✳ P❛r❛ ✈❡r✐✜❝❛r ❛ ❝♦♥t✐♥✉✐❞❛❞❡✱ s❡φ ∈ S✱ s❡❣✉❡ ❞❡ ✭✶✳✺✮ q✉❡
|hTf, φi| ≤ kfkLpkφkLp′
≤ CkfkLpkφkN,1
❉❛ ❡st✐♠❛t✐✈❛ ❛❝✐♠❛✱ ❝♦♥❝❧✉í♠♦s t❛♠❜é♠ q✉❡Lp ֒→ S′✳
❚❡♦r❡♠❛ ✶✳✹✳✷ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✮ ❙❡ f ∈ Lp ❡ g ∈ Lq✱ ❡♥tã♦ ❛ ❝♦♥✈♦❧✉çã♦
f∗g ∈Lr✱ ❝♦♠ 1
r + 1 =
1
p +
1
q✱ ❡ ✈❛❧❡
kf ∗gkLr ≤ kfkLpkgkLq
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✶✵❪✱ ♣á❣✳ ✷✵✳
❉❡✜♥✐çã♦ ✶✳✹✳✸ ❙❡ u∈ S′✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r bu ❞❡ u s❡ ❞❡✜♥❡ ♣♦r
hbu, φi=hu,φbi
P❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✳✶✱ bu ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ ❞❡t❡r♠✐♥❛ ✉♠❛ ♥♦✈❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛✳
▼❛✐s ❛✐♥❞❛✱ F r❡s✉❧t❛ ❝♦♥tí♥✉❛ ❡ ✐♥✈❡rsí✈❡❧ ❡♠ S′✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✶ ❙❡❥❛f ∈ S′(Rd)✳
✭✐✮ ❙❡ f ∈ L1(Rd)✱ ❛ tr❛♥s❢♦r♠❛❞❛ fb❞❡ f ❝♦♠♦ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ ❡ ❝♦♠♦ ❢✉♥çã♦
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✹
✭✐✐✮ ❙❡f ∈ E′(Rd)✱ fbé ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡C∞ ❞❛❞❛ ♣♦r b
f(ξ) = hfx, e−ixξi ✭✶✳✻✮
✭✐✐✐✮ ❙❡ f ∈L2(Rd) ❡♥tã♦ fb∈L2(Rd)✱ ❡ ✈❛❧❡
kfk2
L2 = (2π)−dkfbk2L2 (■❞❡♥t✐❞❛❞❡ ❞❡ ❋♦✉r✐❡r✲P❧❛♥❝❤❡r❡❧)
❉❡♠♦♥str❛çã♦✿ ❱❡r ❬✶✷❪✱ ♣á❣✳ ✽✶✳
❖❜s❡r✈❛çã♦ ✶✳✹✳✶ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ ❞✐str✐❜✉✐çã♦ u ∈ S′ t❛❧ q✉❡ bu ∈ L1(Rd)✳ P❡❧❛
Pr♦♣♦s✐çã♦ ✶✳✹✳✶✱ ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ubé ❛ ❢✉♥çã♦ ❞❛❞❛ ♣♦r bb
u(ξ) =
Z
e−ix.ξbu(x)dx
❈♦♠♦bbu= (2π)du˘✱ ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♥♦s ❞á q✉❡
u(ξ) = 1 (2π)d
Z
eix.ξub(x)dx
❆ss✐♠✱ s❡ bu ∈ L1✱ ♣♦❞❡♠♦s ✧r❡❝✉♣❡r❛r✧ u ♣❡❧❛ ❢ór♠✉❧❛ ❞❡ ✐♥✈❡rsã♦ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛
✶✳✹✳✶✳ P♦r ❛r❣✉♠❡♥t♦ s❡♠❡❧❤❛♥t❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ❞❡ ❘✐❡♠❛♥♥✲▲❡❜❡s❣✉❡✱ ♦❜t❡✲ ♠♦s q✉❡u∈C0
0(Rd)✱ ♦✉ s❡❥❛✱ u é ❝♦♥tí♥✉❛ ❡ ✈❛✐ ❛ ③❡r♦ q✉❛♥❞♦ |ξ| → ∞✳
❚❛♠❜é♠✱ ♣❡❧❛ ♠❡s♠❛ ❡①♣r❡ssã♦✱
|u(ξ)| ≤(2π)−dkbukL1,
♣❛r❛ q✉❛s❡ t♦❞♦ξ ∈Rd✱ ❞♦♥❞❡
kukL∞ ≤(2π)−dkubkL1. ✭✶✳✼✮
✶✳✹✳✶ ❖ ❚❡♦r❡♠❛ ❞❡ P❛❧❡②✲❲✐❡♥❡r
❙❡u ∈ E′(Rd)✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹✳✶✱ bu é ✉♠❛ ❢✉♥çã♦ C∞ ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ ✭✶✳✻✮✳ ❉❡
✉♠❛ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✱ ♣♦❞❡♠♦s ❡st❡♥❞❡r ❛ ❢✉♥çã♦ bu(ξ)❞❡ Rd ❛Cd✳ ◆❡st❡ ❝❛s♦✱ ❝♦♥❢♦r♠❡
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✺
❉❡✜♥✐çã♦ ✶✳✹✳✹ ❙❡❥❛ u∈ E′(Rd)✳ ❆ ❢✉♥çã♦ ✐♥t❡✐r❛ b
u(ζ) =hux, e−ixζi, ζ ∈Cd
é ❞✐t❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ ❞❡ u✳ ❙✉❛ r❡str✐çã♦ ❛ Rd é ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡
❋♦✉r✐❡r ❞❡ u✳
❆ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ ♣r❡s❡r✈❛ ♣r♦♣r✐❡❞❛❞❡s ❛♥á❧♦❣❛s ❛ ❞❡ ❋♦✉r✐❡r ❡♠ r❡✲ ❧❛çã♦ ❛ ❞❡r✐✈❛çã♦ ❡ ❝♦♥✈♦❧✉çã♦✱ ♣♦r ❡①❡♠♣❧♦✳ ❖ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✱ ❝✉❥❛ ♣r♦✈❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✶✷❪✱ ❞á ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ✉♠❛ ❢✉♥çã♦ ❤♦❧♦♠♦r❢❛ ❡♠Cd s❡❥❛ ❛ tr❛♥s❢♦r✲
♠❛❞❛ ❞❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ ❝♦♠ s✉♣♦rt❡ ❝♦♠♣❛❝t♦✳
❚❡♦r❡♠❛ ✶✳✹✳✸ ✭❚❡♦r❡♠❛ ❞❡ P❛❧❡②✲❲✐❡♥❡r✮ ❯♠❛ ❢✉♥çã♦ U(ζ) ✐♥t❡✐r❛ ❡♠ Cd é ❛
tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✲▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ❞✐str✐❜✉✐çã♦u∈ E′(Rn)✱ ❝♦♠S(u)⊂ {x;|x| ≤R}
s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s ❈ ❡ ◆ t❛✐s q✉❡
|U(ζ)| ≤C(1 +|ζ|)Nexp(R|Imζ|) ✭✶✳✽✮
✶✳✺ ❊s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈
◆❡st❡ t❡①t♦✱ ✈❛♠♦s ♥♦s r❡str✐♥❣✐r ❛♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♠♦❞❡❧❛❞♦s ❡♠L2✳ ❊st❡s ❡s♣❛ç♦s
❞❡s❡♠♣❡♥❤❛♠ ✉♠ ♣❛♣❡❧ ❝r✉❝✐❛❧ ♥♦ ❡st✉❞♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✱ ❧✐♥❡❛r❡s ♦✉ ♥ã♦✳ ❖ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ s❡rá ❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳
❉❡✜♥✐çã♦ ✶✳✺✳✶ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ u ♣❡rt❡♥❝❡ ❛♦
❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ í♥❞✐❝❡ s✱ ❞❡♥♦t❛❞♦ ♣♦r Hs(Rd) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ b
u∈L2❧♦❝(Rd) ❡ ub(ξ)∈L2(Rd,(1 +|ξ|2)sdξ)
❊s❝r❡✈❡♠♦s
kuk2Hs =
Z
Rd
(1 +|ξ|2)s|ub(ξ)|2dξ
Pr♦♣♦s✐çã♦ ✶✳✺✳✶ P❛r❛ t♦❞♦ ❡ q✉❛❧q✉❡rs∈R✱ ♦ ❡s♣❛ç♦Hs ❡q✉✐♣❛❞♦ ❝♦♠ ❛ ♥♦r♠❛k·k Hs
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✻
❉❡♠♦♥str❛çã♦✿ ➱ ✐♠❡❞✐❛t♦ q✉❡ ❛ ♥♦r♠❛k · kHs ♣r♦✈é♠ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦
hu, viHs =
Z
Rd
(1 +|ξ|2)sbu(ξ)bv(ξ)dξ
Pr♦✈❡♠♦s ❡♥tã♦ q✉❡ ❡st❡ ❡s♣❛ç♦ é ❝♦♠♣❧❡t♦✳ ❙❡❥❛ (un)n∈N ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠
Hs✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❛ ♥♦r♠❛✱(ub
n)n∈Né ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❈❛✉❝❤② ❡♠L2(Rd,(1+|ξ|2)sdξ)✳
❈♦♠♦ ❡st❡ é ❝♦♠♣❧❡t♦✱ ❡①✐st❡u˜∈L2(Rd,(1 +|ξ|2)sdξ) t❛❧ q✉❡
lim
n→∞kubn−u˜kL2(Rd,(1+|ξ|2)sdξ) = 0. ✭✶✳✾✮
❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ s❡q✉ê♥❝✐❛ (ubn) ❝♦♥✈❡r❣❡ ❛ u˜ ❡♠ S′✳ ❚♦♠❡♠♦s u = F−1u˜✳ ❈♦♠♦ ❛
❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ S′✱ s❡❣✉❡ q✉❡ u∈ S′✳ P♦r ✜♠✱u
n →u❡♠
Hs ❞❡✈✐❞♦ ❛ ✭✶✳✾✮✳
◆♦t❡♠♦s q✉❡ ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ❢♦r♠❛♠ ✉♠❛ ❢❛♠í❧✐❛ ❞❡❝r❡s❝❡♥t❡ ❞❡ ❡s♣❛ç♦s✱ ❝♦♠ r❡s♣❡✐t♦ ❛♦ í♥❞✐❝❡ s✳ ❉❡ ❢❛t♦✱ s ≥ s′ ✐♠♣❧✐❝❛ (1 +|ξ|2)s′
≤ (1 +|ξ|2)s✳ P♦rt❛♥t♦✱ s❡ ✉♠❛
❞✐str✐❜✉✐çã♦ t❡♠♣❡r❛❞❛ f é t❛❧ q✉❡ fb∈L2
❧♦❝(Rd)✱ s❡❣✉❡ q✉❡
kfk2Hs′ =
Z
Rd
(1 +|ξ|2)s′|ub(ξ)|2dξ
≤
Z
Rd
(1 +|ξ|2)s|bu(ξ)|2dξ =kfk2
Hs.
❆ss✐♠✱ Hs(Rd)⊆Hs′
(Rd)❡ t❛❧ ✐♥❝❧✉sã♦ é ❝♦♥t✐♥✉❛✳
❖s t❡♦r❡♠❛s q✉❡ s❡❣✉❡♠ tê♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❝❛r❛❝t❡r✐③❛r ♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♣❛r❛ ❞❡t❡r♠✐♥❛❞♦s ✈❛❧♦r❡s ❞❡s s❡♠ ♦ ✉s♦ ❡①♣❧í❝✐t♦ ❞❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✳
❚❡♦r❡♠❛ ✶✳✺✳✶ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ã♦✲♥❡❣❛t✐✈♦✳ ❖ ❡s♣❛ç♦ Hs(Rd) é ♦ ❡s♣❛ç♦
❞❛s ❢✉♥çõ❡su ♣❡rt❡♥❝❡♥t❡s ❛L2 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦α ❡♠ Nd✱ ❝♦♠|α| ≤st❡♠♦s ∂αu∈L2✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛s∈N✳ P❡❧♦ ❜✐♥ô♠✐♦ ❞❡ ◆❡✇t♦♥✱ t❡♠♦s (1 +|ξ|2)s = s X
i=0
s i
|ξ|2i✳
❆❣♦r❛✱ ✜①❛❞♦ 0≤i≤s ❡ u∈L2✱
|ξ|2i|bu(ξ)|2 = (ξ12+...+ξ2d)i|ub(ξ)|2
= X
|α|=i
cα|ξαbu(ξ)|2
= X
|α|=i
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✼
♣♦✐s∂dαu(ξ) = (iξ)αub(ξ)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✳✶✳ ❆ss✐♠✱
(1 +|ξ|2)s|ub(ξ)|2 =
s X
i=0
X
|α|=i
cα
s i
|∂dαu|2
= X
|α|≤s
˜
cα|∂dαu|2
■♥t❡❣r❛♥❞♦ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❡ ✉t✐❧✐③❛♥❞♦ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❋♦✉r✐❡r✲P❧❛♥❝❤❡r❡❧✱ s❡❣✉❡ q✉❡
Z
(1 +|ξ|2)s|ub(ξ)|2dξ = Z X
|α|≤s
˜
cα|∂dαu|2dξ
= X
|α|≤s
(2π)d˜cαk∂αuk2L2
❚♦♠❛♥❞♦C1 = min
(2π)dc˜
α; |α| ≤s ❡ C2 = max
(2π)d˜c
α; |α| ≤s ✱ ♦❜t❡♠♦s
C1
X
|α|≤s
k∂αuk2
L2 ≤
Z
(1 +|ξ|2)s|ub(ξ)|2dξ≤C 2
X
|α|≤s
k∂αuk2
L2 ✭✶✳✶✵✮
❈♦r♦❧ár✐♦ ✶✳✺✳✶ S é ❝♦♥t✐♥✉❛♠❡♥t❡ ✐♥❝❧✉í❞♦ ❡♠ Hs ✱ ∀s ∈R
❉❡♠♦♥str❛çã♦✿ ❚❡♥❞♦ ❡♠ ✈✐st❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ❥✉♥t❛✲ ♠❡♥t❡ ❝♦♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✭✶✳✺✮✱ s❡s ∈N✱
kuk2
Hs ≤C
X
|α|≤s
k∂αuk2
L2 ≤C
X
|α|≤s
kφk2
N,α,
♣❛r❛ t♦❞❛ ❢✉♥çã♦u∈ S✳ ❆ss✐♠✱ S ֒→Hs✱ s❡ s é ♥❛t✉r❛❧✳
P♦r ✜♠✱ s❡ s ∈ R q✉❛❧q✉❡r✱ ❞❡♥♦t❛♥❞♦ ♣♦r ⌈s⌉ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♠❛✐♦r ♦✉ ✐❣✉❛❧
q✉❡s✱ s❡❣✉❡ q✉❡ S ֒→H⌈s⌉ ֒→Hs✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❡♥❝❛✐①❡✳
Pr♦♣♦s✐çã♦ ✶✳✺✳✷ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ r❡❛❧ ♥♦ ✐♥t❡r✈❛❧♦ (0,1)✳ ❊♥tã♦ ♦ ❡s♣❛ç♦ Hs é ♦
❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s u ❞❡ L2 t❛❧ q✉❡
Z
Rd×Rd
|u(x+y)−u(x)|2
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✽
❆❧é♠ ❞✐ss♦✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡C > 0 t❛❧ q✉❡✱ ♣❛r❛ t♦❞❛ u∈Hs✱
C−1kuk2Hs ≤ kuk2L2 +
Z
Rd×Rd
|u(x+y)−u(x)|2
|y|d+2s dxdy ≤Ckuk
2
Hs
❉❡♠♦♥str❛çã♦✿ ◆♦t❡♠♦s q✉❡✱ ♣❡❧❛ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r✱
F :u(x+y)−u(x)−→ub(ξ)(eiyξ−1).
❡ ❛ss✐♠✱ ♣❡❧❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❋♦✉r✐❡r✲P❧❛♥❝❤❡r❡❧✱
Z
Rd|
u(x+y)−u(x)|2dx= (2π)−d Z
Rd|
eiyξ−1|2|ub(ξ)|2dξ.
❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ ❋✉❜✐♥✐✱ t❡♠♦s
Z
Rd×Rd
|u(x+y)−u(x)|2
|y|d+2s dxdy = (2π)− d
Z Z
|eiyξ−1|2
|y|d+2s
|bu(ξ)|2dξdy =
Z
F(ξ)|bu(ξ)|2dξ,
♦♥❞❡
F(ξ) = (2π)−d
Z
|eiyξ−1|2
|y|2s
dy
|y|d.
P❛r❛ ❝❛❞❛ ξ6= 0✱ ❝♦♥s✐❞❡r❛♠♦s z1 =y.ξ ❡ t♦♠❛♠♦s ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ♦rt♦❣♦♥❛❧ ❝♦♠
❥❛❝♦❜✐❛♥♦|ξ|d✳ ❉❡st❡ ♠♦❞♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♣❛r❛ ✐♥t❡❣r❛✐s✱
F(ξ) = (2π)−d|ξ|2s
Z
|eiz1 −1|2 |z|d+2s dz.
❆ ✐♥t❡❣r❛❧ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ✐♥❞❡♣❡♥❞❡ ❞❡ξ ❡✱ ♣❡❧❛ ❝♦♥❞✐çã♦ s♦❜r❡ s✱ é ✜♥✐t❛✳ ❆ss✐♠✱ F(ξ) = A|ξ|2s, ❝♦♠ A >0.
P♦rt❛♥t♦✱
kuk2
L2 +
Z
Rd×Rd
|u(x+y)−u(x)|2
|y|d+2s dxdy = Z
|ub(ξ)|2dξ+
Z
A|bu(ξ)|2|ξ|2sdξ
(1)
≤ C
Z
(1 +|ξ|2s)|bu(ξ)|2dξ
(2)
≤ C
Z
(1 +|ξ|2)s|ub(ξ)|2dξ = Ckuk2
❈❛♣ít✉❧♦ ✶✳ Pr❡❧✐♠✐♥❛r❡s ✶✾
◆❛ ❞❡s✐❣✉❛❧❞❛❞❡ (1)✱ C = max{1, A} ❡✱ ❡♠ (2)✱ ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❢✉♥çã♦
ρ(ξ) = 1 +|ξ| 2s
(1 +|ξ|2)s
é ❝♦♥tí♥✉❛ ❡ ❧✐♠✐t❛❞❛ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡sM1, M2 ≥0t❛❧ q✉❡M1 ≤ρ(ξ)≤M2✱
♣❛r❛ t♦❞♦ξ ∈Rd✳
❆ ♦✉tr❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡ ♣r♦✈❛ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳
❖ ❡①❡♠♣❧♦ ❛❜❛✐①♦ ✐❧✉str❛ ♦ q✉ã♦ ❛♠♣❧♦ é ♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ t♦♣♦❧ó❣✐❝♦ [
s∈R
Hs✳
❊①❡♠♣❧♦ ✶✳✺✳✶ ❙❡❥❛ u∈ E′(Rd)✳ ❈♦♠♦ub∈C∞(Rd)✱ t❡♠♦s q✉❡ bu∈L2
❧♦❝(Rd)✳
❚❛♠❜é♠✱ ♣❡❧❛ ❡st✐♠❛t✐✈❛ ✭✶✳✽✮ ❞❛❞❛ ♣❡❧♦ ❚❡♦r❡♠❛ ❞❡ P❛❧❡②✲❲✐❡♥❡r✱ ❡①✐st❡♠ C, N > 0
t❛❧ q✉❡
|bu(ξ)| ≤C(1 +|ξ|)N, ∀ξ ∈Rd, ✭✶✳✶✶✮
❡ ❛ss✐♠ ♣❛r❛s∈R ❛ s❡r ❡s❝♦❧❤✐❞♦✱ t❡♠♦s
Z
Rd
(1 +|ξ|2)s|ub(ξ)|2dξ ≤ C
Z
Rd
(1 +|ξ|2)s(1 +|ξ|2)2Ndξ
= C
Z
Rd
(1 +|ξ|2)s+2Ndξ
P❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✷✳✹✱ s❡ ss❛t✐s❢❛③ s <−2N− d
2✱ ❛ ✐♥t❡❣r❛❧ ❛❝✐♠❛ é ✜♥✐t❛✳ ❈♦♥s❡q✉❡♥t❡✲
♠❡♥t❡✱ u♣❡rt❡♥❝❡ ❛♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ Hs✳
❚❡♦r❡♠❛ ✶✳✺✳✷ ❙❡❥❛ s ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❛❧q✉❡r✳
✭✐✮ ❖ ❡s♣❛ç♦ C∞
c (Rd) é ❞❡♥s♦ ❡♠ Hs(Rd)
✭✐✐✮ ❆ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠❛ ❢✉♥çã♦ ❞❡ S é ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥tí♥✉❛ ❞❡ Hs ❡♠ Hs✳
❉❡♠♦♥str❛çã♦✿
✭✐✮ P❛r❛ ❛ ♣r♦✈❛ ❞❡st❡ ✐t❡♠✱ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❞❡♥s✐❞❛❞❡ ❞❡✈✐❞♦ ❛♦ ❈♦r♦❧ár✐♦ ✶✷✳✸ ❞❛ ♣á❣✐♥❛ ✽✽ ❞❡ ❬✷✵❪✿
Pr♦♣r✐❡❞❛❞❡✿ ❙❡❥❛ N ✉♠ ❡s♣❛ç♦ ♥♦r♠❛❞♦✳ ❯♠ s✉❜❡s♣❛ç♦ X ⊂ N é ❞❡♥s♦ ❡♠ N s❡✱ ❡
s♦♠❡♥t❡ s❡✱ ♦ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡N∗ (✐st♦ é✱ ♦ ❝♦♥❥✉♥t♦ ❞❛s ❛♣❧✐❝❛çõ❡s ❧✐♥❡❛r❡s ❡ ❝♦♥tí♥✉❛s
