✣❸■ ❍➴❈ ❍❯➌
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

◆●❯❨➍◆ ◆❍❾❚ ▼■◆❍

❑❍➷◆● ●■❆◆ ❖❘▲■❈❩✲❙❖❇❖▲❊❱
❱⑨ ❳❻P ❳➓ ❇Ð■ ❈⑩❈ ❍⑨▼ ❚❘❒◆

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❚❍❊❖ ✣➚◆❍ ❍×❰◆● ◆●❍■➊◆ ❈Ù❯

❚❤ø❛ ❚❤✐➯♥ ❍✉➳✱ ♥➠♠ ✷✵✶✼


✣❸■ ❍➴❈ ❍❯➌
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
◆●❯❨➍◆ ◆❍❾❚ ▼■◆❍

❑❍➷◆● ●■❆◆ ❖❘▲■❈❩✲❙❖❇❖▲❊❱
❱⑨ ❳❻P ❳➓ ❇Ð■ ❈⑩❈ ❍⑨▼ ❚❘❒◆
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿ ✻✵✳ ✹✻✳ ✵✶✳ ✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
❚❍❊❖ ✣➚◆❍ ❍×❰◆● ◆●❍■➊◆ ❈Ù❯

❈⑩◆ ❇❐ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿

❚❙✳ ❚❘×❒◆● ❱❿◆ ❚❍×❒◆●

❚❤ø❛ ❚❤✐➯♥ ❍✉➳✱ ♥➠♠ ✷✵✶✼

✐


▲❮■ ❈❆▼ ✣❖❆◆
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥✱ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥
❝ù✉ ❝õ❛ tæ✐ ❞÷î✐ sü ❤÷î♥❣ ❞➝♥ trü❝ t✐➳♣ ❝õ❛ t❤➛② ❣✐→♦
❚❙✳ ❚r÷ì♥❣ ❱➠♥ ❚❤÷ì♥❣✳
❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥✱ tæ✐ ✤➣ ❦➳
t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✈➔ ❝→❝
♥❤➔ ❑❤♦❛ ❤å❝ ✈î✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳

◆❣✉②➵♥ ◆❤➟t ▼✐♥❤

✐✐



rữợ t tổ ỷ ớ ỡ rữỡ
ữỡ ỡ ỳ ớ ở ừ
tr sốt q tr ữợ ồ tổ ú
tổ ữủt q ỳ õ t ử ồ t
ự ừ
ổ ữủ tọ ỏ t ỡ qỵ ổ
ợ ồ õ ừ trữớ P ụ
ữ t t t ổ tr trữớ P
sỹ t t sỹ q t ở tổ
tr sốt q tr ồ t tỹ
ổ ụ ỷ ớ ỡ trữớ P Pỏ
ồ trữớ P t tổ t
ổ ồ t ự ừ

ũ õ ố ữ õ tr ọ
ỳ t sõt ữủ ỳ ỵ õ õ
ừ qỵ t ổ ồ ữủ t ỡ
ố ũ tổ s ợ ợ ữớ
t tổ ỳ ữớ ổ st ở ú
ù tổ t
tr trồ ỡ




▼Ö❈ ▲Ö❈
❚r❛♥❣ ♣❤ö ❜➻❛

✐

▲í✐ ❝❛♠ ✤♦❛♥

✐✐

▲í✐ ❝↔♠ ì♥

✐✐✐

▼ö❝ ❧ö❝

✶

▼ð ✤➛✉


✷

✶ ❑❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈

✹

✶✳✶

❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

✣↕♦ ❤➔♠ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✸

❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✵

✷ ❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈

✶✻

✷✳✶


❍➔♠ ❧ç✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✻

✷✳✷

❍➔♠ ❨♦✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✵

✷✳✸

❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹

✷✳✹

❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✸

✷✳✺

▼ët sè t➼♥❤ ❝❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✸ ❳➜♣ ①➾ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈ ❜ð✐ ❝→❝ ❤➔♠ trì♥


✸✽

✸✳✶

✣à♥❤ ♥❣❤➽❛ ♣❤➙♥ ❤♦↕❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✽

✸✳✷

❳➜♣ ①➾ ❜ð✐ ❧î♣ ❤➔♠ trì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

✸✳✸

❳➜♣ ①➾ ❜ð✐ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✸✳✹

❳➜♣ ①➾ ❜ð✐ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❣✐→ ❝♦♠♣❛❝t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

❑➳t ❧✉➟♥

✺✻


❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✺✽
✶


▼Ð ✣❺❯
❍➔♠ sè ❦❤↔ ✈✐ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝✱ ✤➦❝
❜✐➺t ❧➔ tr♦♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❱➔♦ t❤➳ ❦➾ ✷✵✱ ♥❣÷í✐ t❛ t❤➜② r➡♥❣
❦❤æ♥❣ ❣✐❛♥ C 1 ✭❤❛② C 2 ✱✳✳✳✮ ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤ó♥❣ ✤➸ ♥❣❤✐➯♥ ❝ù✉
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❧➔ sü t❤❛② t❤➳ ❝❤♦ ❝→❝
❦❤æ♥❣ ❣✐❛♥ ❝ê ✤✐➸♥ C 1 ✭❤❛② C 2 ✱✳✳✳✮ ❦❤✐ ✤✐ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ ♥➔②✳
❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❝â ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➭♣ ✤➩ ✈➔ ✤÷ñ❝ ù♥❣ ❞ö♥❣
tr♦♥❣ r➜t ♥❤✐➲✉ ❧➽♥❤ ✈ü❝✳ ❚✉② ♥❤✐➯♥✱ ✈î✐ sü ♣❤→t tr✐➸♥ ❦❤æ♥❣ ♥❣ø♥❣ ❝õ❛
t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐✱ ♥❣÷í✐ t❛ ✤➣ ✤÷❛ r❛ ❝→❝ tr÷í♥❣ ❤ñ♣ ✤➦❝ ❜✐➺t tr♦♥❣
♥❤✐➲✉ ❜➔✐ t♦→♥ ❣✐↔✐ t➼❝❤ ❜✐➳♥ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ❚r♦♥❣ q✉→
tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ♥➔② ✤➣ t❤ó❝ ✤➞② ❝→❝ ♥❤➔ t♦→♥ ❤å❝
♥❣❤✐➯♥ ❝ù✉ ❝â ❤➺ t❤è♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈ ♠ð rë♥❣ ❝❤♦ ❝→❝
❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳
◆➳✉ ♥❤÷ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❦❤æ♥❣ ❣✐❛♥ Lp ✤â♥❣ ✈❛✐ trá
q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ✤à♥❤ ♥❣❤➽❛ t❤➻ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈
❧↕✐ ✤÷ñ❝ t❤❛② t❤➳ ❜ð✐ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ LΦ ✳ ❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ✤÷ñ❝
♣❤→t tr✐➸♥ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ ❑r❛s♥♦s❡❧✬s❦✐✐ ✈➔ ❘✉t✐❝❦✐✐ tr♦♥❣ ✤â Φ ❧➔
◆✲❤➔♠✳ ❑❤æ♥❣ ❣✐❛♥ ♥➔② ❝✉♥❣ ❝➜♣ ♠ët t❤✐➳t ❧➟♣ t❤➼❝❤ ❤ñ♣ ❤ì♥ ❦❤æ♥❣
❣✐❛♥ ▲❡❜❡s❣✉❡s Lp tr♦♥❣ ✈✐➺❝ ❣✐↔✐ q✉②➳t ✈î✐ ❝→❝ ❤➔♠ ❙♦❜♦❧❡✈✳ ▲þ t❤✉②➳t
✈➲ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ❝ô♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
✤↕♦ ❤➔♠ r✐➯♥❣✱ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ❧þ t❤✉②➳t ♥❤ó♥❣✱✳✳✳
❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✤➣ ❝❤➾ r❛ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈
✤÷ñ❝ ♠ð rë♥❣ ✤➳♥ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈✳ ❑❤ð✐ ✤➛✉ ✈➔♦ ♥➠♠ ✶✾✼✶ ❜ð✐

❉♦♥❛❧❞s♦♥✲❚r✉❞✐♥❣❡r✱ t✐➳♣ ✤➳♥ ❧➔ ▼❛③✬②❛ ✭✶✾✼✷✴✶✾✼✸✮✱ ●♦ss❡③ ✭✶✾✼✹✮✱
❆❞❛♠s ✭✶✾✼✼✮✳
❍ì♥ ♥ú❛✱ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥
❣✐→ trà ❜✐➯♥ ♣❤✐ t✉②➳♥ ♠↕♥❤✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ♥❤÷ ❇❡♥❦✐r❛♥❡ ✈➔ ●♦ss❡③
❬✹❪✱ ❉♦♥❛❧❞s♦♥ ✈➔ ❚r✉❞✐♥❣❡r❬✽❪✱ ❏❡❛♥✲P✐❡rr❡ ●♦ss❡③ ❬✶✵❪✱✳✳✳ ✤➣ ♥❣❤✐➯♥ ❝ù✉
✷


t➼♥❤ ❝❤➜t ①➜♣ ①➾ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈ t❤❛② ❝❤♦ ❦❤æ♥❣ ❣✐❛♥
❙♦❜♦❧❡✈ ❦❤✐ ❝è ❣➢♥❣ ❧➔♠ ❣✐↔♠ ❜ît ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛♦ ❝❤♦ ✤↔♠ ❜↔♦ t➼♥❤
❣✐↔✐ ✤÷ñ❝ ❝õ❛ ❜➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ❡❧❧✐♣t✐❝ ♣❤✐ t✉②➳♥ ♠↕♥❤ ✤÷ñ❝ ❝❤♦ ❜ð✐✿
A(u) + g(x, u) = f

✈î✐ A ❧➔ t♦→♥ tû tü❛ t✉②➳♥ t➼♥❤ ❡❧❧✐♣t✐❝
|α|

(−1) Dα Aα (x, u, Du, ..., Dm u)

A(u) ≡
|α| m

✈➔ ❤➔♠ g t❤ä❛ ✤✐➲✉ ❦✐➺♥ ❞➜✉ ♥❤÷♥❣ ❤♦➔♥ t♦➔♥ ❦❤æ♥❣ ❤↕♥ ❝❤➳ ❝➜♣ t➠♥❣
t❤❡♦ ❜✐➳♥ u✳
❱î✐ ♠♦♥❣ ♠✉è♥ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ ✈➲ ❝→❝ t➼♥❤ ❝❤➜t✱ ✤➦❝
❜✐➺t ❧➔ ①➜♣ ①➾ ❜ð✐ ❝→❝ ❧î♣ ❤➔♠ trì♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈✱ ❞÷î✐
sü ✤à♥❤ ❤÷î♥❣ ❝õ❛ t❤➛② ❤÷î♥❣ ❞➝♥ ❚❙✳ ❚r÷ì♥❣ ❱➠♥ ❚❤÷ì♥❣✱ tæ✐ ✤➣ ❝❤å♥
✤➲ t➔✐ ✏❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈ ✈➔ ①➜♣ ①➾ ❜ð✐ ❝→❝ ❤➔♠ trì♥✑ ❝❤♦ ❧✉➟♥
✈➠♥ t❤↕❝ s➽ ❝❤✉②➯♥ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤✳
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❞ü❛ tr➯♥ ✈✐➺❝ t❤❛♠ ❦❤↔♦ ❬✸❪✳ ◆❣♦➔✐
❧í✐ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷ñ❝ ❝❤✐❛ ❧➔♠ ❜❛

❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✳ ❑❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳ ❈❤÷ì♥❣ ♥➔② ❞➔♥❤
✤➸ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❧✐➯♥ q✉❛♥ ✤÷ñ❝ ❞ò♥❣ tr♦♥❣ ❧✉➟♥
✈➠♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳

❈❤÷ì♥❣ ✷✳ ❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈✳ ❚r♦♥❣
❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ ❣✐î✐ t❤✐➺✉ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ✈➔ ❦❤æ♥❣ ❣✐❛♥
❖r❧✐❝③✲❙♦❜♦❧❡✈ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥✳

❈❤÷ì♥❣ ✸✳ ❳➜♣ ①➾ ❦❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③✲❙♦❜♦❧❡✈ ❜ð✐ ❝→❝ ❤➔♠ trì♥✳ ❈❤÷ì♥❣
♥➔② ❧➔ ♥ë✐ ❞✉♥❣ ❝èt ❧ã✐ ❝õ❛ ❧✉➟♥ ✈➠♥ ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ✈➲ ♣❤➙♥ ❤♦↕❝❤✱
①➜♣ ①➾ ❜ð✐ ❧î♣ ❤➔♠ trì♥✱ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ✈➔ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❣✐→
❝♦♠♣❛❝t✳

✸


❈❤÷ì♥❣ ✶
❑❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥
❙♦❜♦❧❡✈
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ✈➲ ❝→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ ❦➳t
q✉↔ ❝ì ❜↔♥ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳ ❈→❝ ❦➳t q✉↔ ♥➔② ✤÷ñ❝ sû ❞ö♥❣ ✤➸
①➙② ❞ü♥❣ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦➳t q✉↔ ð ❝❤÷ì♥❣ s❛✉✳

✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ G ⊂ R

n


✈➔ m ∈ N∗ ✳ ❑þ ❤✐➺✉

C m (G) = f : G → R Dα f ❧✐➯♥ tö❝ tr➯♥ G, |α|

m

✭✶✳✶✮

tr♦♥❣ ✤â α = (α1 , α2 , ..., αn ), αi ∈ N∗ ❧➔ ❜ë ✤❛ ❝❤➾ sè ❧➔ ✈❡❝tì ✈î✐ ❝→❝ tå❛
✤ë ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✱
∂ α1 ...∂ αn
f.
|α| = α1 + α2 + ... + αn ✈➔ D f =
∂x1 α1 ...∂xn αn
α

∞

❑þ ❤✐➺✉ C ∞ (G) = ∩ C m (G) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ tr➯♥
m=1

G✳

❱î✐ ♠é✐ ❤➔♠ f ①→❝ ✤à♥❤ tr➯♥ ♠✐➲♥ G ⊂ Rn ✱ ❦þ ❤✐➺✉
supp (f ) := {x ∈ G : f (x) = 0}

✈➔ ❣å✐ ❧➔ ❣✐→ ❝õ❛ ❤➔♠ f tr➯♥ G✳
✣➦t
C0m (G) = {f ∈ C m (G) | supp (f ) ❝♦♠♣❛❝t } .
✹


✭✶✳✷✮


❑❤✐ ✤â
∞

✭✶✳✸✮

C0∞ (G) = ∩ C0m (G) = D(G)
m=1

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ✈➔ ❝â ❣✐→ ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t ❝❤ù❛
tr♦♥❣ G✳ ❑❤æ♥❣ ❣✐❛♥ ♥➔② t❤÷í♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ t❤û✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈❤♦ G ⊂ R ✳ ❑þ ❤✐➺✉
n

CBm (G) = f : G → R Dα f ❜à ❝❤➦♥ tr➯♥ G, 0

|α|

✭✶✳✹✮

m

❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥
f

m (G)

CB

= m❛① sup |Dα f (x)| .
0 |α| m x∈G

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❈❤♦ G ❧➔ ♠ët ♠✐➲♥ ❜à ❝❤➦♥ tr♦♥❣ R

n

✈➔ G ❧➔ ❜❛♦

✤â♥❣ ❝õ❛ G✳ ❑þ ❤✐➺✉
C m (G) = f : G → R Dα f ❧✐➯♥ tö❝ tr➯♥ G, 0

|α|

m .

✭✶✳✺✮

❱➻ C m (G) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ✤â♥❣ ❝õ❛ CBm (G)✳ ❉♦ ✤â C m (G) ❝ô♥❣ ❧➔
❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥ t÷ì♥❣ tü
f

C m (G)

= m❛① sup |Dα f (x)| .
0 |α| m x∈G

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❍➔♠ tr✉♥❣ ❜➻♥❤ ❤â❛

❜ð✐
ω(x) =


1
 Ce− 1− x 2 ; ♥➳✉
0; ♥➳✉



❈❤♦ ω : Rn → R ①→❝ ✤à♥❤
x <1
x

1

s❛♦ ❝❤♦
ω(x)dx =
Rn

ω(x)dx = 1
x <1

1

tr♦♥❣ ✤â x = (x21 + ... + x2n ) /2 ✳
❑❤✐ ✤â✱ ω(x) ❧➔ ♠ët ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❝â ❣✐→ ❝♦♠♣❛❝t tr♦♥❣ Rn ✳
❱î✐ h > 0 tò② þ✳ ❍➔♠
ωh (x) =


x
1
ω
n
h
h
✺


✤÷ñ❝ ❣å✐ ❧➔ ♥❤➙♥ tr✉♥❣ ❜➻♥❤ ❤â❛✳ ❍➔♠ ♥➔② ❝â t➼♥❤ ❝❤➜t s❛✉✿
✶✳ ωh (x) ∈ C ∞ (Rn )❀
✷✳ ωh (x) = 0 ✈î✐ x
✸✳

h❀

ωh (x)dx = 1;
Rn

✹✳ ❱î✐ α tò② þ✱ |α|

0 ✈➔ ∀x ∈ Rn t❛ ❝â |Dα ωh (x)|

❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ❤✳

Cα

tr♦♥❣ ✤â Cα

hn+|α|


▼ët t➟♣ E ⊂ G ✤÷ñ❝ ❣å✐ ❧➔ ❝❤ù❛ ♠ët ❝→❝❤ ❝♦♠♣❛❝t ✤è✐ ✈î✐ ● ♥➳✉
E ⊂ G✱ ❦➼ ❤✐➺✉ E

G✳

❚❛ ❦þ ❤✐➺✉
✭✶✳✻✮

f (y)ωh (x − y)dy, x ∈ Rn , h > 0.

fh (x) =
G

✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ tr✉♥❣ ❜➻♥❤ ❤â❛ ✤è✐ ✈î✐ f (x)✳ ❚ø t➼♥❤ ❝❤➜t ✶ ❝õ❛ ♥❤➙♥
tr✉♥❣ ❜➻♥❤ ❤â❛ ✈➔ f ∈ C(G)✱ t❛ ❝â fh ∈ C ∞ (Rn )✳

✣à♥❤ ❧➼ ✶✳✶✳✺✳ ●✐↔ sû f ∈ C

m

(G) ✈➔ G

G✳ ❑❤✐ ✤â fh − f

C m (G )

→0

❦❤✐ h → 0.

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ h > 0 ✤õ ❜➨✱ h < d(∂G , ∂G)✳ ❑❤✐ ✤â✱ t❤❡♦ t➼♥❤ ❝❤➜t ✶✱
✷✱ ✸ ❝õ❛ ♥❤➙♥ tr✉♥❣ ❜➻♥❤ ❤â❛ ✈î✐ x ∈ G t❛ ✤÷ñ❝
|fh (x) − f (x)| = |

f (y)ωh (x − y)dy − f (x)

ωh (x − y)dy|

Rn

G

=|

f (y)ωh (x − y)dy − f (x)

{ x−y
sup

ωh (x − y)dy|

{ x−y
|f (y) − f (x)|

ωh (x − y)dy

{ x−y { x−y

=

sup

|f (y) − f (x)| .

{ x−y
❱➻ f ❧✐➯♥ tö❝ tr➯♥ G ♥➯♥
fh − f

C(G)

→ 0 ❦❤✐ h → 0.

✻

✭✶✳✼✮


❱î✐ x ∈ G ✱ h ✤õ ❜➨ ✈➔ 0

m t❛ ❝â

|α|

Dxα fh (x) =

f (y)Dxα ωh (x − y)dy

G

= (−1)|α|

f (y)Dyα ωh (x − y)dy

G
α
Dy f (y)ωh (x

=

− y)dy.

G

❚÷ì♥❣ tü✱ t❛ s✉② r❛
|Dxα fh (x) − Dxα f (x)|

sup
{ x−y
|Dα f (y) − Dα f (x)| → 0 ❦❤✐ h → 0.

❉♦ ✤â
lim m❛① sup |Dα fh (y) − Dα f (x)| = 0.

h→0 0 |α| m x∈G

❱➟②

lim fh − f

h→0

C m (G )

= 0.

✶✳✷ ✣↕♦ ❤➔♠ s✉② rë♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ❈❤♦ G ⊂ R

n

✈➔ ❞➣② {ϕk } ⊂ C0∞ (G)✳ ❑❤✐ ✤â

ϕk → ϕ ∈ C0∞ (G) ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö t❤❡♦ ♥❣❤➽❛ D(G) ♥➳✉✿

✭✐✮ tç♥ t↕✐ K

G s❛♦ ❝❤♦ supp(ϕk − ϕ) ⊂ K ✈î✐ ♠é✐ k ✳

✭✐✐✮ lim Dα ϕk (x) = Dα ϕ(x), ∀x ∈ K ✳
k→∞

❍ë✐ tö ♥➔② ✤➲✉ tr➯♥ K ✈î✐ ♠é✐ ✤❛ ❝❤➾ sè α✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳ ❈❤♦ T : D(G) → R t✉②➳♥ t➼♥❤✳ T ✤÷ñ❝ ❣å✐ ❧➔ ❧✐➯♥
tö❝ ♥➳✉ ∀ {ϕk } ⊂ D(G) s❛♦ ❝❤♦ ϕk → ϕ t❤➻ T (ϕk ) → T (ϕ)✳
❑❤✐ ✤â✱ t❛ ❦þ ❤✐➺✉
D (G) = T : D(G) → R t✉②➳♥ t➼♥❤ ✈➔ ❧✐➯♥ tö❝

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣ tr➯♥ G✳

✼


✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ❈❤♦ f : G → R ✈➔ f ∈ L (G)✳ ❑❤✐ ✤â f ✤÷ñ❝ ❣å✐ ❧➔
1

❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ ∀U

G, f ∈ L1 (U )✳ ❑þ ❤✐➺✉ f ∈ L1loc (G)✳

❱î✐ ♠é✐ f ∈ L1loc (G)✱ t❛ ❝â ❤➔♠ s✉② rë♥❣ Tf ∈ D (G) s❛♦ ❝❤♦
f (x)ϕ(x)dx, ∀ϕ ∈ D(G).

Tf (ϕ) =
G

◆❤➟♥ ①➨t ✶✳✷✳✹✳ ●✐↔ sû G ⊂ R ✳ ❚❛ ✤➣ ❜✐➳t ♥➳✉ ❤➔♠ f ∈ C (G) t❤➻ ✈î✐
n

1

❤➔♠ ϕ ∈ C01 (G) t❛ ❝â ✤➥♥❣ t❤ù❝
f (x)ϕ(x)dx = −
G

f (x)ϕ (x)dx.
G

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ❤➔♠ f ❧✐➯♥ tö❝ tr➯♥ G✱ ♥➳✉ tç♥ t↕✐ ❤➔♠ g ∈ L1loc (G)
s❛♦ ❝❤♦ ✈î✐ ♠å✐ ϕ ∈ C01 (G) t❛ ❝â
g(x)ϕ(x)dx = −
G

✭✶✳✽✮

f (x)ϕ (x)dx.
G

❑❤✐ ✤â ❤➔♠ g s➩ trò♥❣ ✈î✐ Df tr➯♥ G ❦❤✐ tç♥ t↕✐ Df ✳ ❱➻ ✈➟②✱ ❤➔♠ g tr♦♥❣
✭✶✳✽✮ ✤÷ñ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝õ❛ ❤➔♠ f ✳ ❚ê♥❣ q✉→t ❤ì♥ ♥➳✉ ❝→❝
❤➔♠ f, g tr♦♥❣ ✭✶✳✽✮ ❦❤æ♥❣ ❧✐➯♥ tö❝ ♠➔ ❝❤➾ ❦❤↔ t➼❝❤ ❤♦➦❝ ❜➻♥❤ ♣❤÷ì♥❣
❦❤↔ t➼❝❤ ✈➔ t➼❝❤ ♣❤➙♥ tr♦♥❣ ✭✶✳✽✮ ❧➔ t➼❝❤ ♣❤➙♥ t❤❡♦ ♥❣❤➽❛ ▲❡❜❡s❣✉❡ t❤➻
t❛ ❝ô♥❣ ♥â✐ ❤➔♠ g ❧➔ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝õ❛ ❤➔♠ f ✳ ❚❛ ❝â ✤à♥❤ ♥❣❤➽❛
tê♥❣ q✉→t s❛✉✿

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✺✳ ●✐↔ sû α = (α , ..., α ) ❧➔ ♠ët ❜ë ❝→❝ sè ♥❣✉②➯♥
1

n

❦❤æ♥❣ ➙♠✳ ❍➔♠ f α ∈ L2loc (G) ✤÷ñ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ α ❝õ❛
❤➔♠ f ∈ L2loc (G) tr♦♥❣ G ♥➳✉ ✈î✐ ♠å✐ ❤➔♠ ϕ ∈ C0|α| (G), |α|

m t❛ ❝â

✤➥♥❣ t❤ù❝
|α|

f α (x)ϕ(x)dx = (−1)
G

✭✶✳✾✮

f (x)Dα ϕ(x)dx.
G

❱➼ ❞ö ✶✳ ❈❤♦ ❤➔♠ f (x) = |x | tr➯♥ G = {x ∈ R

n

1

❑❤✐ ✤â ❝→❝ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ ✶ ❧➔
∂f
= 0; i = 2, ..., n.
∂xi
✽

| x

1}✳

∂f
= Dfx1 (x) = sign(x1 ) ✈➔
∂x1

❚❤➟t ✈➟②
✣➦t G+ = {x ∈ G : x1 > 0} , G− = {x ∈ G : x1 < 0} , G ∩ {x1 = 0} .
❱î✐ ❤➔♠ ϕ ∈ C01 (G)✱ t❛ ❝â
f (x)ϕ x1 (x)dx =

f (x)ϕ x1 (x)dx +
G−

G+

G

f (x)ϕ x1 (x)dx +

x1 ϕ x1 (x)dx −

=

f (x)ϕ x1 (x)dx

G∩{x1 =0}

x1 ϕ x1 (x)dx.

G−

G+

❚❤❡♦ ❝æ♥❣ t❤ù❝ ❖str♦❣r❛❞s❦✐ ✭x1 ϕ = 0 tr➯♥ ❜✐➯♥ ∂G ✈➔ ✈î✐ x1 = 0✮✱ t❛ ❝â
|x1 | ϕ x1 (x)dx =

x1 ϕ(x)ds −
∂G+

G

=−

ϕ(x)dx−

ϕ(x)dx

G−

ϕ(x)dx

G−

G+

=−

∂G−

G+

ϕ(x)dx+

x1 ϕ(x)ds +

sign(x1 )ϕ(x)dx.
G

∂f
= sign(x1 ).
∂x1
2 t❛ ❝â

❱➟② Dfx1 (x) =
❱î✐ i

∂
(|x1 |)ϕ(x)dx =
∂xi

|x1 |ϕ xi (x)dx =
G

❱➟②

G

0.ϕ(x)dx = 0.
G

∂f
(x) = 0; i = 2, ..., n.
∂xi

◆❤➟♥ ①➨t ✶✳✷✳✻✳

✶✳ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❝õ❛ ❤➔♠ f ❧➔ ❞✉② ♥❤➜t t❤❡♦ ♥❣❤➽❛ ❤➛✉ ❦❤➢♣ ♥ì✐✳
✷✳ ◆➳✉ f ∈ C |α| (G) t❤➻ t❛ ❝â
(Dα f (x))ϕ(x)dx = (−1)|α|
G

f (x)Dα ϕ(x)dx
G

✈î✐ ♠å✐ ❤➔♠ ϕ ∈ C0|α| (G).
✸✳ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ α ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ t❤ù tü ❧➜② ✤↕♦ ❤➔♠✳
✾


✹✳ ◆➳✉ ❝→❝ ❤➔♠ fi , i = 1, 2 ❝â ✤↕♦ ❤➔♠ s✉② rë♥❣ Dα fi t❤➻ ❤➔♠ c1 f1 + c2 f2
✈î✐ c1 , c2 ❧➔ ❤➡♥❣ sè✱ ❝ô♥❣ ❝â ✤↕♦ ❤➔♠ s✉② rë♥❣
Dα (c1 f1 + c2 f2 ) = c1 Dα f1 + c2 Dα f2 .

✺✳ ◆➳✉ ❤➔♠ f ∈ L2loc (G) ❝â ✤↕♦ ❤➔♠ s✉② rë♥❣ Dα f = F ✈➔ ❤➔♠ F ❝â
✤↕♦ ❤➔♠ s✉② rë♥❣ Dβ F = G t❤➻ tç♥ t↕✐ ✤↕♦ ❤➔♠ s✉② rë♥❣ Dα+β f
✈➔ Dα+β f = G.
❚❤➟t ✈➟②✱ ❣✐↔ sû ϕ ∈ C0|α+β| (G)✳ ❱î✐ Dβ ϕ ∈ C0|α| (G) t❛ ❝â
|α|

(Dα+β f (x))ϕ(x)dx = (−1)
G

Dα f (x)Dβ ϕ(x)dx
G

|α|

= (−1)

F (x)Dβ ϕ(x)dx

G
|α+β|

Dβ F (x)ϕ(x)dx

= (−1)

G
|α+β|

= (−1)

G(x)ϕ(x)dx.
G

✻✳ ✣↕♦ ❤➔♠ s✉② rë♥❣ ❦❤→❝ ✈î✐ ✤↕♦ ❤➔♠ ❝ê ✤✐➸♥ ❧➔ ♥➳✉ ✤↕♦ ❤➔♠ s✉② rë♥❣
❝➜♣ α ❝õ❛ ❤➔♠ f tç♥ t↕✐ t❤➻ ❝❤÷❛ ❝❤➢❝ ✤➣ ❝â ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ t❤➜♣
❤ì♥✳

✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ❈❤♦ (X, B, µ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✳ ❱î✐ 1

p<

+∞✱ t❛ ❦➼ ❤✐➺✉


Lp (X, µ) = f : X → C ✤♦ ✤÷ñ❝ ,


p

|f | dµ < +∞
X





❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ✤♦ ✤÷ñ❝ ❦❤↔ t➼❝❤ ❜➟❝ p✳
❑❤✐ ✤â✱ Lp (X) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐

 p1
f

p

p
|f | dµ .

=
X

✶✵

❑❤æ♥❣ ❣✐❛♥ L∞ (X) = f : X → C ✤♦ ✤÷ñ❝✱ f ❜à ❝❤➦♥ ❤➛✉ ❦❤➢♣ ♥ì✐
❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈î✐ ❝❤✉➞♥
f

∞

= ess sup |f (x)| =
x∈X

✐♥❢

sup |f (x)| .

E∈B,µE=0 X\E

▼➺♥❤ ✤➲ ✶✳✸✳✷✳ ✭❇➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r✮ ●✐↔ sû f ∈ L

p

(G) , g ∈ Lq (G)
1
1
✈î✐ 1 < p, q < +∞ ❧➔ ❝➦♣ sè ♠ô ❧✐➯♥ ❤ñ♣✱ tù❝ ❧➔ + = 1✳ ❑❤✐ ✤â✱
p
q
1
f g ∈ L (G) ✈➔
|f g| dµ

f

p

g q.

G

▼➺♥❤ ✤➲ ✶✳✸✳✸✳ ✭❇ê ✤➲ ❋❛t♦✉✮ ●✐↔ sû {f } ❧➔ ♠ët ❞➣② ❝→❝ ❤➔♠ ✤♦ ✤÷ñ❝
n

❦❤æ♥❣ ➙♠ tr➯♥ t➟♣ ✤♦ ✤÷ñ❝ G ⊂ R ✳ ❑❤✐ ✤â t❛ ❝â
n

lim fn dx ≤ lim
n→∞

fn dx.

n→∞
G

G

▼➺♥❤ ✤➲ ✶✳✸✳✹✳ ✭✣à♥❤ ❧➼ ▲❡❜❡s❣✉❡ ✈➲ sü ❤ë✐ tö ❜à ❝❤➦♥✮ ●✐↔ sû {f } ❧➔
n

♠ët ❞➣② ❝→❝ ❤➔♠ ✤♦ ✤÷ñ❝ ❤ë✐ tö ❤➛✉ ❦❤➢♣ ♥ì✐ ✤➳♥ ❤➔♠ ✤♦ ✤÷ñ❝ f tr➯♥
t➟♣ ✤♦ ✤÷ñ❝ G ⊂ Rn ✈➔ t❤ä❛ ♠➣♥ |fn (x)| ≤ g (x) ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ G✱

tr♦♥❣ ✤â g ❧➔ ♠ët ❤➔♠ ❦❤↔ t➼❝❤✳ ❑❤✐ ✤â t❛ ❝â
lim

fn dx =

n→∞
G

f (x) dx.
G

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✺✳ ❍➔♠ ✤ì♥ ❣✐↔♥
●✐↔ sû (X, B, µ) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✳ ❍➔♠ ♣❤ù❝ s ①→❝ ✤à♥❤ tr➯♥ X
✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ✤ì♥ ❣✐↔♥ ✤♦ ✤÷ñ❝ ♥➳✉ ♥â ❧➔ ♠ët ❤➔♠ ✤♦ ✤÷ñ❝ tr➯♥ X
✈➔ ❝❤➾ ❧➜② ♠ët sè ❤ú✉ ❤↕♥ ❣✐→ trà ❦❤→❝ ♥❤❛✉✳
●✐↔ sû s ❧➜② ❝→❝ ❣✐→ trà c1 , c2 , ..., cn ✳ ❑❤✐ ✤â s ✤÷ñ❝ ❜✐➸✉ ❜✐➵♥ ❞÷î✐ ❞↕♥❣
n

s=

ck χAk ,
k=1

tr♦♥❣ ✤â χAk ❧➔ ❝→❝ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ t➟♣ ❤ñ♣
Ak = ④ x ∈ X : s(x) = ck ⑥ , k = 1; 2; ...; n.
✶✶


s ỡ ữủ tr X s t tr
X ở ừ t ủ x X : s(x) = 0} ỳ

ủ S ỗ tt ỡ t tr X
trũ t tr Lp (X, à) ợ 1

p <

ự t õ S Lp (X, à) ợ 1

p <

sỷ f Lp (X, à) t ởt ự ữủ ổ
ữợ f = f1 f2 + i(f3 f4 ) tr õ f1 , f2 , f3 , f4 tỹ
ổ ữủ tr X t t f

0 ữủ tr

X õ tỗ t ởt {sn } ỡ ữủ ổ ỡ

t s
lim sn (x) = f (x), x X.

n

0

f (x), x X sn Lp (X, à) ợ ồ n N õ

sn (x)

(f (x) sn (x))p

(f (x))p
lim (f (x) sn (x))p = 0.

n

tt f Lp (X, à) f p L1 (X, à) ử ỵ s
ở tử t ữủ
lim f sn

n

p

= 0.

ợ ộ fk , k = 1, 2, 3, 4 ộ > 0 tỗ t ởt ỡ

t sk s fk sk p < t s = s1 s2 + i(s3 s4 ) t ữủ
4
f s

p

= (f1 s1 ) (f2 s2 ) + i [(f3 s3 ) (f4 s4 )]

p

< .

ỵ ữủ ự

ủ tử tr G R

n

ỵ C0 (G) trũ t ỡ tr Lp (G), 1

õ t

p < .

ự t ủ t trũ t tr
Lp (G) ợ ộ f Lp (G) ộ > 0 tỗ t ởt ỡ

s s f s p
õ t ự ợ s ỡ
2



ε
.
2
❱➻ s ❧➔ ❤➔♠ ✤ì♥ ❣✐↔♥ ❦❤↔ t➼❝❤ ♥➯♥ ❣✐→ ❝õ❛ s ❤ú✉ ❤↕♥✳ ❚❛ ❝â t❤➸ ❣✐↔ t❤✐➳t

❣✐↔♥ tç♥ t↕✐ ❤➔♠ g ∈ C0 (G) s❛♦ ❝❤♦ g − s

p

s(x) = 0 ✈î✐ ♠å✐ x ∈ Rn \G✳ ❚❤❡♦ ✣à♥❤ ❧➼ ▲✉s✐♥ tç♥ t↕✐ ❤➔♠ g ∈ C0 (G)

s❛♦ ❝❤♦
|g(x)|

sup |s(x)| , ∀x ∈ G
x∈Rn

✈➔
µ(④ x ∈ G : s(x) = g(x)⑥ ✮ ❁

⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎
♦❧❞❡r✱ t❛ ✤÷ñ❝

p

|s(x) − g(x)| dµ

✈î✐ 1

G

p
s−g

p

<2 s
p

.

1− pq
1dµ

,

G

∞. ❙✉② r❛
s−g

❱➟② f − g

q

p

 pq 

|s(x) − g(x)| dµ 



G

ε
4supx |s(x)|

f −s

p

∞

∞ (µ(④

ε
4 s

+ s−g

✣à♥❤ ❧➼ ✶✳✸✳✾✳ ❈❤♦ G ⊂ R

n

p

1

x ∈ G : s(x) = g(x)⑥ )) p

∞

ε
= .
2

< ε✳

✈➔ 1

p < ∞✳ ❑❤✐ ✤â✱ ❦❤æ♥❣ ❣✐❛♥ Lp (G) ❧➔

❦❤æ♥❣ ❣✐❛♥ ❦❤↔ ❧②✳
❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠é✐ m ∈ N t❛ ✤➦t
Gm =

x ∈ G : |x|

m ✈➔ d(x, ∂G)

1
m

.

❑❤✐ ✤â✱ Gm ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ●å✐ P ❧➔ t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ tr➯♥ Rn
❝â ❤➺ sè ♣❤ù❝ ❤ú✉ t✛✳ ✣➦t
Pm = {χGm f : f ∈ P } .

❚❤❡♦ ✣à♥❤ ❧➼ ❲❡✐❡rstr❛ss Pm trò ♠➟t tr♦♥❣ C(Gm )✳ ❑❤✐ ✤â✱ t➟♣ ❤ñ♣
∞

∪ Pm ❧➔ t➟♣ ✤➳♠ ✤÷ñ❝✳

m=1

▼➦t ❦❤→❝✱ ✈î✐ ♠é✐ f ∈ Lp (G) ✈➔ ✈î✐ ♠é✐ ε > 0 tç♥ t↕✐ g ∈ C0 (G) s❛♦

ε
❝❤♦ f − g p < ✳
2
✶✸


1
< d(supp, G) t tỗ t h Pm s g h
m
r
1

gh p
g h (à(Gm )) p < .
2



f h

p




1
< (à(Gm )) p .
2

< .

Pm t ữủ Lp (G)
m=1

G R , m N , ||


n

m, 1

p

+

ỵ
m,p (G) = {f : G R ữủ |D f Lp (G)} .



ổ ợ
p1


f
f

=



=

m,p (G)

D f

p
p

ợ 1

p < +.

|| m

m,p (G)

D f



ợ p = +.

|| m

m,p

(G) ổ

ự sỷ (fn ) tr m,p (G)
õ (D fn ) tr Lp (G) ợ s 0

m



Lp (G) ổ ừ tỗ t f f tr
Lp (G) s fn f D fn f tr Lp (G) n

Lp (G) L1loc (G) ợ fn , f Lp (G) tỗ t s rở
Tfn , Tf D (G) s
fn (x)(x)dx; C0 (G).

Tfn () =
G

f (x)(x)dx; C0 (G).

Tf () =
G

ợ C0 (G) tũ ỵ t õ
|Tfn () Tf ()|

|fn (x) f (x)| |(x)|
G





q

fn f

p


1 1
+ = 1.
p q
❈❤♦ n → ∞✱ t❛ ✤÷ñ❝

✈î✐

Tfn (ϕ) → Tf (ϕ), ∀ϕ ∈ C0∞ (G).

✭✶✳✶✶✮

TDα fn (ϕ) → Tfα (ϕ), ∀ϕ ∈ C0∞ (G).

✭✶✳✶✷✮

❚÷ì♥❣ tü✱ t❛ ❝â

❚ø ✭✶✳✶✶✮ ✈➔ ✭✶✳✶✷✮✱ t❛ ❝â
Tfα (ϕ) = lim TDα fn (ϕ)
n→∞

= lim (−1)|α| Tfn (Dα ϕ)
n→∞

= (−1)|α| Tf (Dα ϕ); ∀ϕ ∈ C0∞ (G).

❙✉② r❛ fα = Dα f ✈î✐ f ∈ ❲m,p (G)✳
❉♦ ✤â
lim fn − f

n→∞

❲

❱➟② ❲m,p (G) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳

✶✺

m,p (G)

= 0.


❈❤÷ì♥❣ ✷
❑❤æ♥❣ ❣✐❛♥ ❖r❧✐❝③ ✈➔ ❦❤æ♥❣ ❣✐❛♥

❖r❧✐❝③✲❙♦❜♦❧❡✈

✷✳✶ ❍➔♠ ❧ç✐
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ❍➔♠ Φ : I → R ✭ ✈î✐ I = (a, b), ❬a, b), (a, b❪, ❬a, b❪✱
−∞

a
+∞✮ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❧ç✐ ♥➳✉ Φ t❤ä❛ ♠➣♥

Φ(αx + (1 − α)y)

αΦ(x) + (1 − α)φ(y) ∀x, y ∈ I, α ∈ [0; 1] .

✣à♥❤ ❧➼ ✷✳✶✳✷✳ ◆➳✉ Φ ❧➔ ❤➔♠ ❧ç✐ tr➯♥ (a, b) t❤➻ ❤➔♠ Φ ❧✐➯♥ tö❝ tr➯♥ (a, b)✳
❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ ♠å✐ s, x, y, z ∈ (a, b) s❛♦ ❝❤♦ a < s < x < y < z < b✱ t❛
❝â

Φ(y) − Φ(x)
Φ(z) − Φ(y)
Φ(x) − Φ(s)
= c.
x−s
y−x
z−y
Φ(y) − Φ(x)
❍➔♠
t➠♥❣ ❦❤✐ x → y ♥➯♥ ❦❤✐ x → y t❛ ❝â
y−x
d=

Φ(y) − Φ(x)
x→y
y−x
lim

❚÷ì♥❣ tü✱ ❤➔♠

c.

Φ(y) − Φ(x)
❣✐↔♠ ❦❤✐ y → x ♥➯♥ t❛ ❝â
y−x
Φ(y) − Φ(x)
y→x
y−x
lim

❱➟② Φ ❧✐➯♥ tö❝ t↕✐ ♠å✐ x ∈ (a, b).

✶✻

d.


ú ỵ tr ú (a, b) ổ ú
õ [a, b] ử

0 0 x < 1
(x) =

1 = 1



ỗ ữ ổ tử tr [0, 1]

: R [0; +] tử tọ


x1 + x2
2

1
1
(x1 ) + (x2 ).
2
2



õ ỗ
ự ự x1 , x2 R tọ
(x1 + (1 )x2 )

(x1 ) + (1 )(x2 ), 0



1.

sỷ t tự ổ tọ [0; 1] õ
f () = (x1 + (1 )x2 ) (x1 ) (1 )(x2 )

t tr ỹ tr [0; 1] M > 0
ồ 0 tr ọ t ừ s f (0 ) = M ồ > 0 s
[0 , 0 + ] [0; 1] .
ử t tự
x1 = (0 )x1 + (1 0 + )x2 ; x2 = (0 + )x1 + (1 0 )x2

t ữủ
(0 x1 + (1 0 )x2 )

1
1
(x1 ) + (x2 ).
2
2

r

f (0 ) + f (0 + )
< M.
2
t ợ ồ 0 ú x1 , x2 R
f (0 )

ỗ

ỗ tr [a, b] õ õ
tr t ồ tr ừ [a, b] p (x)


p+ (x).


❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ 0 < h1 < h2 ✱ t❛ ❝â
Φ(x) − Φ(x − h2 )
h2

❙✉② r❛
❤↕♥

Φ(x) − Φ(x − h1 )
h1

Φ(x + h1 ) − Φ(x)
h1

Φ(x + h2 ) − Φ(x)
.
h2

Φ(x) − Φ(x − h)
t➠♥❣ ❦❤✐ h → 0 ✈➔ ❜à ❝❤➦♥ tr➯♥ ♥➯♥ tç♥ t↕✐ ❣✐î✐
h
lim+

h→0

❚÷ì♥❣ tü✱
❣✐î✐ ❤↕♥

Φ(x) − Φ(x − h)
= p− (x).
h

Φ(x + h) − Φ(x)
❣✐↔♠ ❦❤✐ h → 0 ✈➔ ❜à ❝❤➦♥ ❞÷î✐ ♥➯♥ tç♥ t↕✐
h
lim+

h→0

❱➔ t❛ ❝â p− (x)

Φ(x + h) − Φ(x)
= p+ (x).
h

p+ (x).

✣à♥❤ ❧➼ ✷✳✶✳✺✳ ✣↕♦ ❤➔♠ ♣❤↔✐ p

+ (x)

❝õ❛ ❤➔♠ ❧ç✐ Φ(x) ❧➔ ❤➔♠ ❦❤æ♥❣

❣✐↔♠✱ ❧✐➯♥ tö❝ ♣❤↔✐✳

❈❤ù♥❣ ♠✐♥❤✳ ❈❤å♥ x1 < x2 ✳ ❱î✐ h > 0 ✤õ ❜➨✱ t❛ ❝â
Φ(x1 + h) − Φ(x1 )
h

Φ(x2 ) − Φ(x2 − h)
.
h

❈❤✉②➸♥ q✉❛ ❣✐î✐ ❤↕♥ t❛ ✤÷ñ❝ p+ (x1 )

p− (x2 )✳ ❚❤❡♦ ✣à♥❤ ❧➼ ✷✳✶✳✹ t❛ ❝â

p+ (x2 )✳ ❱➟② p+ (x) ✤ì♥ ✤✐➺✉ t➠♥❣✳
Φ(x + h) − Φ(x)
❍ì♥ ♥ú❛✱ t❤❡♦ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ❧➼ ✷✳✶✳✹✱ t❛ ❝â p+ (x)
.
h
❈è ✤à♥❤ h ✈➔ ❝❤♦ x → x+
0 ✱ t❛ ✤÷ñ❝
p− (x2 )

p+ (x2 )✳ ❙✉② r❛ p+ (x1 )

lim+ p+ (x)

x→x0

❈❤♦ h → 0+ t❛ ✤÷ñ❝ lim+ p+ (x)

Φ(x0 + h) − Φ(x0 )

.
h
p+ (x0 ).

x→x0

▼➦t ❦❤→❝✱ ✈î✐ x

x0 t❛ ❝â p+ (x)

p+ (x0 )✳ ❙✉② r❛ lim+ p+ (x)

p+ (x0 ).

x→x0

❱➟② lim+ p+ (x) = p+ (x0 ).
x→x0

❚÷ì♥❣ tü✱ t❛ ❝❤ù♥❣ ♠✐♥❤ p− (x) ✤ì♥ ✤✐➺✉ t➠♥❣ ✈➔ ❧✐➯♥ tö❝ tr→✐✳

✣à♥❤ ❧➼ ✷✳✶✳✻✳ ❍➔♠ ❧ç✐ Φ(x) ❧➔ ❧✐➯♥ tö❝ t✉②➺t ✤è✐ ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥
▲✐♣s❝❤✐t③ tr➯♥ ♠å✐ ❦❤♦↔♥❣ ❤ú✉ ❤↕♥✳
✶✽


❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❦❤♦↔♥❣ (a, b) ❜➜t ❦ý ✈î✐ a < x < y < b✱ t❛ ❝â
Φ(x) − Φ(a)
x−a

Φ(y) − Φ(x)
y−x

Φ(b) − Φ(y)
.
b−y

Φ(y) − Φ(x)
y−x

p− (b).

❈❤✉②➸♥ q✉❛ ❣✐î✐ ❤↕♥ t❛ ✤÷ñ❝
p+ (a)

Φ(y) − Φ(x)
❜à ❝❤➦♥ ∀x, y ∈ (a, b).
y−x
❙✉② r❛ Φ(x) ❧✐➯♥ tö❝ t✉②➺t ✤è✐ tr➯♥ ♠å✐ ✤♦↕♥ ❤ú✉ ❤↕♥✳

✣✐➲✉ ♥➔② ❝❤ù♥❣ tä

✣à♥❤ ❧➼ ✷✳✶✳✼✳ ▼å✐ ❤➔♠ ❧ç✐ Φ(x) t❤ä❛ ✤✐➲✉ ❦✐➺♥ Φ(a) = 0 ❝â t❤➸ ❜✐➸✉
❞✐➵♥ ❞÷î✐ ❞↕♥❣

x

Φ(x) =

p(t)dt,

a

tr♦♥❣ ✤â p(t) ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠ ✈➔ ❧✐➯♥ tö❝ ♣❤↔✐✳
❈❤ù♥❣ ♠✐♥❤✳ ❍➔♠ Φ(x) ❝â ✤↕♦ ❤➔♠ ❤➛✉ ❦❤➢♣ ♥ì✐✳
❚❤➟t ✈➟②✱ ✈î✐ x1 < x2 ✱ t❛ ❝â
p+ (x1 )

❙✉② r❛ p− (x1 )

p+ (x1 )

p− (x2 ) ✈➔ p− (x1 )

p+ (x1 ).

✭✷✳✸✮

p− (x2 )✱ ✈î✐ x1 < x2 ✳

❱➻ p− (x) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉✱ ♥➯♥ ♥â ❧✐➯♥ tö❝ ❤➛✉ ❦❤➢♣ ♥ì✐✳ ●å✐ x1 ❧➔ ✤✐➸♠
❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ p− (x)✱ ❝❤✉②➸♥ q✉❛ ❣✐î✐ ❤↕♥ ❜✐➸✉ t❤ù❝ ✭✷✳✸✮ ❦❤✐ x2 → x1
t❛ ✤÷ñ❝
p− (x1 )

p+ (x1 )

p− (x1 ).

❙✉② r❛ p− (x1 ) = p+ (x1 )✳
❚❤❡♦ ✣à♥❤ ❧➼ ✷✳✶✳✻✱ ❤➔♠ Φ(x) ❧✐➯♥ tö❝ t✉②➺t ✤è✐ tr➯♥ (a, b) ♥➯♥ ∀ε >

0, ∃δε > 0 s❛♦ ❝❤♦ ✈î✐ ♠å✐ ❤å ([ai , bi ))1

i n

⊂ (a, b) ♣❤➙♥ ❜✐➺t ✤æ✐ ♠ët

t❤ä❛ ✤✐➲✉ ❦✐➺♥
n

n

|bi − ai | < δε ⇒
i=1

|Φ(bi ) − Φ(ai )| < ε.
i=1

✶✾


❚❤❡♦ ✣à♥❤ ❧➼ ▲❡s❜❡s❣✉❡✲❱✐t❛❧✐✱ t❛ ❝â
x

Φ(x) =

Φ (t)dt + Φ(a), a

x

b.

a

❱➻ Φ(a) = 0 ♥➯♥ t❛ ❝â t❤➸ ✈✐➳t
x

Φ(x) =

Φ (t)dt.
a

❙✉② r❛ Φ (x) = p− (x) = p+ (x) ❤➛✉ ❦❤➢♣ ♥ì✐✳

✷✳✷ ❍➔♠ ❨♦✉♥❣
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳ ❈❤♦ ❤➔♠ ❧ç✐ Φ : R → R

+

t❤ä❛ ✤✐➲✉ ❦✐➺♥ Φ(−x) =

Φ(x)✱ Φ(0) = 0✱ lim Φ(x) = +∞ ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❨♦✉♥❣✳
x→∞

❱➼ ❞ö ✷✳ ❈❤♦ 1

p < ∞ ✈➔ ❤➔♠ sè Φ(x) = |x| ❧➔ ❤➔♠ ❨♦✉♥❣✳
p

❚❤➟t ✈➟②✱ t❛ ❝â Φ(−x) = Φ(x) ✈➔ Φ(0) = 0✳ ❉♦ 1

p ♥➯♥

p

lim Φ(x) = lim |x| = +∞.

x→∞

x→∞

❉♦ ✤â✱ ❤➔♠ Φ ❧➔ ❤➔♠ ❨♦✉♥❣✳

✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✷✳ ❈❤♦ ❤➔♠ ❨♦✉♥❣ Φ✳ ❚❛ ❣å✐ ❤➔♠ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷
s❛✉
✭✷✳✹✮

Φ(y) = sup {x |y| − Φ(x)} , y ∈ R
x 0

❧➔ ❤➔♠ ❧✐➯♥ ❤✐➺♣ ❝õ❛ ❤➔♠ Φ✳
❑❤✐ ✤â✱ Φ ❝ô♥❣ ❧➔ ❤➔♠ ❧ç✐ ✈➔ Φ(0) = 0, Φ(−y) = Φ(y), lim Φ(y) = +∞✳
y→0

❉♦ ✤â✱ Φ ❝ô♥❣ ❧➔ ❤➔♠ ❨♦✉♥❣✳
❚ø ✭✷✳✹✮ s✉② r❛
xy

Φ(x) + Φ(y), x, y ∈ R

✤÷ñ❝ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣✳

❚✐➳♣ t❤❡♦✱ t❛ ①➨t ♠ët ❧î♣ ❤➔♠ ❨♦✉♥❣ ✤➦❝ ❜✐➺t✳
✷✵

✭✷✳✺✮


✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✸✳ ❈❤♦ ❤➔♠ ❨♦✉♥❣ Φ : R → R

+

❧✐➯♥ tö❝ ✤÷ñ❝ ❣å✐ ❧➔

◆✲❤➔♠ ✭❤❛② ❤➔♠ ◆✐❝❡✮ ♥➳✉
Φ(x)
= 0,
x→0 x

Φ(x)
= +∞.
x→+∞ x

lim

lim

✣à♥❤ ❧➼ ✷✳✷✳✹✳ ❈❤♦ Φ : R → R✱ Φ(x) ❧➔ ◆✲❤➔♠ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
|x|

Φ(x) =

p(t)dt,
0

tr♦♥❣ ✤â p(t) ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ (0; +∞)✱ ❧✐➯♥ tö❝ ♣❤↔✐ ✈➔ t❤ä❛ ✤✐➲✉
❦✐➺♥ p(0) = 0, lim p(t) = ∞.
t→∞

|x|

❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥✳ ❚❤❡♦ ✣à♥❤ ❧➼ ✷✳✶✳✼ t❛ ❝â Φ(x) =

p(t)dt✱
0

✈î✐ p(t) ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ (0; +∞)✱ ❧✐➯♥ tö❝ ♣❤↔✐ ✈➔ Φ(x) ❧➔ ❤➔♠
❝❤➤♥✳ ❍ì♥ ♥ú❛✱ t❛ ❝â
2x

Φ(2x) =

2x

p(t)dt >

p(t)dt > xp(x).
x

0

Φ(2x)

Φ(2x)
=2
. ❈❤✉②➸♥ q✉❛ ❣✐î✐ ❤↕♥ ❦❤✐ x → 0✱ t❛ ✤÷ñ❝
x
2x
Φ(x)
p(0) = lim p(x) = 0. ❱➻ p(x)
✈î✐ x > 0 ♥➯♥ lim p(x) = ∞.
x→∞
x→0
x

❙✉② r❛ p(x) <

|x|

✣✐➲✉ ❦✐➺♥ ✤õ✳ ●✐↔ sû Φ(x) =

p(t)dt ✈➔ p(t) t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥
0

♥❤÷ tr♦♥❣ ✣à♥❤ ❧➼✳

❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ ❤➔♠ Φ(x) ❧ç✐✳ ❱î✐ x1 , x2 > 0, t❛ ❝â
 x1 +x2
x1 +x2
x1

2

Φ

x1 + x2
2

=

p(t)dt

p(t)dt +
0

0
x1

1
p(t)dt + 
2
0
x1

1
=
2

1

2

x1 +x2

2

x2

x1



x1 +x2
2

x2

1
1
p(t)dt = Φ(x1 ) + Φ(x2 ).
2
2
0

✷✶



p(t)dt 


p(t)dt 


p(t)dt +
x1

x1 +x2
2

p(t)dt +
x1

1
p(t)dt +
2
0

2