Một số kết quả về tích hai hàm suy rộng theo nghĩa mikusinski

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷
❑❍❖❆ ❚❖⑩◆
✯✯✯✯✯✯✯✯✯✯✯✯✯✯
P❍Ò◆● ❚❍➚ ❚❍❯

▼❐❚ ❑➌❚ ◗❯❷ ❱➋ ❚➑❈❍ ❍❆■
❍⑨▼ ❙❯❨ ❘❐◆●
❚❍❊❖ ◆●❍➒❆ ▼■❑❯❙■◆❙❑■
❑❍➶❆ ▲❯❾◆ ❚➮❚ ◆●❍■➏P ✣❸■ ❍❖❈
❈❤✉②➯♥ ♥❣➔♥❤ ✿ ●■❷■ ❚➑❈❍
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿
❚✳❙ ❚❸ ◆●➴❈ ❚❘➑
❍⑨ ◆❐■ ✲ ✷✵✶✸

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✶

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤
▲❮■ ❈❷▼ ❒◆

❚r♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥✱ tæ✐ ✤➣ ♥❤➟♥ ✤÷ñ❝ sü ❝❤➾ ❜↔♦ ✈➔ ❣✐ó♣ ✤ï t➟♥

t➻♥❤ ❝õ❛ t❤➛② ❣✐→♦✿ ❚✳❙ ❚❸ ◆●➴❈ ❚❘➑✱ ✤÷ñ❝ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥✲ ❚r÷í♥❣
✣↕✐ ❤å❝ s÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ tr♦♥❣ q✉→ tr➻♥❤ t➻♠ ✈➔ t❤✉ t❤➟♣ t➔✐ ❧✐➺✉
♥❣❤✐➯♥ ❝ù✉✳ ❇➯♥ ❝↕♥❤ ✤â✱ tæ✐ ❝á♥ ♥❤➟♥ ✤÷ñ❝ sü q✉❛♥ t➙♠ ✤ë♥❣ ✈✐➯♥ tø ♣❤➼❛ ❣✐❛ ✤➻♥❤✱
❜↕♥ ❜➧✳
❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tî✐ ❚✳❙ ❚❸ ◆●➴❈ ❚❘➑✱ ♥❣÷í✐ ✤➣ t➟♥ t➻♥❤ ❝❤➾
❜↔♦✱ ❣✐ó♣ ✤ï ✤➸ ❡♠ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✳
❈❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❤ú♥❣ ♥❣÷í✐ ✤➣ ❧✉æ♥ ð ❜➯♥✱ õ♥❣ ❤ë✱ ✤ë♥❣
✈✐➯♥ tæ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✦
❍➔ ♥ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✸
❙✐♥❤ ✈✐➯♥
P❤ò♥❣ ❚❤à ❚❤✉

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✷

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤
▲❮■ ❈❆▼ ✣❖❆◆

❉÷î✐ sü ❤÷î♥❣ ❞➝♥ ❝õ❛ ❚❙✳ ❚❸ ◆●➴❈ ❚❘➑✱ ❝ò♥❣ ✈î✐ sü ❝è ❣➢♥❣ ♥é ❧ü❝ ❝õ❛ ❜↔♥
t❤➙♥✱ tæ✐ ✤➣ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ❑❤â❛ ❧✉➟♥ ❝õ❛ ♠➻♥❤✳ ❚r♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉✱ tæ✐ ❝â
t❤✉ t❤➟♣ ✈➔ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤➣ ♥➯✉ tr♦♥❣ ♠ö❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ♥❤ú♥❣ ❦➳t q✉↔ tr♦♥❣ ❑❤â❛ ❧✉➟♥ ❧➔ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ tæ✐✱
❦❤æ♥❣ trò♥❣ ✈î✐ ❦➳t q✉↔ ❝õ❛ t→❝ ❣✐↔ ❦❤→❝✳ ◆➳✉ s❛✐✱ tæ✐ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠✳
❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✸

❙✐♥❤ ✈✐➯♥

P❤ò♥❣ ❚❤à ❚❤✉

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✸

❑✸✺●✲ ❙P ❚♦→♥


▼ö❝ ❧ö❝
✶ ❑❍➷◆● ●■❆◆ ❍⑨▼ ❚❍Û
✶✳✶ ▼ët ✈➔✐ ❦➼ ❤✐➺✉ ✈➔ ❦❤→✐ ♥✐➺♠✳
✶✳✶✳✶ ▼ët ✈➔✐ ❦➼ ❤✐➺✉✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ ▼ët ✈➔✐ ❦❤→✐ ♥✐➺♠✳ ✳ ✳
✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ t❤û✳ ✳ ✳ ✳

✼
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✷ ❑❍➷◆● ●■❆◆ ❍⑨▼ ❙❯❨ ❘❐◆●

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✶✸

✷✳✶ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✷ ✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ s✉② rë♥❣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✸ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣
❝❤➟♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✸ ❚➑❈❍ ❈❍❾P ❈Õ❆ ❍❆■ ❍⑨▼ ❙❯❨ ❘❐◆● ❚❍❊❖ ◆●❍➒❆ ▼■❑❯❙■◆✲
❙❑■

✷✶
✸✳✶ ❚➼❝❤ ❝❤➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✷ ❚➼❝❤ ❝❤➟♣ ▼✐❦✉s✐♥s❦✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤
▲❮■ ◆➶■ ✣❺❯

✶✳ ▲þ ❞♦ ❝❤å♥ ✤➲ t➔✐✳
❑➸ tø ❦❤✐ r❛ ✤í✐ ✈➔♦ t❤➳ ❦➾ ❳❳✱ ❤➔♠ s✉② rë♥❣ ✤➣ ❣â♣ ♣❤➛♥ q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝
♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤✳ ❉♦ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ♥â✐ ❝❤✉♥❣✱ ✈➔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ♥â✐
r✐➯♥❣✱ t❤÷í♥❣ ❦❤æ♥❣ tç♥ t↕✐ t♦➔♥ ❝ö❝✱ ♥➯♥ ♥❤✉ ❝➛✉ ♠ð rë♥❣ ❦❤→✐ ♥✐➺♠ ♥❣❤✐➺♠ ❝❤♦
♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ♥❣➔② ❝➔♥❣ trð ♥➯♥ ❝➜♣ t❤✐➳t✳
❱î✐ ♥❤ú♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ✤â♥❣ ❣â♣ ❝õ❛ ❏✳▼✐❦✉s✐♥s❦✐ ✈➲ ❝→❝❤ t➼♥❤ t➼❝❤ ❝õ❛ ❤❛✐ ❤➔♠
s✉② rë♥❣✱ ✤➣ ❣✐↔✐ q✉②➳t ✤÷ñ❝ ♣❤➛♥ ♥➔♦ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝õ❛ ❧þ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦
❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤✳ ❚æ✐ ♠♦♥❣ ♠✉è♥ ✤÷ñ❝ t➻♠ ❤✐➸✉ ✈➲ ❦➳t q✉↔ ♠➔ ❏✳▼✐❦✉s✐♥s❦✐ ✤➣
✤↕t ✤÷ñ❝ ✈➲ t➼❝❤ ❝õ❛ ❤❛✐ ❤➔♠ s✉② rë♥❣✳ ❱➻ ✈➟② tæ✐ ✤➣ ❝❤å♥ ✤➲ t➔✐✿

✧▼❐❚ ❑➌❚ ◗❯❷ ❱➋ ❚➑❈❍ ❈Õ❆ ❍❆■ ❍⑨▼ ❙❯❨ ❘❐◆●
❚❍❊❖ ◆●❍➒❆ ▼■❑❯❙■◆❙❑■✧
✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
❇÷î❝ ✤➛✉ ❧➔♠ q✉❡♥ ✈î✐ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❚ø ✤â ❤➻♥❤ t❤➔♥❤ t÷ ❞✉② ❧♦❣✐❝ ✤➦❝
t❤ò ❝õ❛ ❜ë ♠æ♥ ✈➔ t➻♠ ❤✐➸✉ ✈➲ t➼❝❤ ❤❛✐ ❤➔♠ s✉② rë♥❣ t❤❡♦ q✉❛♥ ✤✐➸♠ ❝õ❛ ❏✳▼✐❦✉s✐♥s❦✐✳

✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉

✲ ◆❣❤✐➯♥ ❝ù✉ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣✳
✲ ◆❣❤✐➯♥ ❝ù✉ ✈➲ t➼❝❤ ❝❤➟♣ ✈➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ s✉② rë♥❣ t❤❡♦ ♥❣❤➽❛ ▼✐❦✉s✐♥s❦✐✳
✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
✲ P❤÷ì♥❣ ♣❤→♣ ♣❤➙♥ t➼❝❤✱ tê♥❣ ❤ñ♣✳
✲ ✣å❝ ✈➔ tr❛ ❝ù✉ t➔✐ ❧✐➺✉✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✺

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

✺✳ ❈➜✉ tró❝ ❦❤â❛ ❧✉➟♥✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❦❤â❛ ❧✉➟♥ ❣ç♠ ✸ ❝❤÷ì♥❣✿

❈❤÷ì♥❣ ✶✿ ❑❤æ♥❣ ❣✐❛♥ ❤➔♠ t❤û
✲ ❈❤÷ì♥❣ ♥➔② ❣✐î✐ t❤✐➺✉ ♠ët sè ❦➼ ❤✐➺✉ ✈➔ ❦❤→✐ ♥✐➺♠ ❝â tr♦♥❣ ♥ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥✳
✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ t❤û D (Ω)✳

❈❤÷ì♥❣ ✷✿ ❑❤æ♥❣ ❣✐❛♥ ❤➔♠ s✉② rë♥❣
✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ s✉② rë♥❣✳
✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛ ✈➲ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ s✉② rë♥❣✳
✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ ✈➔ ❝→❝ ❤➔♠ t➠♥❣
❝❤➟♠✳

❈❤÷ì♥❣ ✸✿ ❚➼❝❤ ❝❤➟♣ ❝õ❛ ❤❛✐ ❤➔♠ s✉② rë♥❣ t❤❡♦ ♥❣❤➽❛ ▼✐❦✉s✐♥s❦✐
✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤➨♣ t♦→♥ t➼❝❤ ❝❤➟♣ ❣✐ú❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ✈î✐
❤➔♠ s✉② rë♥❣ ✈➔ ❣✐ú❛ ❝→❝ ❤➔♠ s✉② rë♥❣✳
✲ ◆➯✉ ✤à♥❤ ♥❣❤➽❛ ✈➲ t➼❝❤ ❤❛✐ ❤➔♠ s✉② rë♥❣ ❝õ❛ ❏✳▼✐❦✉s✐♥s❦✐✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✻

❑✸✺●✲ ❙P ❚♦→♥


❈❤÷ì♥❣ ✶
❑❍➷◆● ●■❆◆ ❍⑨▼ ❚❍Û

✶✳✶ ▼ët ✈➔✐ ❦➼ ❤✐➺✉ ✈➔ ❦❤→✐ ♥✐➺♠✳
✶✳✶✳✶ ▼ët ✈➔✐ ❦➼ ❤✐➺✉✳
Zn+ := {x = (x1 , x2 , ..., xn ), xi ∈ Z+ , i = 1, 2, ..., n}✳

Rn := {x = (x1 , x2 , ..., xn ), xi ∈ R, i = 1, 2, ..., n}✳

C(Ω)✿

t➟♣ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ Ω✳
C k (Ω)✿ t➟♣ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ tî✐ ❝➜♣ ❦ tr➯♥ Ω✳
C ∞ (Ω)✿ t➟♣ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥✱ ❧✐➯♥ tö❝ tr➯♥ Ω✳
LP (Ω)✿ t➟♣ ❝→❝ ❤➔♠ ❢ ✤♦ ✤÷ñ❝ t❤❡♦ ♥❣❤➽❛ ▲❡❜❡s❣✉❡ tr♦♥❣ Ω s❛♦ ❝❤♦✿
p

|f (x)|

f =

1
p

<∞

Ω

Lploc (Ω)✿

t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ ❜➟❝ ♣✱ 1 ≤ p ≤ ∞ tr➯♥ Ω ✭❤❛② t➟♣ ❝→❝
❤➔♠ ❢ ①→❝ ✤à♥❤ tr➯♥ Ω s❛♦ ❝❤♦ ✈î✐ ♠å✐ t➟♣ ❱ ❝♦♠♣❛❝t tr♦♥❣ Ω t❤➻ ❢ ❦❤↔ t➼❝❤ tr♦♥❣ ❱✮✳
▼ët ✈❡❝tì ❝â ❞↕♥❣ α = (α1, α2, ..., αn) : αj ∈ Z+, j = 1, 2, ...n ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ✤❛
❝❤➾ sè ✭❤❛② ♥✲ ✤❛ ❝❤➾ sè✮ ✈î✐ ✤ë ❞➔✐ ✭❤❛② ❝➜♣ ❝õ❛ α✮ ❧➔
|α| = α1 + α2 + ... + αn ✳
✼


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❚♦→♥ tû ✈✐ ♣❤➙♥ ❧✐➯♥ ❦➳t ✈î✐ ✤❛ ❝❤➾ sè α ❧➔ ∂ α = ∂1α ∂2α ...∂nα ✱ tr♦♥❣ ✤â ∂j = ∂∂ ✱
√
❤♦➦❝ Dα = D1α D2α ...Dnα ✈î✐ Dj = i∂∂ , j = 1, 2, ...n ✈➔ i = −1✳
❱➼ ❞ö✿ ❈❤♦ ❤➔♠ sè✿ u(x, y) = x2 + y2 + 3xy + x + y, α = (1, 2) ∈ Z2+
❑❤✐ ✤â✱ ∂ α = ∂xα ∂yα = ∂x1∂y2 = 4x + 6y + 2✳

❈❤♦ Ω ❧➔ ♠ët t➟♣ ♠ð ❦❤→❝ ré♥❣ tr♦♥❣ Rn✳ ▼ët ❤➔♠ sè f : Ω → C, x → f (x)✱ ♥➳✉
t♦→♥ tû ✈✐ ♣❤➙♥ ∂ αf tç♥ t↕✐ ✈➔ ❧✐➯♥ tö❝ ✈î✐ ♠å✐ ✤❛ ❝❤➾ sè α ∈ Zn+ t❤➻ t❛ ♥â✐ f ∈ C ∞(Ω)✳
✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔ f ∈ C ∞(Ω) ♥➳✉ f ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ♠å✐ ❝➜♣✳
●✐→ ❝õ❛ ♠ët ❤➔♠ ❧✐➯♥ tö❝ f : Ω → C ❧➔ ❜❛♦ ✤â♥❣ tr♦♥❣ Ω ❝õ❛ t➟♣ ❤ñ♣ {x ∈ Ω : f (x) = 0}
✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ suppf ✳ ❍❛② suppf = cl{x ∈ Ω : f (x) = 0} ⊂ Ω✳
1

2

n

xj

2

1

n

xj

1

2

✶✳✶✳✷ ▼ët ✈➔✐ ❦❤→✐ ♥✐➺♠✳
▼ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ❳ tr➯♥ tr÷í♥❣ P ✭✈î✐ P = C ❤♦➦❝ P = R✮ ❧➔ ♠ët ❦❤æ♥❣
❣✐❛♥ ✈❡❝tì tr➯♥ tr÷í♥❣ P ✤÷ñ❝ tr❛♥❣ ❜à ♠ët tæ♣æ t❤➼❝❤ ❤ñ♣ s❛♦ ❝❤♦ →♥❤ ①↕ (x, y) → x+y
✈➔ (λ, y) → λy ❧➔ ❧✐➯♥ tö❝✳
❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ❳✱ ♠ët t➟♣ ❤ñ♣ E ⊂ X ❣å✐ ❧➔ t➟♣ ❜à ❝❤➦♥ ♥➳✉ ✈î✐ ♠å✐

❧➙♥ ❝➟♥ ❱ ❝õ❛ ❣è❝ θ✱ ❝â ♠ët sè s❃✵ s❛♦ ❝❤♦ ∀t > s t❤➻ E ⊂ tV ✳ ◆➳✉ ❣è❝ θ ❝è ♠ët ❧➙♥
❝➟♥ ❜à ❝❤➦♥ t❤➻ ❦❤æ♥❣ ❣✐❛♥ ❳ ✤÷ñ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥ ✤à❛ ♣❤÷ì♥❣✳
▼ët t➟♣ ❤ñ♣ E ⊂ X ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ❳ ✤÷ñ❝ ❣å✐ ❧➔ t➟♣ ❤ót ♥➳✉ ∀x ∈
X, ∃t = t(x) = 0 s❛♦ ❝❤♦ x ∈ tE ✳ ◆➳✉ ∀α ∈ C ♠➔ |α| < 1✱ t❛ ❝â αE ⊂ E t❤➻ ❊ ✤÷ñ❝
❣å✐ ❧➔ t➟♣ ❝♦♥ ❝➙♥ ✤è✐ ❝õ❛ ❳✳
▼ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ❳ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ♥➳✉ ❝â ♠ët ❝ì sð
❧➙♥ ❝➟♥ ❝õ❛ ❣è❝ θ ❣ç♠ t♦➔♥ ❜ë ♥❤ú♥❣ t➟♣ ❧ç✐✳
▼ët ❦❤æ♥❣ ❣✐❛♥ ❧ç✐ ✤à❛ ♣❤÷ì♥❣ ✤÷ñ❝ ❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ F rechet ♥➳✉ ♥â ❧➔ ❦❤æ♥❣
❣✐❛♥ ♠❡tr✐❝ ✤õ ✈î✐ ♠❡tr✐❝ ❝↔♠ s✐♥❤ d t❤ä❛ ♠➣♥ d(x + z, y + z) = d(x, y)✭d ❜➜t ❜✐➳♥ ✈î✐
♣❤➨♣ tà♥❤ t✐➳♥✮✳
▼ët ❦❤æ♥❣ ❣✐❛♥ ✈❡❝tì tæ♣æ ❳ ❣å✐ ❧➔ ❝â t➼♥❤ ❝❤➜t Heine − borel✱ ♥➳✉ ♠å✐ t➟♣ ❝♦♥
✤â♥❣ ✈➔ ❜à ❝❤➦♥ ❝õ❛ ❳ ✤➲✉ ❧➔ t➟♣ ❝♦♠♣❛❝t✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✽

❑✸✺●✲ ❙P ❚♦→♥


õ tốt

t

ổ tỷ
t t tr R DK ổ ừ tt
f C (Rn ) s suppf K K t
DK = {f C () : suppf K} .

ỹ tổổ tr C () s C () tr t ởt ổ F rechet

õ t t Heine borel DK ởt t õ ừ C () ộ K
ú t ồ t t Kj (j = 1, 2, ...) s Kj intKj+1 = j Kj
ởt ồ ỷ pN tr C () N = 1, 2, ... ữ s
pN = max {|D f (x)| : x KN , || N }

õ pN ởt tổổ ỗ ữỡ tr tr C () ợ ộ
x số x f (x) tử t tổổ DK ừ ổ
t x DK õ tr C ()
ởt ỡ s ữỡ ừ ổ ữủ t ủ
VN =

f C () : pN (f ) <

1
N

, N = 1, 2, ...

{fi} tr C () ố t fi fj VN ợ i, j ừ ợ
õ |Dfi Dfj | < N1 tr KN || N ự tọ Dfi ở tử tr
t t ừ tợ g t fi(x) g0(x) g0 C () s
g = D g0 fi g0 t tổổ ừ C () õ C () ổ F rechet
ụ ú ợ ộ ổ õ DK
sỷ E C () t õ tỗ t MN < s
pN (f ) < MN N = 1, 2, ... ợ ồ f E t tự |Df | MN ú tr
KN || N tử ỗ ừ D f : f E tr KN 1 || N 1

Pũ

P


õ tốt

t

ỵ ascoli ỵ cantor t ồ tr ự ởt
{fi } s D fi ở tử tr t t ừ ợ ộ số õ
{fi } ở tử t tổổ ừ C () ự tọ t t
C () õ t t Heine borel
ủ ừ tt ổ DK tr t tt
t t ừ ồ ổ tỷ ổ ỡ
tr
ởt ổ tỡ ợ ở ợ ổ ữợ
tổ tữớ ừ tr ự
ợ ộ t


N

= max {|D (x)| : x , || N } , N = 1, 2, ...

ồ ừ
ởt t ổ rộ tr Rn
ợ ồ t t K ỵ K tổổ ừ ổ F rechet DK
t tt t ỗ ố W D() s DK W K ợ ồ t
t K
ồ ừ tt ủ ừ t ủ õ + W ợ )

W

ỵ

ởt tổổ ừ ổ D() ởt ỡ s ữỡ ừ
D() ũ ởt ổ tỡ ỗ ữỡ

ự

sỷ V1 , V2 , V1 V2
ự t ự + W V1 V2, W
ừ t tỗ t i D() W s
= i + Wi Vi , (i = 1, 2)

Pũ



P


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❈❤å♥ ❑ s❛♦ ❝❤♦ DK ❝❤ù❛ φ1, φ2, φ✳ ❱➻ DK ∩ Wi ❧➔ ♠ð tr♦♥❣ DK ♥➯♥
φ − φi ∈ (1 − δi )Wi , ∀δi > 0

❉♦ t➼♥❤ ❧ç✐ ❝õ❛ Wi ♥➯♥
φ − φi + δi Wi ⊂ (1 − δi )Wi + δi Wi = Wi

⇒ φi + δi W ⊂ φi + Wi ⊂ Vi , (i = 1, 2)

❉♦ ✤â ✭✯✮ ✤ó♥❣ ✈î✐ W = (δ1W1) ∩ (δ2W2)
❱➟② ✭❛✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❜✮ ●✐↔ sû φ1, φ2 ∈ D(Ω)✳
✣➦t W = {φ ∈ D (Ω) : φ < φ1 − φ2 0}✱ ✈î✐ φ 0 = max
|φ(x)| t❤➻ W ∈ β ✈➔ φ1
x∈Ω
❦❤æ♥❣ ♥➡♠ tr♦♥❣ φ2 + W ✳ ❉♦ ✤â✱ φ1 ❧➔ t➟♣ ✤â♥❣ t÷ì♥❣ ✤è✐ tr♦♥❣ τ ✳
❚❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ♣❤➨♣ t♦→♥ ✤↕✐ sè tr➯♥ D(Ω) t÷ì♥❣ t❤➼❝❤ ✈î✐ tæ♣æ τ ✳
P❤➨♣ ❝ë♥❣ ❧➔ τ −❧✐➯♥ tö❝✱ ✈➻ ✈î✐ ♠å✐ φ1, φ2 ∈ D(Ω) ✈➔ φ1 + φ2 + W ∈ τ ✈î✐ W ∈ β ✳
❉♦ ❲ ❧➔ t➟♣ ❝➙♥ ✤è✐ ♥➯♥ 21 W ∈ β ✱ s✉② r❛✿
1
1
ϕ1 + W ∈ τ ; ϕ2 + W ∈ τ
2
2

❉♦ ✤â✿

1
1
φ1 + W + φ2 + W
2
2

= (φ1 + φ2 ) + W,∀φ1 , φ2 ∈ D(Ω).

❱➟② ♣❤➨♣ ❝ë♥❣ ❤❛✐ ♣❤➛♥ tû tr♦♥❣ D(Ω) ❧➔ ❧✐➯♥ tö❝ t❤❡♦ τ ✳
❱î✐ ♣❤➨♣ ♥❤➙♥ ✈æ ❤÷î♥❣✱ ❝❤å♥ ♠ët ✈æ ❤÷î♥❣ α0 ✈➔ ♠ët φ0 ∈ D(Ω) t❤➻✿

αφ − α0 φ0 = α(φ − φ0 ) + (α − α0 )φ0 .

❱î✐ ♠å✐ W ∈ β ✱ tç♥ t↕✐ δ > 0 s❛♦ ❝❤♦ δφ0 ∈ 12 W✳ ❈❤å♥ ❝❃✵ s❛♦ ❝❤♦ 2c(|α0| + δ) = 1 t❤➻
❞♦ ❲ ❧➔ t➟♣ ❧ç✐ ✈➔ ❝➙♥ ♥➯♥ t❛ ❝â (α − α0)φ0 ∈ W✱ ✈î✐ ♠å✐ |α − α0| < δ ✈➔ φ ∈ φ0 + cW✳
❉♦ ✈➟②✱ ♣❤➨♣ ♥❤➙♥ ✈î✐ ♣❤➛♥ tû ✈æ ❤÷î♥❣ ❧➔ ❧✐➯♥ tö❝ tr♦♥❣ D(Ω) t❤❡♦ tæ♣æ τ ✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✶✶

❑✸✺●✲ ❙P ❚♦→♥


õ tốt

t

ự tọ ổ tỷ D() ổ tỡ tổổ ỡ
ỳ ỏ ồ ổ ỗ ữỡ

r sốt ữỡ t ổ sỷ ởt t ổ rộ ừ
tứ ởt số t q s
ỵ
ởt t ỗ ố ừ D() V
tổổ K ừ DK trũ ợ tổổ ừ ổ DK s tứ D()
ởt t ừ D() t E D() K õ số
MN < s E tọ t tự
MN ; (N = 0, 1, 2, 3, ...)

(j ) ởt tr D() t (j ) DK ợ ồ t t

K
lim

i,j

i j

N

= 0, (N = 0, 1, 2, ...)

(j ) 0 tr tổổ ừ D() t õ ởt t t K õ ự
tt sj Dj ở tử tợ j ợ ồ số
r D() ồ Cauchy ở tử
ỵ sỷ u t t tứ D() ởt ổ ỗ ữỡ
õ s tữỡ ữỡ
u tử
u
(j ) 0 tr D() t uj 0 tr
t u tr t DK D () t uK t uK ổ tử
q ồ t tỷ D ởt tử tứ D() õ

Pũ



P


❈❤÷ì♥❣ ✷

❑❍➷◆● ●■❆◆ ❍⑨▼ ❙❯❨ ❘❐◆●

✷✳✶ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣
D (Ω)

✣à♥❤ ♥❣❤➽❛ ✷✳✶✿ ▼ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤
u : D (Ω) → C
φ → u (φ) = u, φ

❣å✐ ❧➔ ♠ët ❤➔♠ s✉② rë♥❣ ✭t❤❡♦ ♥❣❤➽❛ ❙❝❤✇❛rt③✮ ①→❝ ✤à♥❤ tr➯♥ Ω✱ ♥➳✉ ✈î✐ ♠å✐ t➟♣
❝♦♠♣❛❝t K ⊂ Ω✱ ❝â ♠ët sè t❤ü❝ c ≥ 0 ✈➔ ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ ◆ s❛♦ ❝❤♦✿
sup |Dα φ| , ∀φ ∈ D (Ω)

| u, φ | ≤ c
|α|≤N

x∈Ω

✈î✐ supp φ ⊂ K ✳
❚➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ s✉② rë♥❣ ①→❝ ✤à♥❤ tr➯♥ Ω ❧➟♣ t❤➔♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥✱ ❣å✐ ❧➔
❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ s✉② rë♥❣ tr➯♥ Ω✱ ❦➼ ❤✐➺✉ ❧➔ D (Ω)✳

▼ët sè ✈➼ ❞ö✿
❱➼ ❞ö ✷✳✶✿ ❈→❝ ❤➔♠ sè ❧✐➯♥ tö❝ tr➯♥ Ω ❧➔ ❝→❝ ❤➔♠ s✉② rë♥❣✳
✶✸


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❚❤➟t ✈➟②✱ ❣✐↔ sû f ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ Ω✳ ❑❤✐ ✤â✱ f ❧➔ ♠ët ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥
Ω✳ ❍ì♥ ♥ú❛✱
f (x)φ (x) dx ≤

| f, φ | =
Ω

≤

|f (x)| |φ (x)| dx
Ω

|f (x)| dx . sup |φ (x)| , ∀φ ∈ D (Ω)
Ω

Ω

✣➦t c = Ω |f (x)| dx ≥ 0✱ ❦❤✐ ✤â✿ | f, φ | ≤ c. sup |φ (x)| , ∀φ ∈ D (Ω)
Ω
❱➟② f ❧➔ ❤➔♠ s✉② rë♥❣ ✭t❤❡♦ ♥❣❤➽❛ ❙❝❤✇❛rt③✮✳
❱➼ ❞ö ✷✳✷✿ ❈→❝ ❤➔♠ f tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp(Ω), 1 ≤ p ≤ ∞ ❝ô♥❣ ❧➔ ❝→❝ ❤➔♠ s✉② rë♥❣✱
✈î✐
f (x) φ (x) dx, ∀φ ∈ D (Ω) .

f, φ =
Ω

❱➼ ❞ö ✷✳✸✿ ❍➔♠ δ✲ ❉✐r❛❝✳
δ : D(Rn ) → C, φ → δ, φ = φ (0)

Ð ✤➙②✱ φ ∈ D (Rn) ♥➯♥ φ ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ♠å✐ ❝➜♣✳
❱➻ | δ, φ | = |φ (0)| ≤ 1. sup |φ (x)| , ∀φ ∈ D (Rn) ♠➔ suppφ ⊂ K − compact ⊂ Rn✳ ❱➻
✈➟②✱ δ ❧➔ ♠ët ❤➔♠ s✉② rë♥❣ ✭❣å✐ ❧➔ ❤➔♠ s✉② rë♥❣ ❉✐r❛❝ ❤❛② ❤➔♠ ❉❡❧t❛ ❉✐r❛❝✮✳
❱➼ ❞ö ✷✳✹✿ ❍➔♠ |x|✿
|x|φ (x) dx

|x| : D (R) → C, φ → |x| , φ =
R

✈î✐ suppφ ⊂ K ✱ ❑ ❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ R✳
❚❛ ❝â✿
| |x| , φ | =

|x|φ (x) dx ≤
R

≤

|x| sup |φ (x)| dx = sup φ (x)
R

R

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

|x| |φ (x)| dx
R

|x|dx

R

R

✶✹

= sup |φ (x)|
K

|x|dx
K

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

✣➦t
|x|dx ≥ 0

c=
K

❍❛②
| |x| , φ | ≤ c. sup |φ (x)| , ∀φ ∈ D (R)
K

❱➟② |x| ❧➔ ♠ët ❤➔♠ s✉② rë♥❣✳
✣à♥❤ ❧þ ✷✳✶✿ ▼ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ u ①→❝ ✤à♥❤ tr➯♥ D (Ω) ❧➔ ♠ët ❤➔♠ s✉② rë♥❣
❦❤✐ ✈➔ ❝❤➾ ❦❤✐
lim u, φj = 0

j→∞

✈î✐ ♠å✐ ❞➣② {φj } ❤ë✐ tö tî✐ 0 tr♦♥❣ D (Ω) ❦❤✐ j → ∞✳
✣à♥❤ ♥❣❤➽❛ ✷✳✷✿ ❈❤♦ u ∈ D (Ω)
✶✳ ❍➔♠ s✉② rë♥❣ u ✤÷ñ❝ ❣å✐ ❧➔ ❜➡♥❣ 0 tr➯♥ t➟♣ ♠ð K ⊂ Ω✱ ❦➼ ❤✐➺✉ u|K = 0 ♥➳✉
u, φ = 0, ∀φ ∈ D (K)

✷✳ ●✐→ ❝õ❛ ❤➔♠ s✉② rë♥❣ u ✤÷ñ❝ ❦➼ ❤✐➺✉ suppu ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐✿
suppu = Ω\(∪{K|K

♠ð } ⊂ Ω ✈➔ u|K = 0)

◆➳✉ u ❝â suppu ❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Ω t❤➻ t❛ ♥â✐ u ❧➔ ❤➔♠ s✉② rë♥❣ ❝â ❣✐→ ❝♦♠♣❛❝t✳
❚➟♣ ❤ñ♣ ❝→❝ ❤➔♠ s✉② rë♥❣ ❝â ❣✐→ ❝♦♠♣❛❝t ❦➼ ❤✐➺✉ ❧➔ ε (Ω)

◆❤÷ ✈➟②✱ ❝â t❤➸ t❤➜② D (Ω) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❧✐➯♥ ❤ñ♣ ❝õ❛ D (Ω)✱ tr➯♥ ✤â ✤÷ñ❝
tr❛♥❣ ❜à ❝→❝ ♣❤➨♣ t♦→♥✿
❛✳ P❤➨♣ ❝ë♥❣ ❝→❝ ❤➔♠ s✉② rë♥❣✿
❈❤♦ f, g ∈ D (Ω) t❤➻ f + g ∈ D (Ω) ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ q✉② t➢❝✿
f + g, ϕ = f, ϕ + g, ϕ , ∀ϕ ∈ D (Ω)

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✶✺

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❜✳ P❤➨♣ ♥❤➙♥ ♠ët sè ✈î✐ ♠ët ❤➔♠ s✉② rë♥❣✿
❈❤♦ f ∈ D (Ω) ✈➔ λ ∈ R t❤➻ λf ∈ D (Ω) ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ q✉② t➢❝✿
λf, ϕ = λ f, ϕ , ∀ϕ ∈ D (Ω)

❱î✐ ❤❛✐ ♣❤➨♣ t♦→♥ tr➯♥ t❤➻ D (Ω) trð t❤➔♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤✳

✷✳✷ ✣↕♦ ❤➔♠ ❝õ❛ ❤➔♠ s✉② rë♥❣✳
✣à♥❤ ♥❣❤➽❛ ✷✳✸✿ ❈❤♦ u ∈ D (Ω) t❤➻ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤
∂ α u, φ = (−1)|α| u, ∂ α φ , φ ∈ D (Ω)

α ❧➔ ✤❛ ❝❤➾ sè✱ ✤÷ñ❝ ❣å✐ ❧➔ ✤↕♦ ❤➔♠ s✉② rë♥❣ ❝➜♣ α ❝õ❛ ❤➔♠ s✉② rë♥❣ u ✈➔ ✤÷ñ❝ ❦➼ ❤✐➺✉

❧➔ ∂ αu✳
◆➳✉ | u, φ | ≤ C

φ

✈î✐ ♠å✐ φ ∈ DK ✱ t❤➻
| ∂ α u, φ | ≤ C ∂ α φ

N

≤C φ

N +|α|

❚ø ✤â t❛ ❝â ∂ αu ∈ D (Ω)✳ ❱➔ t❛ ❝â ❝æ♥❣ t❤ù❝
∂ α ∂ β u = ∂ α+β u, ∀u ∈ D (Ω) , ∀α, β

❧➔ ❝→❝ ✤❛ ❝❤➾ sè✳
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ f ❧➔ ❝→❝ ❤➔♠ t❤÷í♥❣ ❦❤↔ ✈✐ t❤➻ t❤➻ ✤↕♦ ❤➔♠ t❤❡♦ ♥❣❤➽❛ s✉② rë♥❣
trò♥❣ ✈î✐ ✤↕♦ ❤➔♠ t❤❡♦ ♥❣❤➽❛ t❤æ♥❣ t❤÷í♥❣✳ ❱➻ ❦❤✐ f ∈ C 1 (Ω) t❤➻
∂j (f φ)dxj =
Ω

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

Ω

∂f
∂φ
φ+
f
∂xj
∂xj

✶✻

=0

❑✸✺●✲ ❙P ❚♦→♥

❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❚❤❡♦ ✤à♥❤ ❧þ ❋✉❜✐♥✐ t❛ ❝â
∂j (f φ)dx1 ...dxn = 0.
Ω

❚ø ✤â s✉② r❛
∂j f φdx = −
Ω

f ∂j φdx
Ω

❱➻ ✈➟②✱ ♥➳✉ f ∈ C 1 (Ω) t❤➻ ❝ô♥❣ ①→❝ ✤à♥❤ ♠ët ❤➔♠ s✉② rë♥❣ f ∈ D (Ω)✱ ✈➔
f, ϕ =

f (x)ϕ (x)dx.
Ω

❱➼ ❞ö ✷✳✺✿ ❍➔♠ ❍❡❛✈✐s✐❞❡
♥➳✉ x ≥ 0
♥➳✉ x < 0


 1
H (x) =
 0

Ð ✤➙② H : R → {0, 1} , Ω = R ✈➔
+∞

H, φ =

+∞

H (x)φ (x) dx =
−∞

1.φ (x) dx.
0

❚❛ ❝â

+∞
1

∂H, φ = (−1) H, ∂φ = −

H (x)∂φ (x) dx = −

−∞

= φ (x)

+∞

+∞

∂φ (x) dx
0

= φ (0) = δ, φ , ∀φ ∈ D (R) .

0

❱➟② ∂H = δ✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✶✼

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❱➼ ❞ö ✷✳✻✿ ❍➔♠ f (x) = ln |x|
❚❛ ❝â f : R\ {0} → R, x → ln |x|✱ ❦❤✐ ✤â f ❧➔ ❤➔♠ ❦❤↔ t➼❝❤ ✤à❛ ♣❤÷ì♥❣ tr➯♥ R✳ ❉♦
✤â f ❧➔ ♠ët ❤➔♠ s✉② rë♥❣ ✈î✐
+∞

f (x)φ (x) dx, ∀φ ∈ D (R)

f, φ =
−∞

❚❛ t➼♥❤ ∂f ✱ t❛ ❝â✿
+∞
1

∂f, φ = (−1) f, ∂φ = −

f (x)∂φ (x) dx

−∞

+∞

=−

ln |x| ∂φ (x) dx

−∞

0

=−

+∞

ln |x| ∂φ (x) dx −

−∞



ln |x| ∂φ (x) dx
0

−ε

− lim+ 

ln |x| ∂φ (x) dx −

ln |x| ∂φ (x) dx

ε→0

−∞

✣➦t u = ln |x| , dv =
✤÷ñ❝✿

ε

∂φ (x) dx

∂f, φ = lim+
ε→0




✱ ✈➔ →♣ ❞ö♥❣ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✱ t❛ t❤✉
−ε



+∞

φ (x)
dx +
x

[φ (ε) − φ (−ε)] ln ε +
−∞



−ε

= lim+ 
ε→0

+∞

φ (x)
dx +
x

−∞

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉



+∞

ε


φ (x) 
dx

x


φ (x) 
dx ,
x

ε

✶✽

❑✸✺●✲ ❙P ❚♦→♥


õ tốt

t

tr õ 0
lim [ () ()] ln = 0 é x1 ổ t
ữỡ tự x1 / L1loc (R) t ổ t s rở x1 ữợ t

t õ t x1 ởt s rở tở D (R)
ừ ln |x| x1 D (R) \L1loc (R) ữ ln |x| = x1
+



(x)
dx +
tự 0
lim
x

t
+

+


(x)
dx
x

tữớ ữủ

+


(x)
dx
x

ữủ ồ

ỵ ỵ trú ừ srt
t ởt s rở ữỡ ừ ởt tử
õ T D () , x0 tỗ t ởt Vx ừ x0 tr tỗ
t f C (Vx ) tỗ t t tỷ r s
0

0

T |Vx0 = f

tr D (Vx )
0

tr õ T |V ởt ừ tr Vx
ứ ỵ s rở t t ởt ổ ọ t tr õ
tử ổ ụ ự tt tở Lploc, p = 1, 2, ...,
x0

0

ổ ổ
s rở t
ổ S(R ) ổ t t ừ C


t ủ số f tr Rn s xD f (x) tr Rn ợ ộ
, Zn+

n


n
0 (R )

S(Rn ) = f C (Rn ) : x D f (x) < C, , , Z+n

ổ S(Rn) ữủ tr ồ
f

Pũ

,

= sup x D f (x) , , Z+n
xRn



P


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

❈→❝ ♣❤➛♥ tû ❝õ❛ S(Rn) ✤÷ñ❝ ❣å✐ ❧➔ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤✳ ❑❤æ♥❣ ❣✐❛♥ S(Rn) ✤÷ñ❝
❣å✐ ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤✳
✣à♥❤ ♥❣❤➽❛ ✷✳✺✳ ❈→❝ ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ tr➯♥ S(Rn) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ s✉②

rë♥❣ t➠♥❣ ❝❤➟♠✳ ❑❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ ✤÷ñ❝ ❦➼ ❤✐➺✉
❧➔ S (Rn)✳
❱➟② T ∈ S (Rn) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ T : S(Rn) → C ❧✐➯♥ tö❝ ✈➔ fn → f tr➯♥ S(Rn)✱ ♥❣❤➽❛
❧➔ ❤➔♠ T (fn) → T (f ) tr➯♥ C✳
❇ê ✤➲ ✷✳✶✳ ❚ø T (fn) − T (f ) = T (fn − f ) ✈➔ fn → f tr➯♥ S(Rn) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
(fn − f ) → 0 tr➯♥ S(Rn )✱ t❛ t❤➜② ♠ët →♥❤ ①↕ t✉②➳♥ t➼♥❤ tr➯♥ S(Rn ) ❧➔ ❧✐➯♥ tö❝ ❦❤✐ ✈➔
❝❤➾ ❦❤✐ ♥â ❧✐➯♥ tö❝ t↕✐ 0 ∈ S(Rn)✳
✣à♥❤ ❧þ ✷✳✸ ⑩♥❤ ①↕ t✉②➳♥ t➼♥❤ T : S(Rn) → S(Rn) t❤♦↔♥ ♠➣♥ |T (f )| ≤ f α,β , ∀f ∈
S(Rn ), ∀α, β ∈ Zn+ t❤➻ T ∈ S (Rn )
❈❤ù♥❣ ♠✐♥❤✿ ⑩♣ ❞ö♥❣ ❜ê ✤➲ tr➯♥✱ ❝❤ó♥❣ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❚ ❧✐➯♥ tö❝ t↕✐ ✵✳
❚❤➟t ✈➟②✱ ♥➳✉ fn → 0 tr➯♥ S(Rn) t❤➻ fn α,β → 0 ✈➔ |T (f )| ≤ f α,β → 0 ❦❤✐ n → ∞✱
♥❣❤➽❛ ❧➔ ❚ ❧✐➯♥ tö❝ t↕✐ ✵✳ ❱➟② T ∈ S (Rn)✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✷✵

❑✸✺●✲ ❙P ❚♦→♥


❈❤÷ì♥❣ ✸
❚➑❈❍ ❈❍❾P ❈Õ❆ ❍❆■ ❍⑨▼
❙❯❨ ❘❐◆● ❚❍❊❖ ◆●❍➒❆
▼■❑❯❙■◆❙❑■

✸✳✶ ❚➼❝❤ ❝❤➟♣
❈❤♦ 1 ≤ p < ∞✱ ✤➦t
Lp (Rn ) = { f ①→❝ ✤à♥❤ ✈➔ ✤♦ ✤÷ñ❝ tr➯♥ Rn : R |f (x)|p dx < ∞}✱ tr♦♥❣ ✤â t➼❝❤
♣❤➙♥ ✤÷ñ❝ ❤✐➸✉ t❤❡♦ ♥❣❤➽❛ ▲❡❜❡s❣✉❡✳ ❑❤✐ tr❛♥❣ ❜à ❝❤✉➞♥
n

p

f

Lp

|f (x)| dx

=

1
p

,

Rn

t❤➻ Lp (Rn) t❤➔♥❤ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
◆➳✉ f, g ∈ Lp (Rn) t❤➻ t➼❝❤ ❝❤➟♣ ❝õ❛ f ✱ g ❦➼ ❤✐➺✉ ❧➔ f ∗ g ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿
(f ∗ g) (x) =

f (y) g (x − y)dy
Rn

✷✶


õ tốt

t

õ t ự ữủ t (f g) (x) tỗ t ỡ f g L1 (Rn)
f g L f L . g L L1 (Rn) tr t ởt số
ữ ổ õ tỷ ỡ õ t t ừ f L1 (Rn)
g Lp (Rn )1 p <
t t tợ t ừ u D (Rn) D (Rn)
u D (Rn) D (Rn) t
1

1

1

( u) (x) = u (y) , (x y) , x Rn

ởt tỷ ừ C (Rn)
r ỵ tt s rở t ừ s rở ởt ổ ử
õ ữủ ữ s
u, v D (Rn) t ồ t ừ u v u v
t t s
u v, = u (y) , v (x) , (x + y) , D (Rn )

ỹ tr t t t u v D (Rn) ỡ ỳ ởt
tr u v õ tr t t t tr t õ u v = v u

ú ỵ

u = u = u, u D (Rn)
t õ tr t

u , = u(y), (x) , (x + y)

= u(y), (y) = u,

u = u tữỡ tỹ t ụ õ u = u
t tr ú ợ trữớ ủ f, g L1 (Rn)
t ợ ồ D (Rn) t t
h (y) =

g (x) (y + x)dx
Rn

Pũ



P


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

t❤➻ h ∈ L1 (Rn)✳ ❍ì♥ ♥ú❛✱
|h (y)| =

|g (x) φ (y + x)|dx =
Rn

|g (t − y) φ (t)|dt

Rn

≤ sup |φ (t)|
t∈sup pφ

|g (t − y)|dt = c g
Rn

L1 , ∀y

∈ Rn .

❚ø ✤â✱
f (y) , g(x), φ(y + x)

f (y)h(y)dy,

= f (y) , h (y) = u, φ =
Rn

✈➔ |f (y) h (y)| ≤ c g
tç♥ t↕✐✳ ❚❛ ❝ô♥❣ ❝â✿

L1

|f (y)|✱

t❛ ❝â sü tç♥ t↕✐ ❝õ❛

f ∗ g, φ =

f (y) , g(x), φ(y + x)

✱ ♥➯♥ f ∗ g

f (y)g(x)φ(y + x)dxdy
Rn ×Rn

=

f (y) g (t − y) dy)φ (t) dt,

(
Rn

Rn

f (y) g (t − y) dy)dt, φ (t)

=
Rn

❱➟② (f ∗ g) (t) =

Rn

f (y) g (t − y) dy)✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✷✸

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

✸✳✷ ❚➼❝❤ ❝❤➟♣ ▼✐❦✉s✐♥s❦✐
❚r÷î❝ ❦❤✐ t➻♠ ❤✐➸✉ ✈➲ t➼❝❤ ❤❛✐ ❤➔♠ s✉② rë♥❣ t❤❡♦ q✉❛♥ ✤✐➸♠ ❝õ❛ ▼✐♥❦✉s✐♥s❦✐✱ ❝❤ó♥❣
t❛ s➩ t➻♠ ❤✐➸✉ ✈➲ t➼❝❤ ❝õ❛ ♠ët ❤➔♠ trì♥ ✈➔ ♠ët ❤➔♠ s✉② rë♥❣✿
✣à♥❤ ♥❣❤➽❛ ✸✳✸ ❈❤♦ ♠ët ❤➔♠ trì♥ f ∈ C ∞ (Ω) ✈➔ ♠ët ❤➔♠ s✉② rë♥❣ u ∈ D (Ω)✳
❚➼❝❤ ❝õ❛ ❤➔♠ f ✈➔ ❤➔♠ u✱ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ f u ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
f u, φ = u, f φ ∀φ ∈ D (Ω)

❑❤æ♥❣ ❦❤â ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✈➳ ♣❤↔✐ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ❧➔ ♠ët ❤➔♠ s✉② rë♥❣✳ ❚❤➟t
✈➙②✱ rã r➔♥❣ ♥➳✉ f ∈ C ∞ (Ω) ✈➔ ∀φ ∈ D (Ω)✱ t❤➻ φ ∈ C ∞ (Ω) ✈➔ suppφ ⊂ K ✱ ❑ ❧➔ t➟♣
❝♦♠♣❛❝t✳ ❉♦ ✤â✱ f φ ∈ C ∞ (Ω) ✈➔ supp(f φ) ⊂ suppφ ⊂ K ⊂ Ω✱ ❤❛②
sup |∂ α (f φ)|.

u, f φ ≤ c
|α|≤N

❱➼ ❞ö ✸✳✶ ◆➳✉ δ ∈ D (R) t❤➻ xδ = 0
❚❤➟t ✈➟②✱ ✈➻

+∞

xδ, φ = δ, xφ =

δ (x). (xφ) (x) dx
−∞

✈➔ δ(x) = 0 ❦❤✐ x = 0, δ (x) = ∞ ❦❤✐ x = 0✱ ♥➯♥
xδ, φ = δ (0) . (xφ) (0) = 0.φ (0) = 0, ∀φ ∈ D (R) .

❱➟② x.δ = 0✳
◆❣♦➔✐ ✈✐➺❝ ❧➜② t➼❝❤ ❝õ❛ ♠ët ❤➔♠ trì♥ ✈➔ ♠ët ❤➔♠ s✉② rë♥❣ ♥❤÷ tr➯♥✱ t❛ ❝â t❤➸ ❧➜②
t➼❝❤ ❤❛✐ ❤➔♠ s✉② rë♥❣ ❜➜t ❦➻ ✈î✐ ✤ì♥ ✈à ❧➔ ✶✳ ❱➔ ❏✳▼✐❦✉s✐♥s❦✐ ✤➣ t➻♠ r❛ ❝→❝❤ t➼♥❤ t➼❝❤
❤❛✐ ❤➔♠ s✉② rë♥❣ ❞ü❛ ✈➔♦ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➼♥❤ q✉② ✈➔ t✐➳♥ q✉❛ ❣✐î✐ ❤↕♥✳

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✷✹

❑✸✺●✲ ❙P ❚♦→♥


❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣

❈❤✉②➯♥ ♥❣➔♥❤ ❣✐↔✐ t➼❝❤

✣à♥❤ ♥❣❤➽❛ ✸✳✹✿ δ−❞➣②
m
▼ët δ−❞➣② ❧➔ ♠ët ❞➣② (δn)+∞
n=1 ❝õ❛ ❝→❝ ♣❤➛♥ tû ❝õ❛ R , m = 1, 2... s❛♦ ❝❤♦✿
❛✮ suppδn ⊂ {x ∈ Rn : x < αn}✱ ✈î✐ αn → 0 ❦❤✐ n → ∞
❜✮ R δn (x) dx = 1 ✭❤♦➦❝ t✐➳♥ tî✐ ✶ ❦❤✐ n → ∞✮✳
◆❤➻♥ ♠ët ❝→❝❤ trü❝ q✉❛♥✱ ♠ët δ−❞➣② ♥❤÷ ❧➔ ♠ët ❞➣② t✐➳♥ tî✐ ❤➔♠ ❉❡❧t❛ ❉✐r❛❝ δ

✭❤❛② ✤ë ✤♦ ❉✐r❛❝✮ t↕✐ ❣è❝ ✵ ❝õ❛ Rm✳ ❚r♦♥❣ ✈➔✐ tr÷í♥❣ ❤ñ♣ t❛ ♣❤↔✐ ✤à♥❤ ♥❣❤➽❛ ❜ê s✉♥❣
t❤➯♠ t➼♥❤ ❝❤➜t ❝õ❛ δ−❞➣②✳
▼ët ✈➔✐ ✈➼ ❞ö ❝õ❛ t➼♥❤ ❝❤➜t ❜ê s✉♥❣ ❦❤✐ ✤à♥❤ ♥❣❤➽❛ δ−❞➣②✱ ❝❤➥♥❣ ❤↕♥✿
c1 ✮ ❚r÷í♥❣ ❤ñ♣ m = 1✱ t➼♥❤ ❝❤➜t ❜ê s✉♥❣ ✭ t❤❡♦ ▼✐❦✉s✐♥s❦✐✮ ❧➔✿
m

sup

xk+1 (δn )k (x) < +∞, ∀k = 0, 1, 2...

x∈R,n∈N

❍♦➦❝
c2 ✮

❚❤❡♦ Antosik, M ikusinskivSikorski :
δn (x) ≥ 0, ∀x ∈ Rm , ∀n ∈ N

❍♦➦❝
c3 ✮

❚❤❡♦ Itano✿
sup
n∈N

Rm

xk ∂ k δn(k) (x) < +∞, ∀k = 0, 1, 2...

❚❛ ❝❤ó þ r➡♥❣✱ ❝❤♦ S, T ∈ D (Rm) ❧➔ ❤❛✐ ❤➔♠ s✉② rë♥❣ ❝❤♦ tr÷î❝✳ ❚ø ❧þ t❤✉②➳t ❝õ❛

❤➔♠ s✉② rë♥❣✱ t➼❝❤ ❝❤➟♣ S ∗ δn ✈➔ T ∗ δn ✤➲✉ t❤✉ë❝ C ∞ (Rm)✳ ❍ì♥ ♥ú❛✱ n→∞
lim δn = δ
tr♦♥❣ D (Rm) ♥➯♥ n→∞
lim T ∗ δn = T ✈➔ lim S ∗ δn = S tr♦♥❣ D (Rm )✳
n→∞

❙✈✿ P❤ò♥❣ ❚❤à ❚❤✉

✷✺

❑✸✺●✲ ❙P ❚♦→♥