Boa postura da "boa" equação de Boussinesq em espaços de Sobolev na reta e no toro

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙ã♦ ❈❛r❧♦s

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❡ ❚❡❝♥♦❧♦❣✐❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❇♦❛ ♣♦st✉r❛ ❞❛ ✧❜♦❛✧ ❡q✉❛çã♦ ❞❡ ❇♦✉ss✐♥❡sq ❡♠

❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♥❛ r❡t❛ ❡ ♥♦ t♦r♦

❆✉t♦r✿ ❘❡♥❛♥ ❞❡ ❈❛r✈❛❧❤♦ ▲♦✉r❡♥ç♦

❖r✐❡♥t❛❞♦r✿ ❘❛❢❛❡❧ ❋❡r♥❛♥❞♦ ❇❛r♦st✐❝❤✐

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦✱

Pr✐♠❡✐r❛♠❡♥t❡ ❛♦ Pr♦❢✳ ❉r✳ ❘❛❢❛❡❧ ❋❡r♥❛♥❞♦ ❇❛r♦st✐❝❤✐✱ ♣❡❧❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❞❡s❡♥✲ ✈♦❧✈❡r ❡st❡ ♣r♦❥❡t♦ ❡ t♦❞♦ ❛✉①í❧✐♦ ❞❛❞♦✳

➚ ❘❡♥❛t❛ ❖❧✐✈❡✐r❛ ❋✐❣✉❡✐r❛✱ ♣♦r t♦❞❛ ❛❥✉❞❛✱ ❡st✉❞♦s ❡ ❛t❡♥çã♦✳

➚ ▼❛r✐❛♥❛ ❙❛❧❣❛❞♦ ▲♦♣❡s✱ ♣♦r t♦❞♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ❛♠♦r✳

❆♦s ♠❡✉s ❡st✐♠❛❞♦s ❛♠✐❣♦s✳

❆♦ ♠❡✉ ❢❛❧❡❝✐❞♦ P❛✐✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜♦❛ ♣♦st✉r❛ ❞❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ♥ã♦ ❧✐♥❡❛r ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❛ ✧❜♦❛✧❡q✉❛çã♦ ❞❡ ❇♦✉ss✐♥❡sq ❡♠ ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈✳ ❆♣r❡s❡♥✲ t❛r❡♠♦s ♦s r❡s✉❧t❛❞♦s ❞❡ ❜♦❛ ♣♦st✉r❛ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ♦ ♣❡r✐ó❞✐❝♦✱ q✉❛♥❞♦ ♦s ❞❛❞♦s ✐♥✐❝✐❛✐s ❡ ❛s s♦❧✉çõ❡s sã♦ ♣❡r✐ó❞✐❝♦s ♥❛ ✈❛r✐á✈❡❧ ❡s♣❛❝✐❛❧✱ ❡ ♦ ♥ã♦ ♣❡r✐ó❞✐❝♦✳

❙✉♠ár✐♦

✶ Pr❡❧✐♠✐♥❛r❡s ✶✶

✷ ❖ ❈❛s♦ P❡r✐ó❞✐❝♦ ✶✾

✷✳✶ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❘❡s✉❧t❛❞♦s Pr✐♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✸ ❖ ❈❛s♦ ❘❡❛❧ ✼✶

✸✳✶ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✷ ❘❡s✉❧t❛❞♦s Pr✐♥❝✐♣❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

■♥tr♦❞✉çã♦

P♦r ✈♦❧t❛ ❞❡ ✶✽✼✵✱ ❇♦✉ss✐♥❡sq ❡♥❝♦♥tr♦✉ ❛❧❣✉♥s ♠♦❞❡❧♦s ♣❛r❛ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛ ❞❡ ♣❡✲ q✉❡♥❛ ❛♠♣❧✐t✉❞❡ ❡ ♦♥❞❛s ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ á❣✉❛✳ ❊st❛s ❡q✉❛çõ❡s ♣♦ss✉❡♠ s♦❧✉çã♦ ♣♦r ♦♥❞❛s ✈✐❛❥❛♥t❡s ❝❤❛♠❛❞❛s ♦♥❞❛s s♦❧✐tár✐❛s q✉❡ sã♦ ❞❡♥♦♠✐♥❛❞❛s ❞❡ss❛ ♠❛♥❡✐r❛ ♣❡❧❛ tr❛♥s❧❛çã♦ ❞❛ ♦♥❞❛ ♥❛ s✉♣❡r❢í❝✐❡✳ ❆ t❡♦r✐❛ ❞❡ ❇♦✉ss✐♥❡sq ❢♦✐ ❛ ♣r✐♠❡✐r❛ ❛ ❞❛r ✉♠❛ ❡①♣❧✐❝❛çã♦ s❛t✐s❢❛tór✐❛ ❡ ❝✐❡♥tí✜❝❛ s♦❜r❡ ♦ ❢❡♥ô♠❡♥♦ ❞❛s ♦♥❞❛s s♦❧✐tár✐❛s ❞❡s❝♦❜❡rt❛s ♣♦r ❙❝♦tt✲❘✉ss❡❧✭✶✽✵✽✲✶✽✽✷✮ ✸✵ ❛♥♦s ❛♥t❡s✳ ➚ ♣r✐♠❡✐r❛ ✈✐st❛✱ ❇♦✉ss✐♥❡sq ❞❡s❡❥❛✈❛ ❞❡♥♦♠✐♥❛r t❛❧ ❡q✉❛çã♦ ❞❡ ❡q✉❛çã♦ ❞❡ ▲②❛♣✉♥♦✈✱ ❞❛❞❛ s✉❛ ❝♦♥❡①ã♦ ❝♦♠ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❛s ♦♥❞❛s s♦❧✐tár✐❛s✳

■r❡♠♦s ❝♦♥s✐❞❡r❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ♣❛r❛ ❛ ❡q✉❛çã♦ ❞❡ ❇♦✉ss✐♥❡sq ✩

✬ ✫ ✬ ✪

utt✁uxx uxxxx ♣f♣uqqxx ✏0, xPT, t→0, ✭❈❛s♦ P❡r✐ó❞✐❝♦✮

u♣0, xq ✏ u0♣xq; ut♣0, xq ✏ u1♣xq

❡

✩ ✬ ✫ ✬ ✪

utt✁uxx uxxxx ♣f♣uqqxx ✏0, xPR, t→0, ✭❈❛s♦ ❘❡❛❧✮

u♣0, xq ✏ u0♣xq; ut♣0, xq ✏ u1♣xq.

❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡✲s❡ ♣❡r❝❡❜❡r q✉❡ ❡①✐st❡♠ ✈ár✐❛s ♠❛♥❡✐r❛s ❞❡ ♦❜s❡r✈❛r♠♦s ❡ ♠♦❞❡❧❛r♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ♣❡q✉❡♥❛ ❛♠♣❧✐t✉❞❡✱ ❜❛st❛ ❝♦❧♦❝❛r♠♦s ❞✐❢❡r❡♥t❡s ❢✉♥çõ❡s ♣❛r❛f♣uq✳ P♦ré♠✱

t♦❞♦s ❡st❡s ♠♦❞❡❧♦s ♣♦ss✉❡♠ ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ♠❛r❝❛♥t❡✱ ❞❡ q✉❡ sã♦ ❛s ♣❡rt✉r❜❛çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ♦♥❞❛ ❧✐♥❡❛r q✉❡ ❝✉✐❞❛♠ ❞❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ❡ ❞✐s♣❡rsã♦✳

P❛rt❡ ❞♦ ♠♦t✐✈♦ ❞❛ r❡❧❛t✐✈❛ ❡s❝❛ss❡③ ❞❡ r❡s✉❧t❛❞♦s r❡❢❡r❡♥t❡s às ❡q✉❛çõ❡s ❞♦ t✐♣♦ ❇♦✉ss✐✲ ♥❡sq ♣♦❞❡♠ s❡r ❞❡✈✐❞❛s ❛♦ ❢❛t♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ♥❡♠ s❡♠♣r❡ é ❣❧♦❜❛❧♠❡♥t❡ ❜❡♠ ♣♦st♦✳ ❊①✐st❡♠ ♣❡r✜s ✐♥✐❝✐❛✐s ❞❡ ♦♥❞❛ ❡ ✈❡❧♦❝✐❞❛❞❡ q✉❡ sã♦ s✉❛✈❡s✱ ♠❛s ♣❛r❛ ♦s q✉❛✐s ❛ s♦❧✉çã♦ ♣❡r❞❡ ❛ r❡❣✉❧❛r✐❞❛❞❡ ❡♠ t❡♠♣♦ ✜♥✐t♦✳

◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r ❜♦❛ ♣♦st✉r❛ ❧♦❝❛❧ ♣❛r❛ ❛ s♦❧✉çã♦ q✉❛♥❞♦f♣uq ✏ u2✳

✶✵ ❙❯▼➪❘■❖ ❖ t✐♣♦ ❞❡ ❡s♣❛ç♦ ♦♥❞❡ ✐r❡♠♦s ❞❡s❡♥✈♦❧✈❡r t❛❧ ♣♦st✉r❛ s❡rá ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ Hs♣Tq ❡ Hs♣Rq✱ _sPR✱ ❡q✉✐♣❛❞♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠ ❛s ♥♦r♠❛s

⑥f⑥Hs_♣Tq ✏ ⑥①n②sfˆ⑥_L2_♣Tq ❡ ⑥f⑥_Hs_♣Rq ✏ ⑥①ξ②sfˆ⑥_L2_♣Rq,

s❡♥❞♦ ①a② ✑ ❛2

1 ⑤a⑤2✳

❆❧é♠ ❞✐ss♦✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ í♥❞✐❝❡s ❞❡ ❙♦❜♦❧❡✈ s → ✁1

4✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❣❛r❛♥t✐r❡♠♦s

q✉❡ ♣❛r❛ s➔ ✁1

4✱ ❛❧❣✉♠❛s ❡st✐♠❛t✐✈❛s ✉s❛❞❛s ❛q✉✐ ♥ã♦ ✈❛❧❡♠✳

❊st❡ tr❛❜❛❧❤♦ ❢♦✐ ❜❛s❡❛❞♦ ♥♦s ❛rt✐❣♦s s♦❜r❡ ❜♦❛ ♣♦st✉r❛ ❞❛ ✧❜♦❛✧❡q✉❛çã♦ ❞❡ ❇♦✉ss✐♥❡sq ♥♦ ❝❛s♦ r❡❛❧ ♣r♦❞✉③✐❞♦ ♣❡❧♦ ♣r♦❢❡ss♦r ❞♦✉t♦r ▲✉✐③ ●✉st❛✈♦ ❋❛r❛❤ ❡ ♥♦ ❝❛s♦ ♣❡r✐ó❞✐❝♦ ❥✉♥t♦ ❝♦♠ ❛ ♣r♦❢❡ss♦r❛ ❞♦✉t♦r❛ ▼ár❝✐❛ ❆ss✉♠♣çã♦ ●✉✐♠❛rã❡s ❙❝✐❛❧♦♠✱ ❝✐t❛❞♦s ❡♠ r1s ❡ r3s✳

❆❧é♠ ❞✐ss♦✱ ❡st❡ t❡①t♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♥♦ ❝❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s q✉❡ s❡rã♦ út❡✐s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡ss❡ tr❛❜❛❧❤♦✱ ❝♦♠ ê♥❢❛s❡ ♥❛ ❞❡✜♥✐çã♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❙♦❜♦❧❡✈ ♣❡r✐ó❞✐❝♦s ❡ ♥ã♦ ♣❡r✐ó❞✐❝♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ♣r✐♠❡✐r❛ s❡çã♦✱ ❛♣r❡s❡♥t❛♠♦s ❛ ❞❡♠♦♥str❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❝♦♥s♦❧✐❞❛r ♥♦ss♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦✱ ❥á ♥❛ s❡❣✉♥❞❛ s❡çã♦✱ ♣♦❞❡♠♦s ❡♥tã♦ ❡♥✉♥❝✐❛r ❡ ❞❡s❡♥✈♦❧✈❡r ❛ ❡st✐♠❛t✐✈❛ ♣r✐♥❝✐♣❛❧✱ ✐♥✐❝✐❛♥❞♦ ❝♦♠ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s ❡ r❡♣❛rt✐♥❞♦ R4 ❡♠ ✻ r❡❣✐õ❡s✱ ❞❡ ♠♦❞♦ q✉❡✱ ❛♦ ♦❜t❡r ❛ ❡st✐♠❛t✐✈❛ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ❢❛r❡♠♦s ✉♠❛ ❡①♣♦s✐çã♦ ❞❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛❜♦r❞❛r✲ ♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ❞❡ss❡ tr❛❜❛❧❤♦✳ ❆s ❞❡♠♦♥str❛çõ❡s ❞❡ss❡s r❡s✉❧t❛❞♦s s❡rã♦ ♦♠✐t✐❞❛s✱ ♦ ❧❡✐t♦r ✐♥t❡r❡ss❛❞♦ ♣♦❞❡rá ❡♥❝♦♥trá✲❧❛s ♥❛s r❡❢❡rê♥❝✐❛s r4s✱r5s ❡ r8s✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❯♠❛ ❢✉♥çã♦ f : R Ñ R é ❞✐t❛ ♣❡r✐ó❞✐❝❛✱ s❡ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ r❡❛❧ _T✱

❝❤❛♠❛❞♦ ♣❡rí♦❞♦ ❞❡ f✱ t❛❧ q✉❡

f♣x Tq ✏f♣xq, ❅xP R_,

♦✉ s❡❥❛✱ q✉❛❧q✉❡r ♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♥❚ ❞❡ ❚ t❛♠❜é♠ é ✉♠ ♣❡rí♦❞♦ ❞❡ ❢✳ ❖ ♠❡♥♦r ✈❛❧♦r ❞❡ ❚ q✉❡ s❛t✐s❢❛③ ❛ ✐❣✉❛❧❞❛❞❡ é ❝❤❛♠❛❞♦ ♣❡rí♦❞♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ f ❡ s❡rá ❞❡♥♦t❛❞♦

♣♦r ❚✳ ◗✉❛❧q✉❡r ♦✉tr♦ ♣❡rí♦❞♦ ❞❡f s❡rá ✉♠ ♠ú❧t✐♣❧♦ ✐♥t❡✐r♦ ❞♦ ♣❡rí♦❞♦ ❢✉♥❞❛♠❡♥t❛❧✳

❊①❡♠♣❧♦ ✶✳✷✳ ❆s ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦ss❡♥♦ sã♦2π✲♣❡r✐ó❞✐❝❛s✳

❉❡✜♥✐çã♦ ✶✳✸✳ ❱❛♠♦s ❞❡♥♦t❛r ♣♦r ①☎② ♦ ♦♣❡r❛❞♦r q✉❡ ❛❣❡ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

①x② ✏. ❛1 ⑤x⑤2_,❅_xP_R.

❉❡✜♥✐çã♦ ✶✳✹✳ ❱❛♠♦s ❞❡♥♦t❛r ♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ♣♦r S1 ⑨R2✱ ♦✉ s❡❥❛✱

S1 ✏ t♣x, yq PR2_;x2 _y2 ✏₁✉_.

✶✷ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ❊①❡♠♣❧♦ ✶✳✺✳ P♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r S1 ❝♦♠ ♦ ✐♥t❡r✈❛❧♦ r0,2πs ✈✐❛ ❛ s❡❣✉✐♥t❡ ❛♣❧✐❝❛çã♦

ϕr0,2πs ÑS1 θ ãÑ ♣cosθ, senθq.

❚❛❧ ❢✉♥çã♦ é s♦❜r❡❥❡t♦r❛✱ ✐♥❥❡t♦r❛ ❛ ♠❡♥♦s ❞❡ ϕ♣0q ✏ϕ♣2πq✳ ❈♦♥s✐❞❡r❛♥❞♦ ❡♥tã♦ t♦♣♦❧♦✲

❣✐❝❛♠❡♥t❡ ♦ q✉♦❝✐❡♥t❡

¯

ϕ : r0,2πs

t0,2π✉ ÑS

1

,

t❡♠♦s q✉❡ ϕ¯ é ✉♠ ❤♦♠❡♦♠♦r✜s♠♦ ❡ ❡♥tã♦

r0,2πs ✓S1.

❉❡✜♥✐çã♦ ✶✳✻✳ ❱❛♠♦s ❞❡✜♥✐r ♦ t♦r♦ n✲❞✐♠❡♥s✐♦♥❛❧ Tn ❝♦♠♦ s❡♥❞♦

Tn✑_{S❧♦♦♦♦♦♦♦♦♦♦♠♦♦♦♦♦♦♦♦♦♦♥}1✂_S1✂ ☎ ☎ ☎ ✂_S1

n ✈❡③❡s

.

❉❡✜♥✐çã♦ ✶✳✼✳ ❱❛♠♦s ❞❡♥♦t❛r ♣♦r C✽♣Tq ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s ❞❡ ❝❧❛ss❡ _C✽ ♠✉♥✐❞♦ ❞❛

♥♦r♠❛

⑥f⑥k ✏sup x_PT

⑤f♣kq♣xq⑤.

❉❡✜♥✐çã♦ ✶✳✽✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♣φjq ❡♠ Cc✽♣Rq ❝♦♥✈❡r❣❡ ❛ ③❡r♦ s❡❣✉♥❞♦ ❛

♥♦r♠❛ s❡

• ❉_K ⑨R ❝♦♠♣❛❝t♦ t❛❧ q✉❡ s✉♣♣ _φj ⑨_K,❅_j P N_;

• _φj ❡ _φ♣kq

j ❝♦♥✈❡r❣❡♠ ✉♥✐❢♦r♠❡♠❡♥t❡ ❛ ③❡r♦ ❅kPN.

❉❡✜♥✐çã♦ ✶✳✾✳ ❯♠❛ ❞✐str✐❜✉✐çã♦2π✲♣❡r✐ó❞✐❝❛ é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦u:C✽♣Tq Ñ C t❛❧ q✉❡

u♣φq✏ ①. u, φ②,

♦✉ s❡❥❛✱ ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s

✶✸

• ①_{u, φj}② Ñ₀s❡♠♣r❡ q✉❡_φj Ñ₀❡♠_C✽♣Tq ❡ ❡st❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♦❝♦rr❡ ✉♥✐❢♦r♠❡♠❡♥t❡

s❡❣✉♥❞♦ ❛ ♥♦r♠❛✳

◆♦t❛çã♦✿ D✶♣Tq ✏ ♣_C✽♣Tqq✝✳

❊①❡♠♣❧♦ ✶✳✶✵✳ ❙❡❥❛ f : T Ñ C ✉♠❛ ❢✉♥çã♦ ♣❡r✐ó❞✐❝❛ ❞❡ ❝❧❛ss❡ _C✽✳ ❉❡✜♥✐♠♦s ♦

♦♣❡r❛❞♦r

Tf : C✽♣Tq ÑC

φ ãÑ

➺2π 0

φ♣xqf♣xqdx,

✐st♦ é

①Tf, φ②✏.

➺2π 0

f♣xqφ♣xqdx.

➱ ❢á❝✐❧ ✈❡r q✉❡Tf ❞❡✜♥❡ ✉♠❛ ❞✐str✐❜✉✐çã♦ 2π✲♣❡r✐ó❞✐❝❛✳ ❆❧é♠ ❞✐ss♦✱ ❞❛❞❛s f, g PC✽♣Tq✱

t❡♠♦s Tf ✏Tg s❡✱ ❡ s♦♠❡♥t❡ s❡✱ f ✏g✳ P♦❞❡♠♦s ❡♥tã♦ ❢❛③❡r ❛ ✐❞❡♥t✐✜❝❛çã♦ Tf ✑f.

❉❡✜♥✐çã♦ ✶✳✶✶✳ ❙❡ uPD✶♣Tq ❡♥tã♦ ❞❡✜♥✐♠♦s

①u✶, φ② ✏ ✁①u, φ✶②,❅ φ PC✽r0,2πs.

P❡r❣✉♥t❛✿ ❙❡ φ é ♣❡r✐ó❞✐❝❛✱ é ♣♦ssí✈❡❧ ❡①♣r❡ss❛r φ ❝♦♠♦ ✉♠❛ s♦♠❛ ❞♦ t✐♣♦

φ♣xq ✏ ➳

kPZ

akeikx,

♣❛r❛ ❝❡rt❛s ❝♦♥st❛♥t❡sak PC❄ ❙❡ ❡ss❛ ❞❡❝♦♠♣♦s✐çã♦ ❢♦r ♣♦ssí✈❡❧ ❡♥tã♦✱ ❅nPZ✱ t❡♠♦s

e✁inxφ♣xq ✏ ➳

kPZ

akei♣k✁nqx.

■♥t❡❣r❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡0 ❛2π✱ t❡♠♦s

➺2π 0

e✁inxφ♣xqdx✏ ➳

kPZ

ak

➺2π 0

ei♣k✁nqxdx✏an2π.

❊♥tã♦ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡✈❡rá s❡r

an✏ 1 2π

➺2π 0

✶✹ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ■ss♦ ♠♦t✐✈❛ ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳

❉❡✜♥✐çã♦ ✶✳✶✷✳ ❙❡ φ PC✽♣Tq✱ ❞❡✜♥✐♠♦s ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ♣❡r✐ó❞✐❝❛ ❞❡ _φ ♣♦r

ˆ

φ♣nq ✏ 1

2π

➺2π 0

e✁inxφ♣xqdx, ❅n PZ_.

❚❡♦r❡♠❛ ✶✳✶✸✳ ❉❡✜♥❛ ♦ ♦♣❡r❛❞♦r Dx ✏ 1

i d

dx. ❊♥tã♦

②

Dxjφ♣nq ✏njφˆ♣nq,❅j PR,❅n PZ.

❉❡✜♥✐çã♦ ✶✳✶✹✳ ❙❡ uP D✶♣Tq ❡♥tã♦ ❞❡✜♥✐♠♦s ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❋♦✉r✐❡r ❞❡ _u ♣♦r

ˆ

u♣nq ✏ 1

2π①u, e

✁inx_②.

❚❡♦r❡♠❛ ✶✳✶✺✳ ❙❡ uPD✶♣Tq ❡♥tã♦

u✏ ➳

nPZ

ˆ

u♣nqeinx.

❱❛❧❡ r❡ss❛❧t❛r q✉❡ ♦ ❚❡♦r❡♠❛ ✶✳✶✺ ♦❝♦rr❡ ♥❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♣♦♥t✉❛❧ ❡♠ D✶♣Tq✳

❉❡✜♥✐çã♦ ✶✳✶✻✳ ❈♦♥s✐❞❡r❡ ♣Ω, A, µq ✉♠ ❡s♣❛ç♦ ❞❡ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛✱ Ω⑨Rn✱ _{A σ}✲ á❧❣❡❜r❛

❡♠ Ω✱ µ ♠❡❞✐❞❛ ♣♦s✐t✐✈❛✱ ❞❡✜♥❛♠♦s

Lp ✏Lpµ♣Ωq ✏ tf : ΩÝÑC;⑥f⑥p ➔ ✽✉,

s❡♥❞♦

⑥f⑥p ✏

✂➺

Ω

⑤f♣tq⑤pdµ♣tq

✡1

p

,1↕p➔ ✽

❡

⑥f⑥_✽✏sup tPΩ ❡ss⑤

f♣tq⑤.

❉❡✜♥✐çã♦ ✶✳✶✼✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ Hs♣Tq✱ _sP R✱ ❝♦♠♦ s❡♥❞♦

Hs♣Tq ✏ t_uPD✶♣Tq_;➳

n_PZ

✶✺ ❙❡ f P Hs♣Tq ❡♥tã♦

⑥f⑥Hs_♣Tq ✏ ✄

➳

n_PZ

♣1 n2q2s⑤φˆ♣nq⑤2

☛1 2

.

❚❡♦r❡♠❛ ✶✳✶✽✳ ⑥.⑥Hs_♣Tq é ✉♠❛ ♥♦r♠❛ ❡ ♣Hs♣Tq,⑥.⑥Hs_♣Tqq é ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt✳ ❚❡♦r❡♠❛ ✶✳✶✾✳ ✭❍ö❧❞❡r✮ ❙❡❥❛♠ 1↕p, q ↕ ✽✱ 1

p

1

q ✏1. ❙❡ f PL

p_{✱ g PL}q _❡♥tã♦

➺

⑤f♣tqg♣tq⑤dµ♣tq ↕ ⑥f⑥p⑥g⑥q.

❚❡♦r❡♠❛ ✶✳✷✵✳ ✭▼✐♥❦♦✇s❦✐✮ ❙❡ 1↕p➔ ✽ ❡ f, gP Lp ❡♥tã♦ f g P Lp ❡

⑥f g⑥p ↕ ⑥f⑥p ⑥g⑥p.

Pr♦♣♦s✐çã♦ ✶✳✷✶✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✮ ❙❡❥❛♠ u, v PL2 ❡♥tã♦

①u, v②2 ↕ ①u, u② ①v, v②,❅u, v P L2.

❙❡♥❞♦ ①u, v② ♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ❞❛❞♦ ♣♦r

①u, v② ✏

➺ R

f♣xq¯g♣xqdx.

❚❡♦r❡♠❛ ✶✳✷✷✳ Lp é ✉♠ ❡s♣❛ç♦ ❞❡ ❇❛♥❛❝❤✳

❉❡✜♥✐çã♦ ✶✳✷✸✳ ❙❡ f P L1♣Rq ❞❡✜♥✐♠♦s ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❞❡ _f ♣♦r

ˆ

f♣ξq ✏

➺ _✽

✁✽

e✁ixξf♣xqdx.

❆❧é♠ ❞✐ss♦✱ ✐r❡♠♦s ♥♦s r❡❢❡r✐r à tr❛♥s❢♦r♠❛❞❛ t♦t❛❧ ❞❡ ❋♦✉r✐❡r q✉❛♥❞♦ ❛ tr❛♥s❢♦r♠❛❞❛ ❢♦r t♦♠❛❞❛ ❡♠ t♦❞❛s ❛s ✈❛r✐á✈❡✐s ❞❛ ❢✉♥çã♦ f✳ ❯s❛r❡♠♦s t❛♠❜é♠ ❛ ♥♦t❛çã♦ F♣_fq♣_ξq_.

❉❡✜♥✐çã♦ ✶✳✷✹✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❙❝❤✇❛rt③ ♣♦r

S♣Rnq ✏ t_f PC✽♣Rnq_{; sup}

xPRn⑥

xjDα_xf♣xq⑥ ➔ ✽,❅j PN_,❅_αPZn✉_,

s❡♥❞♦ α✏ ♣α1, α2,☎ ☎ ☎, αnq✱ x✏ ♣x1, x2,☎ ☎ ☎, xnq ❡ Dαx ✑ 1

i❇

α1

1 ❇ α2

2 ☎ ☎ ☎ ❇ αn

n .

✶✻ ❈❆P❮❚❯▲❖ ✶✳ P❘❊▲■▼■◆❆❘❊❙ ✭✐✮ F é ✐♥✈❡rsí✈❡❧❀

✭✐✐✮ F✁1 é ❝♦♥tí♥✉❛❀

✭✐✐✐✮ ϕ♣xq ✏ ♣2πq✁n

➺ Rn

eixξϕˆ♣ξqdξ.

❚❡♦r❡♠❛ ✶✳✷✻✳ ✭P❧❛♥❝❤❡r❡❧✮ ❙❡ f PL2 ❡♥tã♦ fˆPL2 ❡ ✈❛❧❡

⑥_fˆ_{⑥2 ✏}❄_2π⑥f⑥2.

❉❡✜♥✐çã♦ ✶✳✷✼✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦s ❞❛s ❢✉♥çõ❡s t❡st❡ ♣♦r

Cc✽♣a, bq ✏ tφPC✽♣a, bq; suppφ é ❝♦♠♣❛❝t♦✉,

s❡♥❞♦ suppφ✏ txP ♣a, bq;φ♣xq ✘0✉.

❉❡✜♥✐çã♦ ✶✳✷✽✳ ❯♠❛ ❞✐str✐❜✉✐çã♦ s♦❜r❡ ♣a, bq é ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦u:C_c✽♣a, bq ÝÑ C t❛❧ q✉❡

u♣φq✏ ①. u, φ②,

♦✉ s❡❥❛✱ ✉♠❛ ❛♣❧✐❝❛çã♦ q✉❡ s❛t✐s❢❛③ ✭✐✮ ①u, αφ βψ② ✏α①u, φ② β①u, ψ②;

✭✐✐✮ ①u, φj② Ñ0 s❡♠♣r❡ q✉❡ φj Ñ0 ❡♠ Cc✽♣a, bq.

❯t✐❧✐③❛♠♦s ❛ ♥♦t❛çã♦ D✶♣_{a, b}q ✏ ♣_Cc✽♣_{a, b}qq✶_.

❉❡✜♥✐çã♦ ✶✳✷✾✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ♣φjqjPN ⑨S ❝♦♥✈❡r❣❡ ❛ ③❡r♦ s❡ ⑤xαφ♣_jβq♣xq⑤ Ñ 0,

✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ t♦❞♦ α, β PZ ✳

❉❡✜♥✐çã♦ ✶✳✸✵✳ ❯♠ ❢✉♥❝✐♦♥❛❧ u : S♣Rnq ÝÑ C ❧✐♥❡❛r ❡ ❝♦♥tí♥✉♦ é ❝❤❛♠❛❞♦ ❞❡ ❞✐str✐✲

❜✉✐çã♦ t❡♠♣❡r❛❞❛✳ ◆♦t❛çã♦✿ S✶_.

❉❡✜♥✐çã♦ ✶✳✸✶✳ ❉❡✜♥✐♠♦s ♦ ❡s♣❛ç♦ ❞❡ ❙♦❜♦❧❡✈ ❞❡ ❣r❛✉ sPR ♣♦r

✶✼ ❚❡♦r❡♠❛ ✶✳✸✷✳ ✭❚❡♦r❡♠❛ ❞❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❘✐❡s③✮ ❙❡❥❛ H ✉♠ ❡s♣❛ç♦ ❞❡ ❍✐❧❜❡rt

r❡❛❧ ♦✉ ❝♦♠♣❧❡①♦✱ ♠✉♥✐❞♦ ❞♦ ♣r♦❞✉t♦ ✐♥t❡r♥♦ ①., .②✳ ❙❡❥❛ u ✉♠ ❢✉♥❝✐♦♥❛❧ ❧✐♥❡❛r ❝♦♥tí♥✉♦

❡♠ H ❡♥tã♦ ❡①✐st❡ ✉♠ ✈❡t♦r yPH t❛❧ q✉❡

u♣xq ✏ ①x, y②,❅xPH.

❚❡♦r❡♠❛ ✶✳✸✸✳ ✭❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✮ ❙❡❥❛♠ p, q P R t❛✐s q✉❡ 1 p

1

q ✏ 1 ❡♥tã♦

♣❛r❛ t♦❞♦ ♣❛r ❞❡ ♥ú♠❡r♦s r❡❛✐s ♥ã♦ ♥❡❣❛t✐✈♦sa, b ✈❛❧❡

ab↕ a

p

p bq

q.

❚❡♦r❡♠❛ ✶✳✸✹✳ ✭❚❡♦r❡♠❛ ❞❛ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦♠✐♥❛❞❛✮ s❡❥❛ ♣fnq ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡

❢✉♥çõ❡s ✐♥t❡❣rá✈❡✐s q✉❡ ❝♦♥✈❡r❣❡ q✉❛s❡ s❡♠♣r❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ r❡❛❧ ♠❡♥s✉rá✈❡❧ f✳ ❙❡

❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ✐♥t❡❣rá✈❡❧ g t❛❧ q✉❡ ⑤fn⑤ ↕g✱ ♣❛r❛ t♦❞♦ nPN✱ ❡♥tã♦ f é ✐♥t❡❣rá✈❡❧ ❡

➺

f dµ✏ lim nÑ✽

➺

fndµ.

❋✐♥❛❧✐③❛r❡♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❡♥✉♥❝✐❛♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ ♣♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤✱ ❢✉♥❞❛✲ ♠❡♥t❛❧ ♥❡st❡ tr❛❜❛❧❤♦✳

❉❡✜♥✐çã♦ ✶✳✸✺✳ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♥ã♦ ✈❛③✐♦ ❝♦♠ ✉♠❛ ♠étr✐❝❛d✳ ❯♠❛ ❛♣❧✐❝❛çã♦ f :X ÝÑX é ❞✐t❛ ✉♠❛ ❝♦♥tr❛çã♦ s❡ ❡①✐st✐r 0↕β ➔1 t❛❧ q✉❡

d♣f♣xq, f♣yqq ↕βd♣x, yq,❅x, y PX.

❚❡♦r❡♠❛ ✶✳✸✻✳ ✭P♦♥t♦ ✜①♦ ❞❡ ❇❛♥❛❝❤✮ ❙❡❥❛X ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ♥ã♦ ✈❛③✐♦ ❝♦♠♣❧❡t♦

❈❛♣ít✉❧♦ ✷

❖ ❈❛s♦ P❡r✐ó❞✐❝♦

❊st❡ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❈❛✉❝❤② ♣❛r❛ ❛ ✧❜♦❛✧❡q✉❛çã♦ ❞❡ ❇♦✉ss✐♥❡sq ❝♦♠ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♣❡r✐ó❞✐❝❛s✳

❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ✈❛❧♦r ✐♥✐❝✐❛❧ ♣❡r✐ó❞✐❝♦ ✩

✬ ✫ ✬ ✪

utt✁uxx uxxxx ♣f♣uqqxx ✏0,

u♣0, xq ✏u0♣xq; ut♣0, xq ✏u1♣xq.

✭✷✳✶✮

❝♦♠ f♣uq ✏u✷ ❡ ❛ s❡❣✉✐♥t❡ ♥♦r♠❛

⑥f⑥_Hs_♣T_q ✏ ⑥①n②

s_fˆ⑥ L2_♣Tq.

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ Υ ♦ ❡s♣❛ç♦ ❞❛s ❢✉♥çõ❡s F♣☎q ❝♦♠

✐✮ F :T✂RÑC_;

✐✐✮ F♣x,☎q PS♣Rq _,❅_xPT_;

✐✐✐✮ F♣☎, tq PC✽♣Tq _,❅_tPR_.

P❛r❛ s, bPR✱ ✈❛♠♦s ❞❡♥♦t❛r _Xs,b ♦ ❝♦♠♣❧❡t❛♠❡♥t♦ ❞❡_Υ ❝♦♠ ❛ ♥♦r♠❛

⑥F⑥Xs,b ✏ ⑥①⑤τ⑤ ✁γ♣nq②

b_①n②s_F˜⑥ l2

nL2τ ✏ ✄

➳

nPZ

⑥①⑤τ⑤ ✁γ♣nq②b①n②sF˜⑥L2

τ ☛1

2

✭✷✳✷✮

✏ ✄

➳

nPZ ➺

R

①⑤τ⑤ ✁γ♣nq②b①n②s⑤F˜⑤2dτ

☛1 2

, ✭✷✳✸✮

✷✵ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ s❡♥❞♦ γ ❛ ❢✉♥çã♦ ❞❛❞❛ ♣♦rγ♣xq ✏❄x2 _x4_, ❅_xP_R.

✷✳✶ ❘❡s✉❧t❛❞♦s Pr❡❧✐♠✐♥❛r❡s

◆❡st❛ s❡çã♦✱ ✐r❡♠♦s ❞❡s❡♥✈♦❧✈❡r ❛❧❣✉♥s ❧❡♠❛s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦✲ r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡✳ ❚❛✐s ❧❡♠❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ r1s✱r2s✱r3s✱ r6s ❡ r7s✳

❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♣❛r❝✐❛❧ ❧✐♥❡❛r ✩

✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✪

utt✁uxx uxxxx ✏0,

u♣0, xq ✏φ♣xq, ut♣0, xq ✏ ♣ψ♣xqqx.

✭✷✳✹✮

❆♣❧✐❝❛♥❞♦ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ♣❡r✐ó❞✐❝❛ ❡♠ ✭✷✳✹✮ ♥❛ ✈❛r✐á✈❡❧ x✳ ❚❡♠♦s ❛ s❡✲

❣✉✐♥t❡ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠

♣n4✁n2quˆ♣t, nq ❇_t2uˆ♣t, nq ✏ 0. ✭✷✳✺✮

❆❣♦r❛✱ ♣❡❧♦ ▼ét♦❞♦ ❞❡ ❱❛r✐❛çã♦ ❞♦s P❛râ♠❡tr♦s✱ t❡♠♦s q✉❡ ❛ s♦❧✉çã♦ ❞❡ ✭2.5✮ é ❞❛

❢♦r♠❛

u♣t, nq ✏ c1e

❄

n2 _n4_it

c2e✁

❄

n2 _n4_it

. ✭✷✳✻✮

❈♦♠♦ ❡♠ ✭✷✳✹✮ t❡♠♦su♣0, xq ✏φ♣xq ❡ ut♣0, xq ✏ ♣ψ♣xqqx✱ ❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❞❡ ✭✷✳✺✮

✜❝❛♠ uˆ♣0, nq ✏φˆ♣nq ❡ uˆt♣0, nq ✏ψˆx♣nq. ▲♦❣♦✱

✩ ✬ ✫ ✬ ✪

c1 c2 ✏φˆ♣nq,

❄

n✷ n4_c 1✁

❄

n✷ n4_c

2 ✏ψˆx♣nq.

✭✷✳✼✮

❯s❛♥❞♦ ❛ ♥♦t❛çã♦ ❞❡s❡❥❛❞❛ ❡ ❧❡♠❜r❛♥❞♦ ❛s ♥♦✈❛s ❝♦♥❞✐çõ❡s ✐♥✐❝❛✐s✱ t❡♠♦s ♣❛r❛n ✘0

✩ ✬ ✬ ✫ ✬ ✬ ✪

c1 c2 ✏φˆ♣nq,

c1✁c2 ✏ ˆ

ψx♣nq

γ♣nq .

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✶ ❘❡s♦❧✈❡♥❞♦ ♦ s✐st❡♠❛ ❛❝✐♠❛ t❡♠♦s✱

c1 ✏ ˆ

φ♣nq

2 ˆ

ψx♣nq

γ♣nq , c2 ✏

ˆ

φ♣nq

2 ✁

ˆ

ψx♣nq 2iγ♣nq,

❡ ❛ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ❡ss❡s ✈❛❧♦r❡s ❞❡c1 ❡ c2 ❡♠ ✭✷✳✻✮ t❡♠♦s q✉❡ ❛ s♦❧✉çã♦ ❞❛ ❡❞♦ ✭✷✳✺✮ é

❞❛❞❛ ♣♦r

ˆ

u♣t, nq ✏ ④Vc♣tqφ V④s♣tqψx.

❉❡ss❛ ❢♦r♠❛✱ t♦♠❛♥❞♦✲s❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ✐♥✈❡rs❛✱ ♦❜t❡♠♦s

u♣t, nq ✏ Vc♣tqφ Vs♣tqψx,

s❡♥❞♦

Vc♣tqφ✏

✄

eit❄n✷ n4

e✁it❄n✷ n4

2

ˆ

φ♣nq

☛q

❡

Vs♣tqψx ✏

✄

eit❄n✷✁n4

e✁it❄n✷ n4

2i❄n✷ n4

ˆ

ψx♣nq

☛q

.

P❛r❛ ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ ❞❛ ❊❉P ♥ã♦ ❧✐♥❡❛r✱ ✈❛♠♦s ♣r♦❝✉r❛r ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❢✉♥çõ❡s ❞❡t ❞♦ t✐♣♦

u♣tq ✏ b1♣tqu1♣tq b2♣tqu2♣tq, ♦♥❞❡ u1 ✏eitγ♣nq ❡ u2♣tq ✏ e✁itγ♣nq.

❉❛ t❡♦r✐❛ q✉❛❧✐t❛t✐✈❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s✱ t❡♠♦s

b1♣tq ✏ ✁

➺

u2♣tq♣u 2_q

xx

W♣u1, u2q

dt c1 ❡b2♣tq ✏

➺

u1♣tq♣u 2_q

xx

W♣u1, u2q

dt c2.

❯♠❛ ✈❡③ q✉❡✱

W♣u1, u2q♣tq ✏ ❉❡t

✔

✕ eit γ♣nq

e✁itγ♣nq iγ♣nqeitγ♣nq ✁iγ♣nqeitγ♣nq

✜

✷✷ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ ♦❜t❡♠♦s

b1♣tq ✏ c1 ✁

➺t 0

e✁itγ♣nq 2iγ♣nq♣u

2

qxx♣tqdt ❡ b2♣tq ✏c2

➺t 0

e✁itγ♣nq 2iγ♣nq♣u

2

qxx♣tqdt.

❙✉❜st✐t✉✐♥❞♦ ❛s ❝♦♥st❛♥t❡s c1 ❡ c2 ♣❡❧♦ q✉❡ ♦❜t❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❝❤❡❣❛♠♦s ♥❛ s♦❧✉çã♦

❞❛ ❡q✉❛çã♦ ♥ã♦ ❧✐♥❡❛r ❞❛❞❛ ♣♦r

u♣tq ✏Vc♣tqφ Vs♣tqψx

➺t 0

Vs♣t✁t✶q♣u 2

qxx♣t✶qdt✶.

❙❡❥❛ ❛❣♦r❛ θ P C_c✽♣Rq ✉♠❛ ❢✉♥çã♦ ❞❡ ❝♦rt❡ t❛❧ q✉❡✱ ₀ ↕ _θ ↕ ₁✱ _θ ✑ ₁ ❡♠ r✁_1,1s✱

supp♣θq ⑨ r✁2,2s✳ P❛r❛ 0➔T ➔1✱ ❞❡✜♥❛ θT♣tq ✏θ♣

t

Tq✳ ❆ss✐♠✱

u♣tq ✏θ♣tq

✂

Vs♣tqψx Vc♣tqφ θT♣tq

➺t 0

Vs♣t✁t✶q♣u✷qxx♣t✶qdt✶

✡

, ✭✷✳✾✮

é ✉♠❛ s♦❧✉çã♦ ❞❡ ✭✷✳✹✮ ♦❜t✐❞❛ ❡♠ r0, Ts✳

▲❡♠❛ ✷✳✷✳ ❙❡❥❛ u♣tq ✉♠❛ s♦❧✉çã♦ ❞❡

✩ ✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✪

utt✁uxx uxxxx✏0,

u♣0, xq ✏ φ♣xq, ut♣0, xq ✏ ♣ψ♣xqqx,

✭✷✳✶✵✮

❝♦♠ φPHs ❡ ψ PHs✁1✳ ❊♥tã♦ ❡①✐st❡ c→0 ❞❡♣❡♥❞❡♥❞♦ ❞❡ θ, s, b t❛❧ q✉❡

⑥θu⑥Xs,b ↕c♣⑥φ⑥Hs ⑥ψ⑥Hs✁1q.

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛u♣tq ✏Vc♣tqφ Vs♣tqψx ❡θ♣tq ❞❛❞♦s ❝♦♠♦ ❛♥t❡r✐♦r♠❡♥t❡✳ ❊♥tã♦

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✸ ❚♦♠❛♥❞♦✲s❡ ❛ tr❛♥s❢♦r♠❛❞❛ t♦t❛❧ ❞❡ ❋♦✉r✐❡r ❞❡ θu✱ ♦❜t❡♠♦s

①

θu♣τ, ξq ✏

➺ R

e✁itτ①θu♣t, ξqdt✏

➺

eitτθ

✄

eitγ♣ξq

✓

ˆ

φ♣ξq

2

ξψ♣ˆξq

2γ♣ξq

✛

e✁itγ♣ξq

✓

ˆ

φ♣ξq

2 ✁

ξψ♣ˆξq

2γ♣ξq

✛☛

dt

✏ ➺

eit♣τ γ♣ξqqθ♣tq

✓

ˆ

φ♣ξq

2

ξψˆ♣ξq

2γ♣ξq

✛

dt

➺

e✁it♣τ✁γ♣ξqqθ♣tq

✓

ˆ

φ♣ξq

2 ✁

ξψ♣ˆξq

2γ♣ξq

✛

dt

✏ θˆ♣τ γ♣ξqq

2

✓

ˆ

φ♣ξq ξ

ˆ

ψ♣ξq γ♣ξq

✛

ˆ

θ♣τ ✁γ♣ξqq

2

✓

ˆ

φ♣ξq ✁ ξ

ˆ

ψ♣ξq γ♣ξq

✛

.

❙❡❥❛♠ ❛❣♦r❛ h1♣ξq ✏ φˆ♣ξq

ξψˆ♣ξq

γ♣ξq ❡ h2♣ξq ✏

ˆ

φ♣ξq ✁ ξ

ˆ

ψ♣ξq

γ♣ξq ✳ ❆ss✐♠✱

⑥θu⑥Xs,b ✏ ⑥①⑤τ⑤ ✁γ♣ξq②

b_①

ξ②sθu①⑥L2

τ,ξ

✏✎✎✎✎_✎①⑤τ⑤ ✁γ♣ξq②b①ξ②s

✄

h1♣ξqθˆ♣τ ✁γ♣ξqq 2

h2♣ξqθˆ♣τ γ♣ξqq 2 ☛✎✎ ✎✎ ✎ L2 τ,ξ

↕✎✎✎✎_✎①⑤τ⑤ ✁γ♣ξq②b①ξ②sh1♣ξq

ˆ

θ♣τ ✁γ♣ξqq

2 ✎✎ ✎✎ ✎ L2 τ,ξ ✎✎ ✎✎

✎①⑤τ⑤ ✁γ♣ξq②b①ξ②s

h2♣ξqθˆ♣τ γ♣ξqq 2 ✎✎ ✎✎ ✎ L2 τ,ξ . ▲♦❣♦✱

I✷ ✏

✎✎ ✎✎

✎①⑤τ⑤ ✁γ♣ξq②b①ξ②s

h1♣ξqθˆ♣τ ✁γ♣ξqq 2 ✎✎ ✎✎ ✎ 2 L2 τ,ξ ✏ ➺ R ➺ R

①ξ②2s①⑤τ⑤✁γ♣ξq②2b⑤h1♣ξq⑤✷

✞✞ ✞✞ ✞

ˆ

θ♣τ ✁γ♣ξqq

4 ✞✞ ✞✞ ✞ 2 dτ dξ.

❯♠❛ ✈❡③ q✉❡✱

⑤⑤τ⑤ ✁ ⑤γ♣ξq⑤⑤ ✏ ⑤⑤τ⑤ ✁γ♣ξq⑤ ↕ ♠í♥ t⑤τ ✁γ♣ξq,⑤τ γ♣ξq⑤✉,

t❡♠♦s✱

I✷ ↕

➺ R

➺ R

①ξ②2s①τ✁γ♣ξq②2b⑤h1♣ξq⑤✷

✞✞ ✞✞ ✞

ˆ

θ♣τ ✁γ♣ξqq

4 ✞✞ ✞✞ ✞ 2 dτ dξ.

❋❛③❡♥❞♦ ❛ ♠✉❞❛♥ç❛α ✏τ ✁γ♣ξq,t❡♠♦s dα ✏dτ ❡

I✷ ↕

➺ R

➺ R

①ξ②2s①α②2b⑤h1♣ξq⑤✷

✞✞ ✞✞ ✞

ˆ

θ♣αq

4 ✞✞ ✞✞ ✞ 2 dαdξ ✏ ➺ R

①ξ②2s⑤h1♣ξq⑤✷dξ

➺ R

①α②2b

✞✞ ✞✞ ✞

ˆ

θ♣αq

✷✹ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ ❖❜s❡r✈❡♠♦s ❛❣♦r❛ q✉❡

➺ R

①α②2b

✞✞ ✞✞ ✞

ˆ

θ♣αq

4

✞✞ ✞✞ ✞

2

dα↕C_θ,b2 ,

✉♠❛ ✈❡③ q✉❡ _θˆ_PS s❡♠♣r❡ q✉❡ _θ P_Cc✽♣Rq✳ P♦rt❛♥t♦✱ I✷ ↕C_θ,b2

➺ R

①ξ②2s⑤h1♣ξq⑤✷dξ ✏C 2 θ,b ⑥①ξ②

2

h1⑥ 2 L2

ξ.

❆♥❛❧♦❣❛♠❡♥t❡✱ ♦❜t❡♠♦s ✉♠ r❡s✉❧t❛❞♦ ♣❛r❛ ♣IIq✷✱ ✐st♦ é✱

♣IIq✷ ↕C_θ,b2 ⑥①ξ②2h2⑥ 2 L2

ξ.

❏✉♥t❛♥❞♦ t♦❞❛s ❛s ♣❛rt❡s✱ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡

⑥θu⑥Xs,b ↕ ♣Iq ♣IIq ↕ 2Cθ,b ✁

⑥①ξ②2h1♣ξq⑥ 2 L2

ξ ⑥①ξ②

2

h2♣ξq⑥ 2 L2

ξ ✠

.

❖❜s❡r✈❡ ❡♥tã♦ q✉❡

⑥①ξ②2h1♣ξq⑥ 2 L2

ξ ⑥①ξ②

2

h2♣ξq⑥ 2 L2

ξ ✏ ✎✎ ✎✎ ✎①ξ②s

✄

♣_φˆ_♣ξq ξψˆ♣ξq

γ♣ξq

☛✎✎ ✎✎ ✎ L2 ξ ✎✎ ✎✎ ✎①ξ②s

✄

♣_φˆ_{♣ξq ✁} ξψ♣ˆξq

γ♣ξq

☛✎✎ ✎✎ ✎

L2

ξ

↕2⑥①ξ②sφˆ♣ξq⑥L2_ξ 2

✎✎ ✎✎ ✎①ξ②s

ξψˆ♣ξq γ♣ξq

✎✎ ✎✎ ✎ L2 ξ . ▼❛s✱ ✎✎ ✎✎ ✎①ξ②s

ξψˆ♣ξq γ♣ξq

✎✎ ✎✎ ✎ 2 L2 ξ ✏ ➺ R

①ξ②s ⑤ξ⑤

2

γ♣ξq2⑤ ˆ

ψ♣ξq⑤2dξ ✏

➺ R

①ξ②s 1

①ξ②2⑤ ˆ

ψ♣ξq⑤2dξ,

♣♦✐s✱

ξ2

ξ2 _ξ4 ✏ 1

1 ξ2 ✏

1

①ξ②2.

❊♠ ✈✐st❛ ❞❛s ♦❜s❡r✈❛çõ❡s ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦ ✎✎

✎✎ ✎①ξ②s

ξψˆ♣ξq γ♣ξq

✎✎ ✎✎ ✎

L2

ξ

✏ ⑥ψ⑥Hs✁1,

❡

⑥θu⑥Xs,b ↕Cθ,b♣⑥φ⑥Hs ⑥ψ⑥Hs✁1q.

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✺

▲❡♠❛ ✷✳✸✳ ❙❡❥❛♠ ✁1

2 ↕b

✶ _{↕0↕b ↕b}✶ _{1 ❡ 0➔T ↕1. ❊♥tã♦}

✭✐✮ ✎✎✎✎θT♣tq

➺t 0

f♣t✶qdt✶✎✎✎✎

Hb t

↕T1✁♣b✁b✶q⑥f⑥_Hb✶ t

✭✐✐✮ ✎✎✎✎θT♣tq

➺t 0

Vs♣t✁t✶qf♣uq♣t✶qdt✶

✎✎ ✎✎

Xs,b

↕T1✁♣b✁b✶q

✎✎ ✎✎ ✎F✁

1

✄

˜

f♣uq♣τ, nq

2iγ♣nq

☛✎✎ ✎✎ ✎ Xs,b . ❉❡♠♦♥str❛çã♦✿

✭✐✮ ❱❛♠♦s r❡❡s❝r❡✈❡r ♣iq ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

θT♣tq t

➺

0

f♣t✶qdt✶ ✏ 1

2πθT♣tq

t ➺ 0 ✽ ➺ ✁✽

eit✶τf♣♣τqdτ dt✶

✏ 1

2πθT♣tq

✽

➺

✁✽

♣

f♣τq

t

➺

0

eit✶τdt✶dτ

✏ 1

2πθT♣tq

✽

➺

✁✽

♣

f♣τq

✂

eitτ_✁1

iτ

✡

dτ.

❆❣♦r❛✱ ✈❛♠♦s s❡♣❛r❛r ❛ ✐♥t❡❣r❛❧ ❡♠ ❞✉❛s r❡❣✐õ❡s ⑤τ⑤T ↕1 ❡ ⑤τ⑤T →1✱

θT♣tq t

➺

0

f♣t✶qdt✶ ✏ θT♣tq

➺

⑤τ⑤T↕1

♣

f♣τq

✂

eitτ_✁1

iτ

✡

dτ θT♣tq

➺

⑤τ⑤T→1

♣

f♣τq

✂

eitτ_✁1

iτ

✡

dτ

✏ ♣■q ♣■■q ♣■■■q

❝♦♠

♣■q ✏ θT♣tq

✽

➳

k_✏1

tk k!

➺

⑤τ⑤T↕1

♣iτqk✁1f♣♣τqdτ

♣■■q ✏ θT♣tq

➺

⑤τ⑤T→1

♣

f♣τqeitτ♣iτq✁1dτ

♣■■■q ✏ ✁θT♣tq

➺

⑤τ⑤T→1

♣

f♣τq♣iτq✁1dτ.

❆q✉✐✱ ✉s❛♠♦s ♦ s❡❣✉✐♥t❡ ❢❛t♦eitτ✁1✏

✽

➳

k✏1

♣itτqk

k! ✳ ❱❛♠♦s ❛❣♦r❛ ❡st✐♠❛r ❝❛❞❛ ♣❛r❝❡❧❛✳

✷✻ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ ❡♥tã♦

⑥♣■q⑥Hb t ↕

✽

➳

k✏1 1 k! ✎✎ ✎✎ ✎✎ ✎

tkθT♣tq

➺

⑤τ⑤T↕1

♣iτqk✁1f♣♣τqdτ

✎✎ ✎✎ ✎✎ ✎ Hb t ↕ ➳✽

k_✏1 1

k!T 1✁k

✎✎ ✎✎ ✎✎ ✎

tkθT♣tq

➺

⑤τ_⑤T_↕1

♣

f♣τqdτ

✎✎ ✎✎ ✎✎ ✎ Hb t ✏ ➳✽

k✏1

T1✁k k!

✞✞ ✞✞ ✞

➺

⑤τ_⑤T_↕1

♣

f♣τqdτ

✞✞ ✞✞

✞✎✎tkθT♣tq✎✎_Hb t .

❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ ♦❜s❡r✈❛♠♦s q✉❡ ✞✞

✞✞ ✞✞ ✞

➺

⑤τ⑤T↕1

♣

f♣τqdτ

✞✞ ✞✞ ✞✞ ✞✏ ✞✞ ✞✞ ✞✞ ✞ ➺

⑤τ⑤T↕1

♣

f♣τq①τ②b✶①τ②✁b✶dτ

✞✞ ✞✞ ✞✞ ✞↕ ⑥

f⑥_Hb✶ ☎ ✝ ✆

➺

⑤τ⑤T↕1

①τ②✁2b✶dτ

☞ ✍ ✌ 1 2 ,

♦ q✉❡ ✐♠♣❧✐❝❛

⑥♣■q⑥Hb

t ↕ ⑥f⑥Hb✶

✽

➳

k✏1

T1✁k k! ✎✎t

k_θ T✎✎_Hb

✄ ➺

⑤τ_⑤T_↕1

①τ②✁2b✶dτ

☛1 2

.

▲♦❣♦✱ é s✉✜❝✐❡♥t❡ ♣r♦✈❛r q✉❡

✽

➳

k_✏1

T1✁k

k! ✎✎t k_θ

T✎✎_Hb ✄ ➺

⑤τ⑤T↕1

①τ②✁2b✶dτ

☛1 2

↕CT1✁♣b✁b✶q. ✭✷✳✶✶✮

❚❡♠♦s ✎✎tkθT✎✎_Hb ✏ ✎✎

✎①τ②bt②k_θ T♣τq✎✎✎

L2 ❝♦♠

②

tk_θ

T♣τq ✏

✽

➺

✁✽

e✁itτtkθT♣tqdt✏

✽

➺

✁✽

e✁itτtkθ

✂ t T ✡ dt ✏ ✽ ➺ ✁✽

e✁i♣T sqτ♣T sqkθ♣sqT ds✏Tk 1

✽

➺

✁✽

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✼ ✐st♦ é✱ ②tk_θ

T♣τq ✏Tk 1✑①

tk_θ♣_{T τ}q✙✳ ❆ss✐♠✱

✎✎tkθT✎✎_Hb ✏T

k 1

☎ ✆

✽

➺

✁✽

①τ②2b✞✞✞ ①tk_θ♣_{T τ}q✞✞_✞ 2

dτ

☞ ✌

1 2

✏Tk 1

☎ ✆

✽

➺

✁✽

①T✁1ρ②2b✞✞✞ ①tk_θ♣_ρq✞✞_✞T✁1

dρ

☞ ✌

1 2

.

❈♦♠♦ T✁2 ➙T✁1 ➙1✱ t❡♠♦s q✉❡

①T✁1ρ②2 ✏1 T✁2ρ2 ↕T✁2♣1 ρ2q ✏ T✁2①ρ②2,

♦ q✉❡ ✐♠♣❧✐❝❛ ♦ s❡❣✉✐♥t❡ ❢❛t♦

✎✎tkθT✎✎ ↕Tk 1

T✁12

☎ ✆

✽

➺

✁✽

T✁2b①ρ②2b✞✞✞ ①tk_θ♣_ρq✞✞_✞ 2

dρ

☞ ✌

1 2

✏Tk 12T✁b✎✎tkθ✎✎

Hb. ✭✷✳✶✷✮

❯s❛♥❞♦ q✉❡ b ↕1 b✶ ↕1✱ ♦❜t❡♠♦s

✎✎tkθ✎✎_Hb ↕✎✎t

k_θ✎✎

H1 ✏✎✎t

k_θ✎✎

L2 ✎✎❇t♣t

k_θq✎✎

L2 ✏✎✎t

k_θ✎✎

L2 ✎✎kt

k✁1

θ✎✎_L2 ✎✎t

k_θ✶✎✎ L2.

❈♦♠♦ s✉♣♣ ♣θq ⑨ r✁2,2s✱ ♣♦❞❡♠♦s tr❛❜❛❧❤❛r ❝♦♠ ❧✐♠✐t❛çõ❡s ❞♦ s❡❣✉✐♥t❡ t✐♣♦

⑤tkθ♣tq⑤ ↕2k⑤θ♣tq⑤, ⑤ktk✁1θ♣tq⑤ ↕ 2k✁1k⑤θ♣tq⑤ ❛♥❞ ⑤tkθ✶♣tq⑤ ↕2k⑤θ✶♣tq⑤,

♣❛r❛ q✉❛✐sq✉❡r tPR ❡ _k➙1✳ ❊♥tã♦✱

✎✎_tk_θ✎✎

Hb ↕k2

k_♣2⑥θ⑥

L2 ⑥θ✶⑥_L2q ✏k2kC_θ, ❅k ➙1.

❉❛ ❡q✉❛çã♦ ✭✷✳✶✷✮ t❡♠♦s ✎✎_tk_θ

T✎✎_Hb ↕CθT✁

b_Tk 12k2k.

❙❡♥❞♦ ❛ss✐♠✱

✽

➳

k✏1

T1✁k

k! ✎✎t k_θ✎✎

Hb ↕CθT

1✁b_T1 2

✽

➳

k✏1 2k

♣k✁1q! ↕CθT 1✁b_T1

✷✽ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖

♦♥❞❡ ✉s❛♠♦s q✉❡ ➳✽

k_✏1 2k

♣k✁1q! ✏ 2e 2

➔ ✽✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ b✶ ↕0 ❡ T✁2 ➙ 1✱

t❡♠♦s ✄ ➺

⑤τ⑤T↕1

①τ②✁2b✶dτ

☛1 2

✏

✄ ➺

⑤τ⑤T↕1

♣1 τ2q✁b✶dτ

☛1 2

↕ ✄ ➺

⑤τ⑤T↕1

♣2T✁2q✁b✶dτ

☛1 2

↕ 2✁b2✶Tb✶

✄ ➺

⑤τ_⑤T_↕1 1dτ

☛1 2

✏Cb✶Tb

✶

T✁12.

❊♥tã♦✱

✽

➳

k_✏1

T1✁k

k! ✎✎t k_θ

T✎✎_Hb ✄ ➺

⑤τ_⑤T_↕1

①τ②✁2b✶dτ

☛1 2

↕CθT

1✁b_T12C

b✶Tb

✶

T✁12 ✏C

θ,b✶T

1✁♣b✁b✶q_,

❝♦♥❝❧✉✐♥❞♦ ❛ss✐♠ q✉❡ ✭✷✳✶✶✮ ❛❝♦♥t❡❝❡✳ ❊st✐♠❛t✐✈❛ ♣❛r❛ ✭■■✮✿ ❖❜s❡r✈❡♠♦s q✉❡

⑥♣■■q⑥Hb t ✏

✎✎ ✎✎ ✎✎ ✎

θT♣tq

➺

⑤τ⑤T→1

♣iτq✁1f♣♣τqdτ

✎✎ ✎✎ ✎✎ ✎ Hb t ✏ ✞✞ ✞✞ ✞✞ ✞ ➺

⑤τ⑤T→1

♣iτq✁1f♣♣τqdτ

✞✞ ✞✞ ✞✞ ✞⑥

θT⑥_Hb. ✭✷✳✶✸✮

❯s❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❡♠♦s ✞✞

✞✞ ✞✞ ✞

➺

⑤τ⑤T→1

♣iτq✁1f♣♣τqdτ

✞✞ ✞✞ ✞✞ ✞✏ ✞✞ ✞✞ ✞✞ ✞ ➺

⑤τ⑤T→1

①τ②✁b✶♣iτq✁1①τ②b✶f♣♣τqdτ

✞✞ ✞✞ ✞✞ ✞↕

✄ ➺

⑤τ⑤T→1

①τ②✁2b✶⑤τ⑤✁2dτ

☛1 2

⑥f⑥_Hb✶ ,

❝♦♠ ➺

⑤τ⑤T→1

①τ②✁2b✶⑤τ⑤✁2dτ ✏ 2

✽

➺

T✁1

♣1 τ2q✁b✶τ✁2dτ ✏2

✽

➺

1

♣1 T✁2s2q✁b✶T2s✁2T✁1ds

↕ T2b✶ 1

✽

➺

1

♣1 s2q✁b✶s✁2ds,

♣♦✐s T✁2 ➙1 ❡ ✁b✶ ➙0✳ ❆❧é♠ ❞✐ss♦✱

✽

➺

1

♣1 s2q✁b✶s✁2ds↕

✽

➺

1

♣2s2q✁b✶s✁2ds ✏2✁b✶

✽

➺

1

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✷✾ ♣♦✐s 2b✶ 2→1✳ ❊♥tã♦✱

✄ ➺

⑤τ⑤T→1

①τ②✁2b✶⑤τ⑤✁2dτ

☛1 2

⑥f⑥_Hb✶ ↕Cb✶Tb

✶ 1 2 ⑥g⑥

Hb✶ .

P♦r ♦✉tr♦ ❧❛❞♦✱ ⑥θT⑥_Hb ✏✎①✎✎ τ②bθ♣T✎✎✎

L2 ❝♦♠

♣

θT♣τq ✏

➺

e✁itτθ

✂

t T

✡

dt✏

➺

e✁isT τθ♣sqT ds✏T♣θ♣T τq.

❆ss✐♠✱

⑥θT⑥_Hb ✏ ✄ ➺

①τ②2bT2⑤♣θ♣T τq⑤2dτ

☛1 2

✏T

✄ ➺

①T✁1ρ②2b⑤♣θ♣ρq⑤2T✁1dρ

☛1 2

↕ T12

✄ ➺

①T✁1ρ②2b⑤♣θ♣ρq⑤2dρ

☛1 2

↕T12

✄ ➺

T✁2b①ρ②2b⑤♣θ♣ρq⑤2dρ

☛1 2

✏ T12✁b⑥θ⑥

Hb,

♣♦✐s ①T✁1ρ②2b ✏ ♣1 T✁2ρ2qb ↕ T✁2b♣1 ρ2qb ✏ T✁2b①ρ②2b✳ ❱♦❧t❛♥❞♦ à ❡q✉❛çã♦

✭✷✳✶✸✮✱ ❝♦♥❝❧✉í♠♦s q✉❡

⑥♣■■q⑥Hb

t ↕ Cb✶T

b✶ 12T 1 2✁b⑥θ⑥

Hb⑥g⑥_Hb✶ ✏ Cθ,b,b✶T

1✁♣b✁b✶q_⑥g⑥ Hb✶.

❊st✐♠❛t✐✈❛ ♣❛r❛ ✭■■■✮✿ ❱❛♠♦s ❡s❝r❡✈❡r ♣■■■q ✏θT♣tqJ♣tq✱ s❡♥❞♦

J♣tq ✏

➺

⑤τ_⑤T_→1

eitτ♣iτq✁1g♣♣τqdτ.

❚❡♠♦s ❡♥tã♦ q✉❡

⑥♣■■■q⑥Hb t ✏

✎✎

✎①τ②b♣④θT ☎Jq✎✎✎

L2 ✏C

✎✎

✎①τ②bθ①T ✝ ♣J♣τq✎✎✎ L2.

❖❜s❡r✈❡♠♦s q✉❡

✸✵ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ ❉❡ ❢❛t♦✱

⑤τ⑤2 ↕ ♣⑤τ ✁y⑤ ⑤y⑤q2 ✏ ⑤τ ✁y⑤2 2⑤τ ✁y⑤⑤y⑤ ⑤y⑤2 ↕2♣⑤τ ✁y⑤2 ⑤y⑤2q,

♣♦✐s 2⑤τ ✁y⑤⑤y⑤ ↕ ⑤τ ✁y⑤2 ⑤y⑤2✳ ❆ss✐♠✱ ❝♦♠♦ b➙0✱

①τ②b ✏ ♣1 ⑤τ⑤2qb2 ↕ 1 2♣⑤τ ✁y⑤2 ⑤y⑤2q✟

b

2 ↕

22b♣1 ⑤τ✁y⑤2 ⑤y⑤2q

b

2.

P❛r❛ ♣r♦✈❛r♠♦s ❛ ❡q✉❛çã♦ ✭✷✳✶✹✮✱ é s✉✜❝✐❡♥t❡ ❣❛r❛♥t✐r q✉❡

♣α βq2b ↕α

b

2 β

b

2 ✭✷✳✶✺✮

é ✈❡r❞❛❞❡ ♣❛r❛ t♦❞♦ α, β ➙ 0✳ ❙✉♣♦♥❤❛♠♦s q✉❡ α ✘ 0 t❡♠♦s ❡♥tã♦ q✉❡ ✭✷✳✶✺✮

♦❝♦rr❡✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✂

1 β

α

✡b

2

↕1

✂

β α

✡b

2

.

❱❛♠♦s ❝♦♥s✐❞❡r❛r ❛ ❢✉♥çã♦g♣tq ✏ 1 t2b ✁ ♣1 tq

b

2✳ ❈♦♠♦ g♣0q ✏ 0❡

g✶♣tq ✏ b

2

✁

tb2✁1✁ ♣1 tq

b

2✁1

✠ ➙0,

♣♦✐s 0 ↕ b

2 ➔ 1✱ t❡♠♦s g♣tq ➙ 0 ♣❛r❛ t♦❞♦ t ➙ 0✳ Pr♦✈❛♠♦s ❛ss✐♠ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

✭✷✳✶✺✮✱ ♦ q✉❡ ✜♥❛❧✐③❛ ❛ ♣r♦✈❛ ❞❡ ✭✷✳✶✹✮✳ ❯s❛♥❞♦ ✭✷✳✶✹✮✱ ♦❜t❡♠♦s

①τ②b①θT ✝ ♣J♣τq ✏

➺

①τ②bJ♣♣τ✁yq①θT♣yqdy

↕ C

➺

①τ ✁y②bJ♣♣τ ✁yq①θT♣yq J♣♣τ ✁yq⑤y⑤b①θT♣yqdy

✏ C

✁

♣①τ②bJ♣q ✝ ①θT J♣✝ ♣⑤τ⑤b①θTq

✠

.

❆ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❨♦✉♥❣✱ t❡♠♦s

⑥♣■■■q⑥Hb ↕ C✁✎✎✎♣①τ②

b♣

Jq ✝ ①θT✎✎✎ L2

✎✎

✎ ♣J✝ ♣⑤τ⑤b①θTq✎✎✎ L2

✠

↕ C

✁

⑥θT⑥L1✎✎✎①τ②

b♣

J✎✎✎

L2

✎✎

✎⑤τ⑤b♣θT✎✎✎ L1

✎✎ ✎ ♣J✎✎✎

L2

✠

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✶ ❙❡♥❞♦ χ ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ♥♦ ❝♦♥❥✉♥t♦ tτ PR_; ⑤_τ⑤_T →₁✉✱ t❡♠♦s ❡♥tã♦

J♣tq ✏

➺

eitτ

✁

χ♣τq♣iτq✁1f♣♣τq

✠

dτ ✏F✁1

τ

✁

χ♣τq♣iτq✁1f♣♣τq

✠ ♣tq,

♠♦str❛♥❞♦ q✉❡ ♣J♣τq ✏χ♣τq♣iτq✁1f♣♣τq✳ ❆ss✐♠✱

✎✎

✎①τ②bJ♣♣τq✎✎✎

L2 ✏

✎✎

✎①τ②bχ♣τq♣iτq✁1f♣♣τq✎✎✎

L2 ✏

✄ ➺

⑤τ_⑤T_→1

⑤τ⑤✁2①τ②2b⑤ ♣f♣τq⑤2dτ

☛1 2

✏

✄ ➺

⑤τ⑤T→1

⑤τ⑤✁2①τ②2♣b✁b✶q①τ②2b✶⑤ ♣f♣τq⑤2τ

☛1 2

↕ ✄

sup

⑤τ⑤T→1

⑤τ⑤✁2①τ②2♣b✁b✶q

☛1 2

⑥f⑥_Hb✶.

❙❡❣✉❡ ❞❡ 0↕b✁b✶ ↕1 ❡ T✁1 ➙1 q✉❡

⑤τ⑤✁2①τ②2♣b✁b✶q ✏ ⑤τ⑤✁2♣1 τ2qb✁b✶ ✏ ⑤τ⑤✁2♣b✁b✶q✁1 ⑤τ⑤2♣1✁♣b✁b✶q✁1q✟b✁b✶

↕ T2♣b✁b✶q✁1 T✁2♣1✁♣b✁b✶q✁1q✟b✁b✶

✏ T2♣1 T✁2qb✁b✶ ↕2b✁b✶T2♣1✁♣b✁b✶qq,

♣❛r❛ t♦❞♦ ⑤τ⑤ →T✁1✳ ❖❜t❡♠♦s ❡♥tã♦

✎✎

✎①τ②bJ♣♣τq✎✎✎

L2 ↕Cb,b

✶T1✁♣b✁b✶q⑥f⑥_Hb✶. ❆❧é♠ ❞✐ss♦✱ ❧❡♠❜r❡♠♦s q✉❡ ①θT♣tq ✏ Tθ♣♣T τq✳ ▲♦❣♦

✎✎ ✎①θT✎✎✎

L1 ✏

➺

⑤♣θ♣T τq⑤T dτ ✏

➺

⑤♣θ♣sq⑤ds ✏✎✎✎♣θ✎✎✎

L1.

❊♥tã♦✱

✎✎ ✎①θT✎✎✎

L1

✎✎

✎①τ②bJ♣♣τq✎✎✎

L2 ↕Cθ,b,b✶T

1✁♣b_✁b✶_q⑥

g⑥_Hb.

❘❡st❛ ❛❣♦r❛ ❡st✐♠❛r ❛♣❡♥❛s ✎✎✎⑤τ⑤b①θT✎✎✎ L1

✎✎ ✎ ♣J✎✎✎

L2✳ ❚❡♠♦s

✎✎

✎⑤τ⑤bθ①T✎✎✎ L1 ✏

➺

⑤τ⑤b⑤♣θ♣T τqT dτ ✏

➺

⑤T✁1s⑤b⑤♣θ♣sq⑤ds✏T✁b

➺

✸✷ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ ♣♦✐s ♣θP S✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❧❡♠❜r❡♠♦s q✉❡ ♣_J♣_τq ✏_χ♣_τq♣_iτq✁1♣_g♣_τq✳ ❙❡♥❞♦ ❛ss✐♠

✎✎ ✎ ♣J✎✎✎

L2 ✏

✄ ➺

⑤τ⑤T→1

⑤τ⑤✁2⑤ ♣f♣τq⑤2dτ

☛1 2

✏ ✄ ➺

⑤τ⑤T→1

⑤τ⑤✁2①τ②✁2b✶①τ②2b✶⑤ ♣f♣τq⑤2dτ

☛1 2

↕ ✄

sup

⑤τ⑤T→1

⑤τ⑤✁2①τ②✁2b✶

☛1 2

⑥f⑥_Hb✶.

❈♦♠♦ 0↕ ✁b✶ ➔1 ❡T✁1 ➙1✱ t❡♠♦s

⑤τ⑤✁2①τ②✁2b✶ ✏ ⑤τ⑤✁2♣1 τ2q✁b✶ ✏ τ2♣b✶q✁1 τ2♣1 ♣b✶q✁1✟✁b✶ ↕ T✁2♣b✶q✁1 T✁2♣1 ♣b✶q✁1q✟✁b✶

✏ T2♣1 T✁2qb✶ ↕2✁b✶T2♣1 b✶q,

♣❛r❛ t♦❞♦ ⑤τ⑤ →T✁1✳ ▲♦❣♦✱

✎✎

✎⑤τ⑤b①θT✎✎✎ L1

✎✎ ✎ ♣J✎✎✎

L2 ↕Cθ,bT

✁b_2✁b✶

2T1 b✶ ✏C

θ,b,b✶T

1_✁♣b_✁b✶_q,

♦ q✉❡ ✜♥❛❧✐③❛ ❛ ❡st✐♠❛t✐✈❛ ♣❛r❛ ♣■■■q ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ ♣r♦✈❛ ❞❡ ✭✷✳✸✮♣iq✳

✭✐✐✮ ❚❡♠♦s q✉❡ Vs♣tq é ♦ ♦♣❡r❛❞♦r ♣s❡✉❞♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❛❞♦ ♣♦r

F♣_Vs♣_tq_ϕ♣_ξq ✏ ④_Vs♣_tq_ϕ♣_ξq ✏

✂

eitγ♣ξq_✁e✁itγ♣ξq 2iγ♣ξq

✡

ˆ

ϕ♣ξq.

◆♦t❡♠♦s ❛❣♦r❛ q✉❡

F_ξ

✂

θT♣tq

➺t 0

Vs♣t✁t✶qf♣uq♣t✶qdt✶

✡

♣t, ξq ✏θT♣tq

➺t 0

F_ξ♣_Vs♣_t✁_t✶q_f♣_uq♣_t✶qq_dt✶

✏θT♣tq

➺t 0

✂

ei♣t✁t✶qγ♣ξq_✁e✁i♣t✁t✶qγ♣ξq 2iγ♣ξq

✡

F_ξ♣_f♣_uqq♣_t✶_{, n}q_dt✶

✏eitγ♣nq

✂

θ♣T♣tq

➺t 0

e✁it✶γ♣nq_F

ξ♣f♣uqq♣t✶, nq

2iγ♣nq dt

✶

✡

✁e✁itγ♣nq

✂

θ♣T♣tq

➺t 0

eit✶γ♣nq_F

ξ♣f♣uqq♣t✶, nq

2iγ♣nq dt

✶

✡

✏eitγ♣nq

✂

θT♣tq

➺t 0

h1♣t✶, nqdt✶

✡

✁e✁itγ♣nq

✂

θT♣tq

➺t 0

h2♣t✶, nqdt✶

✡

.

❉❡✜♥❛✿

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✸ ❚♦♠❛♥❞♦ ❛❣♦r❛✱ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ❋♦✉r✐❡r ❡♠ r❡❧❛çã♦ ❛t✱ ♥♦s ❞á

A♣τ, nq ✏

➺ R

e✁itτF_x♣_A♣_{t, n}qq_dt

✏ ➺

R

e✁it♣τ✁γ♣nqqF_x♣_w1♣_{t, n}qq_dt✁

➺ R

e✁it♣τ γ♣nqqF_x♣_w2♣_{t, n}qq_dt

✏w˜1♣τ✁γ♣nq, nq ✁w˜2♣τ γ♣nq, nq.

❆❣♦r❛✱ ✉s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡Xs,b

⑥A♣t, xq⑥Xs,b ✏ ⑥①⑤τ⑤ ✁γ♣nq②

b_①n②s_A˜⑥ l2

nL2τ

↕ ⑥①⑤τ⑤ ✁γ♣nq②b①n②sw˜1♣τ✁γ♣nq, nq⑥_l2

nL2τ ⑥①⑤τ⑤ ✁γ♣nq②

b_①n②s_w˜

2♣τ γ♣nq, nq⑥_l2

nL2τ.

❆ss✐♠✱

⑥A♣t, xq⑥Xs,b ↕ ✄

➳

n_PZ ①n②2s

➺ R

①⑤τ⑤ ✁γ♣nq②2b⑤w˜1♣τ ✁γ♣nq, nq⑤ 2

dτ

☛1 2

✄ ➳

nPZ ①n②2s

➺ R

①⑤τ⑤ ✁γ♣nq②2b⑤w˜2♣τ γ♣nq, nq⑤ 2

dτ

☛1 2

✏ ✄

➳

nPZ ①n②2s

➺ R

①⑤ρ γ♣nq⑤ ✁γ♣nq②2b⑤w˜1♣ρ, nq⑤ 2

dρ

☛1 2

✄ ➳

nPZ ①n②2s

➺ R

①⑤ρ✁γ♣nq⑤ ✁γ♣nq②2b⑤w˜2♣ρ, nq⑤ 2

dρ

☛1 2

.

❈♦♠♦γ♣nq ➙0, ❅n PR✱ t❡♠♦s

♠á① t⑤ρ✁γ♣nq⑤ ✁γ♣nq,⑤ρ γ♣nq⑤ ✁γ♣nq✉ ↕ ⑤ρ⑤,

♣♦✐s✱

⑤ρ⑤ ➙ ⑤ρ γ♣nq⑤ ✁γ♣nq ❡ ⑤ρ⑤ ➙γ♣nq ✁ ⑤ρ✁γ♣nq⑤,

❡♥tã♦

⑤ρ⑤ ➙ ⑤⑤ρ γ♣nq⑤ ✁γ♣nq⑤.

P♦r ♦✉tr♦ ❧❛❞♦✱

✸✹ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ ❡ ❞❛í✱

⑤ρ⑤ ➙ ⑤⑤ρ✁γ♣nq⑤ ✁γ♣nq⑤.

▲♦❣♦✱

⑥A♣t, xq⑥Xs,b ↕ ✄

➳

nPZ ①n②2s

➺ R

①ρ②2b⑤w˜1♣ρ, nq⑤ 2

dρ

☛1

2 ✄➳

nPZ ①n②2s

➺ R

①ρ②2b⑤w˜2♣ρ, nq⑤ 2 dρ ☛1 2 ✏ ✄ ➳

nPZ ①n②2s

➺ R

①ρ②2b⑤F_ρ♣F_x♣_w1♣_{ρ, n}qqq⑤2_dρ

☛1

2 ✄➳

nPZ ①n②2s

➺ R

①ρ②2b⑤F_ρ♣F_x♣_w2♣_{ρ, n}qqq⑤2_dρ

☛1 2

✏ ✄

➳

n_PZ

①n②2s⑥F_x♣_w1q⑥_Hb t

☛1

2 ✄➳

n_PZ

①n②2s⑥F_x♣_w2q⑥_Hb t

☛1 2

.

❆❣♦r❛✱ ❧❡♠❜r❡♠♦s q✉❡

F_x♣_w1♣_{t, n}qq ✏_θT♣_tq

➺t 0

h1♣t✶, nqdt✶ ❡ Fx♣w2♣t, nqq ✏θT♣tq

➺t 0

h2♣t✶, nqdt✶.

P❡❧❛ ♣❛rt❡ ♣iq t❡♠♦s

⑥F_x♣_w1♣_{t, n}qq⑥

Hb t ↕T

1✁♣b✁b✶q_⑥h

1♣t✶, nq⑥_Hb✶ t

❡

⑥F_x♣_w2♣_{t, n}qq⑥

Hb t ↕T

1✁♣b✁b✶q_⑥h

2♣t✶, nq⑥_Hb✶ t .

▲♦❣♦✱

⑥A♣t, xq⑥Xs,b ↕T

1✁♣b✁b✶q

✔ ✕

✄ ➳

nPZ

①n②2s⑥h1♣t✶, nq⑥ 2 Hb✶

t ☛1

2 ✄➳

nPZ

①n②2s⑥h2♣t✶, nq⑥ 2 Hb✶

t ☛1 2 ✜ ✢, s❡♥❞♦

h1♣t, nq ✏

e✁itγ♣nq_F

x♣f♣uq♣t, nqq

2iγ♣nq ❡ h2♣t, nq ✏

eitγ♣nq_F

x♣f♣uq♣t, nqq

2iγ♣nq .

❊♥tã♦

F_x♣_h1♣_{τ, n}qq ✏

➺ R

e✁itτh1♣t, nqdt✏ ˜

f♣uq♣τ γ♣nq, nq

2iγ♣nq

❡

F_x♣_h2♣_{τ, n}qq ✏

➺ R

e✁itτh2♣t, nqdt ✏ ˜

f♣uq♣τ ✁γ♣nq, nq

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✺ P♦rt❛♥t♦✱

⑥A♣t, xq⑥Xs,b ↕T

1✁♣b✁b✶q

☎ ✆➳

nPZ ①n②2s

➺ R

①τ②2b✶

✞✞ ✞✞ ✞

˜

f♣uq♣τ ✁γ♣nq, nq

2iγ♣nq

✞✞ ✞✞ ✞ 2 dτ ☞ ✌ 1 2

T1✁♣b✁b✶q

☎ ✆➳

nPZ ①n②2s

➺ R

①τ②2b✶

✞✞ ✞✞ ✞

˜

f♣uq♣τ γ♣nq, nq

2iγ♣nq

✞✞ ✞✞ ✞ 2 dτ ☞ ✌ 1 2 .

❋❛③❡♥❞♦ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s

τ✁γ♣nq ✏ ρ ❡ τ γ♣nq ✏ ρ,

t❡♠♦s

⑥A♣t, xq⑥Xs,b ↕T

1✁♣b✁b✶q

☎ ✆➳

nPZ ①n②2s

➺ R

①ρ γ♣nq②2b✶

✞✞ ✞✞ ✞

˜

f♣uq♣ρ, nq

2iγ♣nq

✞✞ ✞✞ ✞ 2 dρ ☞ ✌ 1 2

T1✁♣b✁b✶q

☎ ✆➳

nPZ ①n②2s

➺ R

①ρ✁γ♣nq②2b✶

✞✞ ✞✞ ✞

˜

f♣uq♣ρ, nq

2iγ♣nq

✞✞ ✞✞ ✞ 2 dρ ☞ ✌ 1 2 .

❖❜s❡r✈❡♠♦s ❛❣♦r❛ q✉❡

⑤⑤ρ⑤ ✁γ♣nq⑤ ↕ ♠í♥ t⑤ρ✁γ♣nq⑤,⑤ρ γ♣nq⑤✉,

❞♦♥❞❡ s❡q✉❡ q✉❡

⑤⑤ρ⑤ ✁γ♣nq⑤2b✶ ➙ ♠á① t⑤ρ✁γ♣nq⑤2b✶,⑤ρ γ♣nq⑤2b✶✉,

♣♦✐sb✶ ↕0✳

❉❛í✱

⑥A♣t, xq⑥Xs,b ↕T

1✁♣b✁b✶q✎✎✎✎

✎F✁

1

✄

˜

f♣uq♣ρ, nq

2iγ♣nq

☛✎✎ ✎✎ ✎ Xs,b , ♦✉ s❡❥❛✱ ✎✎ ✎✎θT♣tq

➺t 0

Vs♣t✁t✶qf♣uq♣t✶qdt✶

✎✎ ✎✎

Xs,b

↕T1✁♣b✁b✶q

✎✎ ✎✎ ✎F✁

1

✄

˜

f♣uq♣τ, nq

2iγ♣nq

☛✎✎ ✎✎ ✎

Xs,b

✸✻ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖

❆ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ❡stá ❝♦♠♣❧❡t❛✳ ✆

▲❡♠❛ ✷✳✹✳ ❊①✐st❡ c→0 t❛❧ q✉❡

1

c ↕x,ysup➙0

1 ⑤x y⑤

1 ⑤x✁❛y✷ y⑤ ↕c.

❉❡♠♦♥str❛çã♦✿ ❆✜r♠❛♠♦s q✉❡

y↕❛y2 _y↕_y 1

2, ❅y➙0.

❉❡ ❢❛t♦✱

y✏ ⑤y⑤ ✏❛y2 ↕❛_y2 _y, ❅_y➙_0,

❛❧é♠ ❞✐ss♦✱

♣y 1

2q 2 _✏

y2 y 1

4 ➙y

2

y.

❈♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡

❛

y2 _y↕_y 1 2,

❞❡♠♦♥str❛♥❞♦ ❛ ❛✜r♠❛çã♦✳ ❉❛í✱

1 ⑤x✁❛y2 _y⑤ ➙₁ ⑤_x⑤ ✁ ⑤❛_y2 _y⑤ ✏_{1 x}✁❛_y2 _y➙_{1 x}✁_y✁1

2 ✏

1

2 ♣x✁yq,

t❡♠♦s t❛♠❜é♠✱

1 ⑤x✁❛y2 _y⑤ ➙₁ ❛_y2 _y✁_x➙₁ ♣_y✁_xq ➙ 1

2 ♣y✁xq,

❧♦❣♦✱

1 ⑤x✁❛y2 _y⑤ ➙ 1

2 ⑤x✁y⑤ ➙ 1 2

❡♥tã♦

1 ⑤x✁y⑤

1 ⑤x✁❛y2 _y⑤ ↕

1 2

1 ⑤x✁❛y2 _y⑤

1 2

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✼ P♦r ♦✉tr♦ ❧❛❞♦✱

1 ⑤x✁y⑤ ➙1 ♣x✁yq ➙ 1 x✁❛y2 _y ➙ 1

2 x✁

❛

y2 _y

❡

1 ⑤x✁y⑤ ➙1 ♣y✁xq ➙1 ❛y2 _y✁1

2 ✁x✏

1 2

❛

y2 _y✁_x,

♦ q✉❡ ✐♠♣❧✐❝❛

1 ⑤x✁y⑤ ➙ 1

2 ⑤x✁

❛

y2 _y⑤ ➙ 1

2♣1 ⑤x✁

❛

y2 _y⑤q

❡

1 ⑤x✁y⑤

1 ⑤x✁❛y2 _y⑤ ➙

1 2.

P♦rt❛♥t♦✱ ♣❛r❛ c✏2 ❝♦♥❝❧✉í♠♦s ♦ r❡s✉❧t❛❞♦ ❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❧❡♠❛ ❡stá ❝♦♠♣❧❡t❛✳ ✆

▲❡♠❛ ✷✳✺✳ ❙❡❥❛♠ 0↕a_✁↕a ✱ a a_✁ → 1

2, ❡ a ✘

1 2 ❡♥tã♦

J♣sq ✏

➺ R

1

①y✁s②2a ①_{y s}②2a✁dy↕

c

①s②α, ❅sPR,

s❡♥❞♦ α✏2a_✁✁ r1 2a s ❡

rλs ✏

✩ ✬ ✬ ✬ ✬ ✬ ✫ ✬ ✬ ✬ ✬ ✬ ✪

λ s❡ λ→0

ǫ, λ✏0

0 s❡ λ➔0.

✭✷✳✶✻✮

❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s q✉❡

①y②2 ✏ ♣1 ⑤y⑤2q.

❉❛í

①y②2 ↕1 ♣⑤y✁s⑤ ⑤s⑤q2 ✏1 ⑤y✁s⑤2 ⑤s⑤2 2⑤y✁s⑤⑤s⑤,

❡ ❝♦♠♦

✸✽ ❈❆P❮❚❯▲❖ ✷✳ ❖ ❈❆❙❖ P❊❘■Ó❉■❈❖ t❡♠♦s

①y②2 ↕2♣1 ⑤y✁s⑤2 ⑤s⑤2q ↕2♣1 ⑤s⑤2q♣1 ⑤y✁s⑤2q ✏2①s②2①y✁s②2

❡

①y② ↕❄2①y✁s②①s②,

✐st♦ é✱

1

①y✁s② ↕

❄

2①s②

①y② .

❆♥❛❧♦❣❛♠❡♥t❡✱

1

①y s② ↕

❄

2①s②

①y② ,

❡ ❡♥tã♦

1

①y✁s②2a ①_{y s}②2a_✁ ↕

❄

22♣a a✁q①s②2♣a a✁q

①y②2♣a a_✁q ↕

2①s②

①y②2♣a a_✁q,

♣♦✐s 2♣a a_✁q →1✳ ▲♦❣♦✱

J♣sq ↕

➺ R

2①s②

①y②2♣a a✁qdy➔ ✽, ❅sP R.

❱❛♠♦s ❛❣♦r❛ ❞✐✈✐❞✐r J♣sq ❡♠ 3 r❡❣✐õ❡s✱ s❡♥❞♦ ❡❧❛s

A1 ✏ tyPR ; 0↕y↕2s✉

A2 ✏ tyPR ; ✁2s↕y↕0✉

A3 ✏ tyPR ; ⑤y⑤ ➙2s✉.

❉❡ss❛ ❢♦r♠❛✱ J♣sq ✏ JA1♣sq JA2♣sq JA3♣sq✳ ◆♦ss❛ ♠❡t❛ ❛❣♦r❛ é ❡st✐♠❛r J♣sq ❛tr❛✈és

❞❡st❛s ✐♥t❡❣r❛✐s✳ ❱❛♠♦s tr❛❜❛❧❤❛r ♣r✐♠❡✐r♦ ❝♦♠ ❛ ✐♥t❡❣r❛❧ JA1♣sq✱ t❡♠♦s

JA1♣sq ✏

➺2s 0

1

①y✁s②2a ①_{y s}②2a✁dy.

◆❡st❡ ❝❛s♦✱0↕y↕s✱ ❞♦♥❞❡ s❡❣✉❡ q✉❡ s↕y s ❡ ①s② ↕ ①y s②✱ ❡ ❛ss✐♠✱

1

①y s②2 ↕ 1

✷✳✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙ ✸✾ ▲♦❣♦

JA1♣sq ↕

1

①s②2a_✁

➺2s 0

1

①y✁s②2a dy ✏ 1

①s②2a_✁

➺2s 0

1

♣1 ⑤y✁s⑤2qa dy

✏ 1

①s②2a_✁

➺s

✁s 1

♣1 u2qa du✏ 2

①s②2a_✁

➺s 0

1

♣1 u2qa du.

❖❜s❡r✈❡ ❛❣♦r❛ q✉❡

1 u2 ↕1 2u u2 ✏ ♣1 uq2 ↕2♣1 uq2,

♣♦✐s0↕ ♣1✁uq2 ❡2u↕1 u2✳ ❉❛í✱

1

1 u2 ➙

1

♣1 uq2

❡

2

①s②2a_✁

➺s 0

1

♣1 u2qa du↕ 2c

①s②2a_✁

➺s 0

1

♣1 uq2a du

✏ 2c ①s②2a_✁

➺s 0

1

♣1 uq2a du

✏ ✩ ✬ ✬ ✫ ✬ ✬ ✪ 2c

①s②2a_✁

✂

♣1 uq✁2a 1

✁2a 1

✡✞✞ ✞✞s

0

, s❡ a ✘ 1

2 2c

①s②2a_✁ln⑤1 u⑤ ⑤ s

0, s❡ a ✏

1 2.

▲♦❣♦✱ ❝♦♠♦a ✘ 1

2 ❡♥tã♦ 1✁2a →0 ❡

JA1♣sq ↕

2c

①s②2a✁

✂

1

♣1✁2a q♣1 sq1✁2a ✁ 1 1✁2a

✡

↕ _① 2c

s②2a_✁

1

♣1 sq2♣1 2✁a✁q

↕ _① c

s②2a_✁ 1

①s②1✁2a

✏c①s②✁2a✁ ♣1✁2a _{q ↕}

c①s②✁2a✁ r1✁2a _s

,

♣♦✐s ①s② ➙1❡ 1✁2a ↕ r1✁2a s ✳

❖❜s❡r✈❡ t❛♠❜é♠ q✉❡

1

♣1 sq2♣1

2✁a✁q

↕ 1

♣1 s2q1₂✁a_✁ ✏ 1

①s②12✁a✁