(Luận văn thạc sĩ) một số kết quả về tính bị chặn của tích phân dao động

ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN - Nguyễn Thị Xâm MỘT SỐ KẾT QUẢ VỀ TÍNH BỊ CHẶN CỦA TÍCH PHÂN DAO ĐỘNG LUẬN VĂN THẠC SĨ KHOA HỌC Hà Nội - Năm 2019 ĐẠI HỌC QUỐC GIA HÀ NỘI TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN - Nguyễn Thị Xâm MỘT SỐ KẾT QUẢ VỀ TÍNH BỊ CHẶN CỦA TÍCH PHÂN DAO ĐỘNG Chun ngành: Tốn Giải Tích Mã số: 8460101.02 LUẬN VĂN THẠC SĨ KHOA HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC:TS VŨ NHẬT HUY Hà Nội - Năm 2019 ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✸ ✹ P❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❚➼❝❤ ❝❤➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹✳✶ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✹✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L1(Rn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷ ✣→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❙t❡✐♥✲❲❛✐♥❣❡r ✶✹ ✸ ìợ ữủ t tỷ t ✷✻ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✵ ✹✵ ✷✳✶ ữợ t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ✣→♥❤ ❣✐→ ❝➟♥ tr➯♥ ❝õ❛ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✶ ❇ê ✤➲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✈ỵ✐ ❤➔♠ ♣❤❛ ❧❛✐ ✤❛ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỡ rữợ tr ♥ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ❝õ❛ ♠➻♥❤ tỵ✐ ❚❙✳ ❱ơ ◆❤➟t ❍✉②✱ ✈➻ sü ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤✱ ❝ị♥❣ ♥❤ú♥❣ ❧í✐ ✤ë♥❣ ✈✐➯♥ ✈ỉ ❝ị♥❣ þ ♥❣❤➽❛ ❝õ❛ ❚❤➛② tr♦♥❣ s✉èt q✉→ tr➻♥❤ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ tèt ♥❣❤✐➺♣✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝→♠ ì♥ sü ❣✐ó♣ ✤ï ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✈➔ ❑❤♦❛ ❙❛✉ ✤↕✐ ❤å❝✱ ✤➣ ♥❤✐➺t t➻♥❤ tr✉②➲♥ t❤ư ❦✐➳♥ t❤ù❝ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❈❛♦ ❤å❝✳ ❚æ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❦❤✉②➳♥ ❦❤➼❝❤✱ ❣✐ó♣ ✤ï tỉ✐ r➜t ♥❤✐➲✉ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤å❝ t➟♣✳ ▼➦❝ ❞ị ✤➣ ❝è ❣➢♥❣ r➜t ♥❤✐➲✉ ✈➔ ♥❣❤✐➯♠ tó❝ tr♦♥❣ q✉→ tr ự ữ ợ q ợ ổ t→❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❚→❝ ❣✐↔ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ þ ❦✐➳♥ ✤â♥❣ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❍➔ ◆ë✐✱ ♥➠♠ ✷✵✶✾ ◆❣✉②➵♥ ❚❤à ❳➙♠ ✷ ▼ð ✤➛✉ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ t❤✉ ❤ót ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ t ỵ tứ ❦❤✐ ①✉➜t ❤✐➺♥ ❝æ♥❣ tr➻♥❤ ❚❤➨♦r✐❡ ❆♥❛❧②t✐q✉❡ ❞❡ ❧❛ ❈❤❛❧❡✉r s rr t ỵ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ❤➻♥❤ ❤å❝ ✤↕✐ sè✱ ỵ tt st ỵ tt số t ✈➲ q✉❛♥❣ ❤å❝✱ ➙♠ ❤å❝✱ ❝ì ❤å❝ ❧÷đ♥❣ tû✱✳✳✳ ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣✳ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ sû ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ❦❤→❝ ♥❤❛✉ ✈➔ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ tø ❝→❝ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ❬✸✲✻❪✳ ◆❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ✤➣ r➜t ♥é ❧ü❝ ữợ t trỹ t tr t ✤ë♥❣ ✈➔ tè❝ ✤ë s✉② ❣✐↔♠ ❝õ❛ ❝❤✉➞♥ ❝õ❛ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ✭①❡♠ ❬✸✱ ✺✱ ✻❪ ✮✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✿ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ ❈❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣ ✈➔ ♠ët sè ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn ) ✈➔ L1 (Rn )✳ ❈❤÷ì♥❣ ✷✿ ✣→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❙t❡✐♥✲❲❛✐♥❣❡r✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➲ ✈✐➺❝ ✤→♥❤ ❣✐→ ❝➟♥ tr➯♥ ✈➔ ❝➟♥ ữợ t ý eiP (x) I() = R dx , x ữợ ữủ tr ữợ tổ q ✤❛ t❤ù❝ P (x)✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✹❪✳ ❈❤÷ì♥❣ ✸✿ ✣→♥❤ ❣✐→ ❝❤✉➞♥ ❝õ❛ t♦→♥ tû ❞❛♦ ✤ë♥❣✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ t➻♠ ❤✐➸✉ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ❞↕♥❣✿ eiλS(x,y) ψ(x, y)φ(y)dy, (Tλ φ)(x) = R tr♦♥❣ ✤â S(x, y) ❧➔ ♠ët ❤➔♠ ♣❤❛ ♥❤➟♥ ❣✐→ trà t❤ü❝✱ ψ(x, y) ❧➔ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❝â ❣✐→ ❝♦♠♣❛❝t ✈➔ λ ❧➔ ♠ët t❤❛♠ sè✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✸❪✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣ ✈➔ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳ ✶✳✶ P❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❈❤♦ Ω ❧➔ ♠ët t➟♣ ❤đ♣ tr♦♥❣ Rn✳ ▼ët ❤å ✤➳♠ ✤÷đ❝ ❝→❝ ❝➦♣ {(Ωj , ϕj )}∞j=1✱ tr♦♥❣ ✤â Ωj ❧➔ t➟♣ ♠ð tr♦♥❣ Rn✱ ϕj ❧➔ ❤➔♠ t❤✉ë❝ ❧ỵ♣ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈ỉ ❤↕♥ tr➯♥ Rn✱ ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ❝õ❛ t➟♣ Ω ♥➳✉ ❝→❝ t➼❝❤ ❝❤➜t s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ∞ {Ωj }∞ j=1 ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ Ω, Ω ⊂ Uj=1 Ωj ✱ ≤ ϕj (x) ≤ 1, x ∈ Ω, j = 1, 2, , ∞ ϕj ∈ C0∞ (Rn ), supp ϕj ⊂ Ωj , j = 1, 2, , j=1 ϕj (x) = 1, x ∈ Ω✳ ❚❛ ❝á♥ ❣å✐ {ϕj }∞j=1 ❧➔ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ù♥❣ ✈ỵ✐ ♣❤õ ♠ð {Ωj }∞j=1 ❝õ❛ t➟♣ Ω✳ ❚❛ ❝â ✤à♥❤ ỵ s ỡ ỵ ❈❤♦ ❑ ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Rn✱ ❤å ❤ú✉ ❤↕♥ {Uj }Nj=1 ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ ❑✳ ❑❤✐ õ tỗ t ởt ỳ ✈✐ ✈æ ❤↕♥ {ϕj }Nj=1 ①→❝ ✤à♥❤ ♠ët ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à ù♥❣ ✈ỵ✐ ♣❤õ ♠ð {Uj }Nj=1 ❝õ❛ t➟♣ rữợ ự ỵ t t ρ : Rn → R ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ ρ(x) := Ce 0, x −1 , ♥➳✉ x < ♥➳✉ x ≥ tr♦♥❣ ✤â✱ C ❧➔ ❤➡♥❣ sè s❛♦ ❝❤♦ ρ(x)dx = Rn ✹ ❍➔♠ ρ ❝â ❝→❝ t➼♥❤ ❝❤➜t ✿ ρ ∈ C0∞ (Rn ), s✉♣♣ρ = B[0, 1] = x ∈ Rn x ≤ , ρ(x) ≥ 0, ρ(x)dx = 1, Rn ✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ x ✳ ❱ỵ✐ ♠é✐ > 0✱ t❛ ①➨t ❤➔♠ ρ ♥❤÷ s❛✉ ρ (x) = −n x ρ ❍➔♠ ρ ❝ô♥❣ ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ρ, ❝ö t❤➸ ❧➔ ρ ∈ C0∞ (Rn ), s✉♣♣ρ = B[0, ] = x ∈ Rn x ≤ , ρ (x) ≥ 0, ρ (x)dx = 1, Rn ✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ x ✳ ❱ỵ✐ ♠é✐ ❤➔♠ f ∈ L1loc (Rn )✱ ✤➦t f (x) = (f ∗ ρ ) (x) = f (y)ρ (x − y)dy Rn ❱✐➺❝ ✤➦t ♥➔② ❝â ♥❣❤➽❛ ✈➻ f (y)ρ (x − y)dy = Rn f (x − y)ρ (y)dy = Rn f (y)ρ (x − y)dy B[x, ] ▼➺♥❤ ✤➲ ✶✳✶✳ ❈❤♦ f ∈ L1loc(Rn)✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ❦➳t ❧✉➟♥ s❛✉✳ ✭✐✮ f ∈ C ∞(Rn)✳ ✭✐✐✮ ◆➳✉ supp f = K ⊂ Rn t❤➻ f ∈ C0∞ (Rn )✱ supp f ⊂ K tr♦♥❣ ✤â K = K + B[0, ] = x ∈ Rn d(x, K) ≤ ✭✐✐✐✮ ◆➳✉ f ∈ C(Rn), lim →0 + ❈❤ù♥❣ ♠✐♥❤✳ sup |f (x) − f (x)| = 0, K ⊂ Rn ✳ x∈K ✭✐✮ ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ tø ✤➥♥❣ t❤ù❝ s❛✉ Dxα f (y)ρ (x − y)dy f (y)Dxα ρ (x − y)dy = Rn Rn ✭✐✐✮ ❉♦ supp f = K ♥➯♥ f (y)ρ (x − y)dy = f (x) Rn f (y)ρ (x − y)dy Rn ❱ỵ✐ ♠é✐ x ∈ / K ❝â x − y > , ∀y ∈ K ✳ ▼➔ supp ρ = B[0, 1] ♥➯♥ ρ (x − y) = 0, ∀y ∈ K ✳ ❉♦ ✤â✱ f (x) = ❦❤✐ x ∈ / K ❤❛② supp f ⊂ K ✳ ✭✐✐✐✮ ❉➵ t❤➜② f (x) − f (x) = (f (x − y) − f (x)) p(y)dy Rn ✺ (f (x − y) − f (x)) p(y)dy = B(0,1) ♥➯♥ |f (x) − f (x)| ≤ sup |f (x − y) − f (x)| y∈B[0,1] ▼➔ f ∈ C(Rn ) ♥➯♥ f ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ tø♥❣ t➟♣ ❝♦♠♣❛❝t K ⊂ Rn ✳ ❉♦ ✤â lim sup |f (x) − f (x)| = 0, K ⊂ Rn →0+ x∈K ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ▼➺♥❤ ✤➲ ✶✳✷✳ ❈❤♦ t➟♣ K ⊂ Rn✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ ❝â ❤➔♠ ϕ ∈ C0∞(Rn) t❤ä❛ ♠➣♥ ✈➔ ϕ(x) = 1, ∀x ∈ K /2✳ >0 ≤ ϕ(x) ≤ ∀x ∈ Rn ✱ supp ϕ ⊂ K ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t χ(x) ❧➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ t➟♣ K3 /4 ✱ tù❝ ❧➔ 1, ♥➳✉ x ∈ K3 0, ♥➳✉ x ∈ / K3 χ(x) := /4 , /4 ❈â χ ∈ L1 (Rn ) ⊂ L1loc (Rn ), supp χ = K3 /4 ✱ ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶ ❝â χ∗ρ /4 ∈ C0∞ (Rn ), supp(λ ∗ ρ /4 ) ⊂ K , ≤ (χ ∗ ρ /4 )(x) ∀x ∈ Rn ▼➔ (χ ∗ ρ /4 )(x) χ(x − y)ρ = B /4 (y)dy /4(0) ♥➯♥ (χ ∗ ρ /4 )(x) ≤ ρ B /4 (y)dy =1 ∀x ∈ Rn , /4(0) ✈➔ (χ ∗ ρ /4 )(x) = ρ B /4 (y)dy = 1, x ∈ K /2 /4(0) ◆❤÷ ✈➟② ❤➔♠ ❝➛♥ t➻♠ ❧➔ ϕ(x) = χ ∗ ρ (x) ✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❈❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ ứ tt K t t {Uj }N j=1 ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ K t❛ ❝â W1 := K \ ∪N J=2 Uj ⊂ U1 tỗ t > s W1 ⊂ W1 + B(0, ) ⊂ U1 ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ❤➔♠ ψ1 ∈ C0∞ (Rn ; [0; 1]) s❛♦ ❝❤♦ V1 := W1 + B(0, ) ⊂ supp ψ1 ⊂ W1 + B(0, ) ⊂ U1 , ψ1 (x) = 1, x ∈ V1 ▲↕✐ ❝â✱ W1 := K \ ∪N J=2 Uj ⊂ V1 ♠➔ V1 ❧➔ t➟♣ ♠ð ♥➯♥ W2 := K \ V1 ∪ ∪N J=3 Uj ❉♦ ✤â✱ tỗ t U2 > s W2 ⊂ W2 + B(0, ) ⊂ U2 ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ♠ët ❤➔♠ ψ2 ∈ C0∞ (Rn ; [0; 1]) s❛♦ ❝❤♦ V2 := W2 + B(0, 2 ) ⊂ supp ψ2 ⊂ W2 + B(0, ) ⊂ U2 , ψ2 (x) = 1, x ∈ V2 N ❈ù ♥❤÷ t❤➳ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❞➣② ❝→❝ ❤➔♠ {ψj}N j=1 ✈➔ ❝→❝ t➟♣ {Vj , Wj }j=1 t❤ä❛ ♠➣♥ ψj ∈ C0∞ (Rn ; [0; 1]) , Vj := Wj + B(0, j ) ⊂ supp ψj ⊂ Wj + B(0, j ) ⊂ Uj N ψj (x) > 0, x ∈ ∪N j=1 Vj (⊃ K) , ψj (x) = 1, x ∈ Vj , j=1 ✈➔ N ψj (x) < N + 1, x ∈ Rn j=1 ❈â K N j=1 Vj tỗ t số > s❛♦ ❝❤♦ K ⊂ K + B(0, ) ⊂ ∪N j=1 Vj ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷ ❝â ❤➔♠ ❦❤æ♥❣ ➙♠ φ t❤ä❛ ♠➣♥ φ ∈ C0∞ (Rn ), K ⊂ K + B(0, /2) ⊂ s✉♣♣φ ⊂ K + B(0, ) ⊂ ∪N j=1 Vj , ✈➔ ≤ φ(x) ≤ 1, x ∈ Rn , φ(x) = 1, x ∈ K + B(0, /2) ✣➦t ψj (x) ϕj (x) := φ(x) N k=1 ψk (x) + (1 − φ(x)) N + − ✼ N k=1 ψk (x) ❝â ≤ ϕj (x) ≤ 1, x ∈ K, j = 1, 2, , N, ϕj ∈ C0∞ (Rn ), supp ϕj ⊂ Uj , j = 1, 2, , N, ✈➔ N ϕj (x) = 1, x ∈ K j=1 ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ✶✳✷ ❚➼❝❤ ❝❤➟♣ ◆➳✉ f, g ∈ L1 (Rn ) t❛ ✤à♥❤ ♥❣❤➽❛ f ∗ g(x) = f (x − y)g(y)dy = f (y)g(x − y)dy Rn Rn ①→❝ ✤à♥❤ ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ❚❛ ❣å✐ f ∗ g ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ f t❤❡♦ ❤➔♠ g ✳ ❘ã r➔♥❣✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ f t❤❡♦ ❤➔♠ g ✈➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ g t❤❡♦ ❤➔♠ f ữ ứ ỵ õ |f ∗ g(x)| d(x) = f (x − y)g(y)dy dx Rn Rn |f (x − y)| dx dy ≤ f |g(y)| ≤ Rn Rn L1 (Rn ) g L1 (Rn ) ♥➯♥ f ∗ g ∈ L1 (Rn ) ✈➔ f ∗g L1 (Rn ) ≤ f L1 (Rn ) g L1 (Rn ) ❚ê♥❣ q✉→t✱ ✈ỵ✐ f ∈ L1 (Rn ), g ∈ Lp (Rn )(1 ≤ p ≤ ∞) t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣ ♥❤÷ s❛✉ f ∗g p ≤ f p g ✶✳✸ ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn) ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ❑❤æ♥❣ ❣✐❛♥ S (Rn) ❧➔ t➟♣ ❤ñ♣ S (Rn ) = {ϕ ∈ C ∞ (Rn ) : sup xα Dβ ϕ (x) < ∞ x∈Rn ✽ ∀α, β ∈ Zn+ } ❚❛ ❦➼ ❤✐➺✉ M := sup{|x| + |y| (x, y) ∈ s✉♣♣ψ}, −1 |x| ≤ λ α }, C2 := sup{x : C1 := sup{x : −1 λ α ≤ |x| ≤ 2M} D1 = {y ∈ M : |x − y| < |x + y|}; D2 = {y ∈ M : |x − y| ≥ |x + y|} −1 −1 α α B± ; B± := {y ∈ M : |x ± y| < λ := {y : |x ± y| ≥ λ } ✭✸✳✹✮ ❚❛ t❤➜② |K(x, y)|q dy = M |K(x, y)|q dy + |K(x, y)|q dy := I1 + I2 D1 D2 ❚r♦♥❣ ❝→❝ ✤→♥❤ ❣✐→ s❛✉✱ ♠ët ❤➡♥❣ sè ❝❤✉♥❣ C s ữủ sỷ tr tt ữợ t➼♥❤ ❝õ❛ K(x, y)✳ ❚❛ ❝â ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✸✳✶✳ ❈❤♦ x, y ∈ R ❧➔ ❝→❝ sè t❤ü❝✱ ✈➔ ❝❤♦ j ∈ N✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✭✶✮ ✭✷✮ ◆➳✉ j ❧➔ sè ❧➫✱ t❤➻ 2j−1 ≤ |xj − y j | |x − y|j ◆➳✉ j ❧➔ sè ❝❤➤♥✱ t❤➻ ❈❤ù♥❣ ♠✐♥❤✳ 2m k=0 ✭✸✳✻✮ ✣➛✉ t✐➯♥✱ t❛ ❝â ✿ m m−1 2k 2m−2k x y k=1 m = x = j 2 −1 ≤ j |xj − y j | |x2 − y | xk y 2m−k − (x2m + y 2m ) = = = ✭✸✳✺✮ x + x y x2k−1 y 2m−2k+1 +2 k=0 k=1 m−1 x 2k−2 2m−2k y +y k=1 m m 2k 2m−2k−2 x y k=1 x2k−2 y 2m−2k + 2xy k=0 m x2k−2 y 2m−2k + y (x + y)2 m 2k 2m−2k k=1 m x2k−2 y 2m−2k + 2xy k=1 x2k−2 y 2m−2k k=1 m x2k−2 y 2m−2k ≥ k=1 ≥0 ✷✼ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ 2m k=0 ❉➵ ❞➔♥❣ t❤➜② xk y 2m−k ≥ (x2m + y 2m ) x2m + y 2m ≥ ❉♦ ✤â 2m |x| + |y| xk y 2m−k ≥ k=0 • 2m |x| + |y| 2m ✭✸✳✼✮ ✭✸✳✽✮ ❈❤ù♥❣ ♠✐♥❤ ✭✸✳✺✮ ✈ỵ✐ j := 2m + 1✳ ❙û ❞ư♥❣ ✭✸✳✽✮✱ t❛ ❝â 2m |x 2m+1 −y 2m+1 xk y 2m−k ≥ |x − y| | = |x − y| k=0 ≥ |x − y| x−y 2m =2 x−y x2m + y 2m ✭✸✳✾✮ 2m+1 , t❤♦↔ ♠➣♥ ã ự ợ j := 2m + 2✳ ❙û ❞ö♥❣ ✭✸✳✽✮✱ t❛ t❤➜② m |x 2m+2 −y 2m+2 m+1 | = |(x ) m+1 − (y ) 2 x2k y 2m−2k | = |x − y | k=0 m−1 = |x2 − y | x2m + y 2m + x2k y 2m−2k k=1 2 ≥ |x − y | ≥2 x2m x2 − y 2 + y 2m x2 + y ≥ |x − y | 2 m ✭✸✳✶✵✮ m+1 ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❇➜t ✤➥♥❣ t❤ù❝ ✭✸✳✺✮ ❤♦➦❝ ✭✸✳✻✮ trð t❤➔♥❤ ✤➥♥❣ t❤ù❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y ✱ ❤♦➦❝ x = −y, ❍➺ q✉↔ ✸✳✶✳ ●✐↔ sû x, y ð ố t tỗ t ởt số C s❛♦ ❝❤♦    C , |x − y| ≤ C |xj − y j |   , |x − y | ✷✽ ♥➳✉ j ❧➔ sè ❧➫✳ ♥➳✉ j ❧➔ sè ❝❤➤♥✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❝â ♠ët sè ❞÷ì♥❣ δ s❛♦ ❝❤♦ |x| ≥ δ ✈➔ |y| ≥ δ ✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✾✮✱ ✭✸✳✶✵✮ t❛ ❝â✿ x2m + y 2m ≥ δ 2m |x − y|; x2m + y 2m |x2m+2 − y 2m+2 | ≥ |x2 − y | ≥ δ 2m |x2 − y | |x2m+1 − y 2m+1 | ≥ |x − y| t ự ữủ ữợ t C = −2m ✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❇ê ✤➲ t✐➳♣ t❤❡♦ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❦➳t q✉↔ ❝❤➼♥❤ ð ❜➔✐ ✈✐➳t ♥➔②✳ ❇ê ✤➲ ✸✳✷✳ ❈❤♦ ≤ p ≤ 2✱ r = 2q = p−1p ✳ ●✐↔ sû |K(x, y)|q dy)1/q , (sup CK,q = max{(sup x∈R |K(x, y)|q dx)1/q } < ∞ y∈R R R ◆➳✉ φ ∈ Lp(R)✱ t❤➻ Tλ φ 2 ≤ Tλ∗ Tλ φ r φ p ✭✸✳✶✶✮ ≤ CK,q φ 2p ◆â✐ r✐➯♥❣ ✶✳ ①→❝ ✤à♥❤ ♠ët t♦→♥ tû r➡♥❣ ❜✉ë❝ tø Lp(R) ✤➳♥ Lr (R) ✈ỵ✐ ♠ët ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❝❤✉➞♥ ❧➔ CK,q ✳ Tλ∗ Tλ ✷✳ ①→❝ ✤à♥❤ ♠ët t♦→♥ tû r➡♥❣ ❜✉ë❝ tø Lp(R) tỵ✐ L2(R) ✈ỵ✐ ♠ët ❣✐ỵ✐ ❤↕♥ tr➯♥ ❝õ❛ ❝❤✉➞♥ ❧➔ (CK,q )1/2✳ Tλ ❈❤ù♥❣ ♠✐♥❤✳ r = p + q ❚ø ❣✐↔ t❤✐➳t✱ t❛ s✉② r❛ r➡♥❣ ≤ q, r ≤ ∞, (1/r) + (1/p) = 1, ✈➔ − ❳➨t φ ∈ Lp (R)✳ ❇➡♥❣ ❝→❝❤ ✤ê✐ ❜✐➳♥ sè tr♦♥❣ t➼❝❤ ♣❤➙♥ ✈➔ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r ❝❤♦ (1/r) + (1/p) = 1, t❛ ❝â eiλS(x,y) ψ(x, y)φ(y)dy Tλ φ(x)dx Tλ φ(x)Tλ φ(x)dx = R R R eiλS(x,y) ψ(x, y)Tλ φ(x)dx φ(y)dy = R R Tλ∗ Tλ φ(y)φ(y)dy ≤ φ = p Tλ∗ Tλ φ r R ❉♦ ✤â ✱ Tλ φ 2 ≤ Tλ∗ Tλ φ ✷✾ r φ p ✭✸✳✶✷✮ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ t♦→♥ tû Tλ ✈➔ Tλ∗ t❛ t❤➜② (Tλ∗ Tλ φ)(x) = e−iλS(z,x) ψ(z, x)Tλ φ(z)dz R e−iλS(z,x) ψ(z, x) = R R e−iλ(S(z,x)−S(z,y)) ψ(z, x)ψ(z, y)dz dy φ(y) = eiλS(z,y) ψ(z, y)φ(y)dy dz R R ❚ø ✤â✱ t♦→♥ tû Tλ∗ Tλ ❧➔ ♠ët t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ K(x, y) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ ✭✸✳✶✸✮ e−iλ(S(z,x)−S(z,y)) ψ(z, x)ψ(z, y)dz K(x, y) = R ✈➔ (Tλ∗ Tλ φ)(x) = φ(y)K(x, y)dy R ❑➳t ❤ñ♣ ✭✸✳✸✮✱ ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ ❨♦✉♥❣ tr♦♥❣ ❝❤÷ì♥❣ ✶ ✈➔ Tλ∗ Tλ φ r ≤ CK,q φ r = p + q − t❛ ✤÷đ❝ ✭✸✳✶✹✮ p ❑➳t ❤đ♣ ✭✸✳✶✷✮ ✈➔ ✭✸✳✶✹✮ ✤÷đ❝ t❤♦↔ ♠➣♥ ✭✸✳✶✶✮✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ✸✳✷ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✈ỵ✐ ❤➔♠ ♣❤❛ ❧❛✐ ✤❛ t❤ù❝ ❚❛ ✤➦t Q(x,y) (z) := S(x, z) − S(y, z), ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ❤➔♠ ❝õ❛ ❜✐➳♥ z ∈ M ✈ỵ✐ ❝→❝ t❤❛♠ sè x, y ∈ M✳ ❚❛ q✉❛♥ t➙♠ ✤➳♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ tr♦♥❣ ❤➔♠ sè Q(x,y) (z)✳ dk1 Q (z) ≥ C1 |xj1 − y j1 | ✈ỵ✐ ❤➡♥❣ sè C1 > 0, dz k1 (x,y) ✈➔ t❤➯♠ ✈➔♦ ✤â Q(x,y) (z) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ ❦❤✐ k1 = dk2 Q(x,y) (z) ≥ C2 |xj2 − y j2 | ✈ỵ✐ ❤➡♥❣ sè C2 > 0, k dz ✈➔ t❤➯♠ ✈➔♦ ✤â Q(x,y) (z) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ ❦❤✐ k2 = ✭✸✳✶✺✮ ✭✸✳✶✻✮ ❚❛ ①→❝ ✤à♥❤ β t❤æ♥❣ q✉❛ ❝→❝ sè k1 , j1 , k2 , j2 ♥❤÷ s❛✉ β := ❤♦➦❝ ❧➔ ( kq2 − j2 ) − ( kq1 − j1 ) 2(j1 k2 − j2 k1 ) 1 β := max ; 2k1 2k2 ✸✵ ; ✭✸✳✶✼✮ ỵ p 2, ✈➔ r = 2q = p−1p ✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✭✶✮ ◆➳✉ ➼t ♥❤➜t ♠ët tr♦♥❣ ✷ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✺✮ ✈ỵ✐ j1 < kq ✱ ✈➔ t❤♦↔ ♠➣♥✱ t❤➻ Tλ Lp →L2 ✭✸✳✶✻✮ ✈ỵ✐ j2 < kq ữủ ợ j2 = kq ữủ = O λ−β , t↕✐ ✤â β ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❝æ♥❣ t❤ù❝ ✭✸✳✶✽✮✳ ✭✷✮ ◆➳✉ ➼t ♥❤➜t ♠ët tr♦♥❣ ✷ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✺✮ ✈ỵ✐ j1 = kq ✱ ✈➔ t❤♦↔ ♠➣♥✱ t❤➻ Tλ t↕✐ ✤â β ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ✭✸✮ ◆➳✉ ❝↔ ✭✸✳✶✺✮✲✭✸✳✶✻✮ ✭✸✳✶✽✮✳ ✤➲✉ ✤ó♥❣ ✈ỵ✐ ( kq Tλ t↕✐ ✤â β ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ✭✹✮ ◆➳✉ ❝↔ ✭✸✳✶✺✮✲✭✸✳✶✻✮ = O λ−β log λ , Lp →L2 − j1 )( kq2 − j2 ) < 0✱ Lp →L2 t❤➻ = O λ−β , ✭✸✳✶✼✮✳ ✤➲✉ ✤ó♥❣ ✈ỵ✐ ( kq − j1 )( kq2 − j2 ) = 0✱ t❤➻ Tλ t↕✐ ✤â β ✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t γ1 = k1 q , γ2 = O λ−β log λ , Lp →L2 ✭✸✳✶✽✮✳ = k2 q ❚❛ ❜✐➳t r➡♥❣✱ ❤↕t ♥❤➙♥ K(x, y) ❝õ❛ t♦→♥ tû Tλ T∗λ ✤÷❛ r❛ ❜ð✐ ✭✸✳✸✮ ♥❤÷ s❛✉ eiλ.Q(x,y) (z) ψ(x, z)ψ(y, z)dz K(x, y) = M ❈❤ù♥❣ ♠✐♥❤ ❝❤♦ ✭✶✮✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ tr÷í♥❣ ủ ữủ t ợ j1 < ⑩♣ ❞ö♥❣ ✭✸✳✶✺✮ ✈➔ ❜ê ✤➲ ❱❛♥❞❡r ❈♦r♣✉t✱ t❛ ❝â |K(x, y)| ≤ C λ k1 1 |xj1 − y j1 | k1 ❉♦ ✤â |K(x, y)|q ≤ C λ γ1 ✸✶ 1 |xj1 − y j1 | ã rữớ ủ j1 ❧➔ sè ❧➫✳ ❙û ❞ö♥❣ ❜ê ✤➲ ✸✳✶ t❛ ❝â C |K(x, y)|q dy ≤ λ M dy γ1 M ≤ j1 γ |xj1 − y | C λ dy γ1 ✭✸✳✷✵✮ j1 M |x − y| γ1 ❚❛ t❤➜②✱ ❞♦ j1 < γ1 ✱ ♥➯♥ dy M |x − y| dy ≤ j1 γ1 ≤ C j1 t γ1 2M ❑➳t ❤ñ♣ ✤✐➲✉ ợ t ữủ C |K(x, y)|q dy ã M rữớ ủ ✷✳ j1 ❧➔ sè ❝❤➤♥✳ ⑩♣ ❞ö♥❣ ✭✸✳✶✾✮ ✈➔ ❜ê ✤➲ ✸✳✶✱ t❛ ❝â |K(x, y)|q dy ≤ C λ M dy γ1 M x2 − y2 C ≤ j1 2γ1 λ dy γ1 x−y M j1 2γ1 x+y j1 2γ1 ✭✸✳✷✷✮ ❚❛ t❤➜②✱ ❞♦ j1 < γ1 ✱ ♥➯♥ dy M x−y j1 2γ1 x+y j1 2γ1 dy ≤ M |x − y| j1 γ1 dy + M |x − y| j1 γ1 dy ≤2 2M t j1 γ1 ≤ C ❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ✭✸✳✷✷✮✱ t❛ ♥❤➟♥ ✤÷đ❝ C |K(x, y)|q dy ≤ ✭✸✳✷✸✮ λ γ1 M ❚â♠ ❧↕✐✱ ✈ỵ✐ ♠å✐ j ∈ N✱ t❛ ❧✉æ♥ ❝â C |K(x, y)|q dy ≤ λ γ1 M ❚÷ì♥❣ tü ✤→♥❤ ❣✐→ ♥➔② ❝ơ♥❣ ✤ó♥❣ ❦❤✐ t❛ t❤❛② M |K(x, y)|q dy ❜ð✐ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✷ t❛ s✉② r❛ Tλ Lp →L2 ≤ C λ 2γ1 ✭✶✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ ✭✷✮✳ ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ sup |K(x, y)| ≤ |ψ(x, z)ψ(y, z)|dz ≤ C (x,y)∈R2 R ✸✷ M |K(x, y)|q dx t ủ ợ t ữủ −1 −1 |K(x, y)|q ≤ C 1, λ γ1 |xγ1 − y γ1 | γ1 • ✭✸✳✷✹✮ ∀x, y ∈ R ❚r÷í♥❣ ❤đ♣ ✶✳ γ1 ❧➔ sè ❧➫✳ ⑩♣ ❞ö♥❣ ✭✸✳✷✹✮✱ t❛ ❝â −1 |K(x, y)|q dy ≤ C B− M −1 λ γ1 |xγ1 − y γ1 | γ1 dy 1.dy + C B− ≤ C.λ − γ1 −1 |xγ1 − y γ1 | γ1 dy 1+ B− ❙û ❞ö♥❣ ❇ê ✤➲ ✸✳✶ t❛ ✤÷đ❝ −1 |x − y|−1 dy ≤ C log(λ) |xγ1 − y γ1 | γ1 dy ≤ C B− B− ❉♦ ✤â |K(x, y)|q dy ≤ C.λ log() M ã rữớ ủ ❧➔ sè ❝❤➤♥✳ ⑩♣ ❞ư♥❣ ✭✸✳✷✹✮✱ t❛ t❤✉ ✤÷đ❝ λ γ1 |xγ1 − y γ1 | γ1 dy 1dy + C B± M −1 −1 |K(x, y)|q dy ≤ C − B+ ∩B2 −1 − γ1 ≤ Cλ |xγ1 − y γ1 | γ1 dy 1+ − B+ ∩B2 ❚ø ❇ê ✤➲ ✸✳✶ t❛ ❝â −1 |xγ1 − y γ1 | γ1 dy ≤ C |x − y| − B+ ∩B2 −1 |x + y| −1 dy − B+ ∩B2 |x + y|−1 dy + C ≤C − B+ ∩B2 |x − y|−1 dy ≤ C log(λ) − B+ ∩B2 ❉♦ ✤â |K(x, y)|q dy ≤ C.λ − γ1 log(λ) M ❚â♠ ❧↕✐✱ ✈ỵ✐ ♠å✐ j ∈ N✱ t❛ ❧✉æ♥ ❝â C |K(x, y)|q dy ≤ log(λ) λ γ1 M ❚÷ì♥❣ tü ✤→♥❤ ❣✐→ ♥➔② ❝ơ♥❣ ✤ó♥❣ ❦❤✐ t❛ t❤❛② M |K(x, y)|q dy ❜ð✐ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✷ t❛ s✉② r❛ Tλ Lp →L2 ≤ C λ ✸✸ 2γ1 log (λ) M |K(x, y)|q dx✱ ✈➔ ✭✷✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ ✭✸✮✳ ❈â ❤❛✐ ❦❤↔ ♥➠♥❣ j1 < γ1 ✈➔ j2 > γ2 ✱ ❤♦➦❝ j1 > γ1 ✈➔ j2 < γ2 ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ tr÷í♥❣ ❤đ♣ ❦❤✐ j1 < γ1 ✈➔ j2 > γ2 ✱ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❝â t❤➸ ❝♦✐ ❧➔ t÷ì♥❣ tü✳ ❚❤❡♦ ❣✐↔ ✤à♥❤ ✭✸✳✶✺✮✲✭✸✳✶✻✮✿ |Q(γ1 ) (z)| ≥ C1 |xj1 − y j1 |; |Q(γ2 ) (z)| ≥ C2 |xj2 − y j2 |, ♥➯♥ →♣ ❞ö♥❣ ❇ê ✤➲ ❱❛♥❞❡r ❈♦r♣✉t✱ t❛ ♥❤➟♥ ✤÷đ❝ −1 −1 −1 −1 |K(x, y)| ≤ C λ k1 |xj1 − y j1 | k1 , λ k2 |xj2 − y j2 | k2 ❉♦ ✤â −1 −1 −1 −1 ✭✸✳✷✺✮ |K(x, y)|q ≤ C λ γ1 |xj1 − y j1 | γ1 , λ γ2 |xj2 − y j2 | γ2 ❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✭✸✳✷✻✮ |K(x, y)|q dy ≤ Cλ−2β M ❜➡♥❣ ❝→❝❤ ❝❤✐❛ ❧➔♠ ✹ tr÷í♥❣ ❤đ♣✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔②✱ t❛ sû ❞ö♥❣ ❇ê ✤➲ ✸✳✶ ✤➸ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ s❛✉ −1 |K(x, y)|q ≤ C λ γ1 |x ± y| −j1 γ1 −1 , λ γ2 |x ± y| −j2 γ2 ✭✸✳✷✼✮ , ✈ỵ✐ ♠å✐ x ∈ M✱ ✈➔ ✈ỵ✐ ♠å✐ y tr♦♥❣ D1 ✈➔ D2 , ❞ü❛ tr➯♥ t➼♥❤ ❝❤➤♥ ❤♦➦❝ ❧➫ ❝õ❛ j1 , j2 ✳ ❈ư t❤➸ ♥❤÷ s ã rữớ ủ j1, j2 số ❑➳t ❤ñ♣ ✭✸✳✷✺✮ ✈➔ ❇ê ✤➲ ✸✳✶ t❛ ❝â −1 |K(x, y)|q ≤ C λ γ1 |x − y| −j1 γ1 −1 ; λ γ2 |x − y| −j2 γ2 ✭✸✳✷✽✮ ◆➯♥ t❛ ❝â |K(x, y)|q dy = M |K(x, y)|dy + B− |K(x, y)|dy B− −1 |x − y| ≤ C λ γ1 −j1 γ1 −1 |x − y| dy + λ γ2 B− −1 ≤ C λ γ1 B− |t| −j1 γ1 −1 C1 −1 ≤ C λ γ1 λ |t| dt + λ γ2 −j2 γ2 C2 −(γ1 −j1 ) α.γ1 −1 + λ γ2 λ ✸✹ −(γ2 −j2 ) α.γ2 dt −j2 γ2 dy ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ − |K(x, y)|q dy ≤ C λ α+γ1 −j1 α.γ1 +λ α+γ2 −j2 α.γ2 − ✭✸✳✷✾✮ M ❚❛ ✤→♥❤ ❣✐→ tr♦♥❣ ✭✸✳✷✾✮ ❜➡♥❣ ❝→❝❤ ❝❤å♥ α t❤ä❛ ♠➣♥ − λ ❉♦ ✤â α= α+γ1 −j1 α.γ1 j1 γ2 − j2 γ1 ; γ2 − γ1 − =λ α+γ2 −j2 α.γ2 α + γ1 − j1 α + γ2 − j2 = α.γ1 α.γ2 2β := ❱➻ ✈➟②✱ |K(x, y)|q dy C2 M ã rữớ ủ j1, j2 ❧➔ sè ❝❤➤♥✳ ❙û ❞ö♥❣ ❇ê ✤➲ ❱❛♥❞❡r ❈♦r♣✉t ✈➔ ❇ê ✤➲ ✸✳✶ t❛ ❝â −j1 −1 −j2 −1 ✭✸✳✸✵✮ |K(x, y)|q ≤ C λ γ1 |x2 − y | 2γ1 ; λ γ2 |x2 − y | 2γ2 , ✈ỵ✐ ♠å✐ x, y ∈ M ❱ỵ✐ y ∈ D1 t❛ ❝â ✤→♥❤ ❣✐→ −1 |K(x, y)|q ≤ C λ γ1 |x − y| −j1 γ1 −1 ; λ γ2 |x − y| −j2 γ2 ❑❤✐ ✤â |K(x, y)|q dy = |K(x, y)|dy + D1 ∩B− D1 |K(x, y)|dy D1 ∩B− −1 ≤ C λ γ1 |x − y| −j1 γ1 −1 B− −1 −j1 γ1 −1 −1 |t| dt + λ γ2 C1 ≤ C λ γ1 λ −j2 γ2 dy B− |t| ≤ C λ γ1 |x − y| dy + λ γ2 −j2 γ2 dt C2 −(γ1 −j1 ) α.γ1 −1 + λ γ2 λ −(γ2 −j2 ) α.γ2 ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ − |K(x, y)|q dy ≤ C λ α+γ1 −j1 α.γ1 B1 ❚è✐ ÷✉ ❤â❛ ✤→♥❤ ❣✐→ tr➯♥✱ t❛ ✤÷đ❝ I1 ≤ Cλ−2β ✸✺ − +λ α+γ2 −j2 α.γ2 ✭✸✳✸✶✮ ❱ỵ✐ y ∈ D2 t❛ t❤➜② −1 |K(x, y)|q ≤ C min{λ γ1 |x + y| −j1 γ1 −1 ; λ γ2 |x + y| −j2 γ2 } ❑❤✐ ✤â |K(x, y)|q dy = |K(x, y)|dy + |K(x, y)|dy D2 ∩B− D2 D2 ∩B− −1 |x + y| ≤ C λ γ1 −j1 γ1 −1 B− −1 ≤ C λ γ1 −1 −j2 γ2 dy B− |t| −j1 γ1 −1 |t| dt + λ γ2 C1 ≤ C λ γ1 λ |x + y| dy + λ γ2 −j2 γ2 dt C2 −(γ1 −j1 ) α.γ1 −1 + λ γ2 λ −(γ2 −j2 ) α.γ2 ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ − |K(x, y)|q dy ≤ C λ α+γ1 −j1 α.γ1 − +λ α+γ2 −j2 α.γ2 ✭✸✳✸✷✮ B2 ❚è✐ ÷✉ ❤â❛ ✤→♥❤ ❣✐→ tr➯♥ t❛ ✤÷ñ❝ I2 ≤ Cλ−2β ❉♦ ✤â |K(x, y)|q dy ≤ I1 + I2 C2 M ã rữớ ủ j1 sè ❧➫✱ j2 ❧➔ sè ❝❤➤♥✳ ❚ø ❇ê ✤➲ ✸✳✶ t❛ ❝â j2 j2 xj2 − y j2 ≥ C|x − y| |x + y| , ✈ỵ✐ ♠å✐ x, y ∈ N✱ ✈ỵ✐ y ∈ D1 t❛ ❝â |K(x, y)|q ≤ C λ − γ1 j − γ1 |x − y| ;λ − γ1 j − γ2 |x − y| ❚÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✷✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ I1 ≤ Cλ−2β ❱ỵ✐ y ∈ D2 ✱ t❛ ❝â |K(x, y)|q ≤ C λ − γ1 j − γ1 |x − y| ✸✻ ;λ − γ1 j − γ2 |x + y| ❚÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✷ t❛ ❝â I2 ≤ Cλ−2β ❉♦ ✈➟② |K(x, y)|q dy ≤ I1 + I2 C2 M ã rữớ ủ j1 ❧➔ sè ❝❤➤♥✱ ✈➔ j2 ❧➔ sè ❧➫✳ ❇➡♥❣ ✈✐➺❝ ❤♦→♥ ✤ê✐ ✈❛✐ trá ❝õ❛ j1 ✈➔ j2 t❛ ❝â t❤➸ tr❛♥❤ ❝ù ❝❤♦ tr÷í♥❣ ❤đ♣ ♥➔② t÷ì♥❣ tü ♥❤÷ rữớ ủ õ ợ tt trữớ ❤ñ♣ ❝õ❛ j1 , j2 t❛ ❝â |K(x, y)|q dy ≤ I1 + I2 ≤ Cλ−2β M ❚÷ì♥❣ tü ✤→♥❤ ❣✐→ ♥➔② ❝ơ♥❣ ✤ó♥❣ ❦❤✐ t❛ t❤❛② |K(x, y)|q dy ❜ð✐ M M |K(x, y)|q dx✱ ✈➔ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✷ t❛ s✉② r❛ Tλ Lp →L2 C ≤ λ 2γ1 ✭✸✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ù♥❣ ♠✐♥❤ ✭✹✮ ❈❤ó♥❣ t❛ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ tr÷í♥❣ ❤đ♣ j1 < γ1 ✈➔ j2 = γ2 ✈➔ ❝→❝ tr÷í♥❣ ❤đ♣ ỏ õ t ữủ ự tữỡ tỹ ã ❚r÷í♥❣ ❤đ♣ ✶✳ j1, j2 ❧➔ sè ❧➫✳ ❑❤✐ ✤â −1 |K(x, y)|q ≤ C λ γ1 |x − y| −j1 γ1 −1 ; λ γ2 |x − y| −j2 γ2 ✣✐➲✉ ♥➔② ❝❤♦ t❛ −1 |K(x, y)|q dy ≤ C λ γ1 |x − y| −j1 γ1 −1 B− M |x − y| dy + λ γ2 B− −1 |x − y|−1 dy ≤ C λ−2β + λ γ2 B− = C λ−2β + λ ✈➻ −1 γ2 log λ ≤ Cλ−2β log(λ), α + γ2 − j2 = 2β αγ2 ❉♦ ✤â |K(x, y)|q dy ≤ Cλ−2β log λ M ✸✼ −j2 γ2 dy ❚÷ì♥❣ tü ✤→♥❤ ❣✐→ ♥➔② ❝ơ♥❣ ✤ó♥❣ ❦❤✐ t❛ t❤❛② M |K(x, y)|q dy ❜ð✐ M |K(x, y)|q dx✱ ✈➔ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✷ t❛ s✉② r❛ Tλ Lp →L2 ≤ C λ 2γ1 log λ ❦➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✸✳✷ t❛ ❝❤ù♥❣ ♠✐♥❤ ữủ trữớ ủ ãrữớ ủ j1, j2 ❈→❝ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✈➔ t✐➳♥ ❤➔♥❤ t÷ì♥❣ tü t❤❡♦ ❝→❝ tr÷í♥❣ ❤đ♣ ð ✭✸✮✱ ♥❤÷♥❣ ❝→❝ ②➳✉ tè ❧♦❣❛r✐t ♥➯♥ ✤÷đ❝ t❤➯♠ ✈➔♦ tr♦♥❣ ữợ t ỵ ữủ ự ự ữủ t ú ỵ rồ t ỵ õ ợ ❣✐↔ t❤✐➳t ✭✸✳✶✺✮✲ ✭✸✳✶✻✮ t❤ä❛ ♠➣♥ ✈ỵ✐ ( kq − j1 )( kq2 − j2 ) ≤ β− 2k1 ❇➡♥❣ ♣❤➨♣ t➼♥❤ ✤ì♥ ❣✐↔♥ t❛ ❝â (k2 − k1 )2 ( kq1 − j1 )( kq2 − j2 ) ≤ β− = 2k2 4k1 k2 (j1 k2 − j2 k1 )2 ✣✐➲✉ ✤â ♥❣❤➽❛ ❧➔✿ 1 ; 2k1 2k2 ≤ β ≤ max 1 ; 2k1 2k2 ≤ ✭✸✳✸✸✮ ❉♦ ✤â✱ ♥➳✉ ✷ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✺✮✲✭✸✳✶✻✮ ❝â t❤➸ ❦➳t ❤đ♣✱ tè❝ ✤ë ❤ë✐ tư ❝õ❛ ❝❤✉➞♥ t♦→♥ tû tỵ✐ ✤÷đ❝ ✤→♥❤ ❣✐→ ❝❤➼♥❤ ①→❝ ❤ì♥✳ ❚❤❡♦ ✤â ❦➳t q tự ỗ t ữủ ð ✣à♥❤ ❧➼ ✸✳✶✳ ❚❛ ①➨t ❝→❝ ❤➔♠ ♣❤❛ ✤❛ tự ỗ t ữợ k0 S1 (x, y) = a2j x2n−2j y 2j ; ✭✸✳✸✹✮ a2j+1 x2n−2j+1 y 2j+1 ; ✭✸✳✸✺✮ a2j x2n−2j+1 y 2j ✭✸✳✸✻✮ j=j0 k0 S2 (x, y) = j=j0 k0 S3 (x, y) = j=j0 Ð ✤➙② ❝â t❤➸ ❣✐↔ ✤à♥❤ r➡♥❣ a2j0 a2k0 = 0, ✈➔ a2j0 +1 a2k0 +1 = ●✐↔ sû ♠å✐ ❤➺ sè tr♦♥❣ S(x, y) ❝ò♥❣ ❞➜✉✱ tù❝ ❧➔✱ a a +2 ≥ ✈ỵ✐ t➜t ❝↔ ✸✽ = j0 , ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ r➡♥❣ tø♥❣ ✤❛ t❤ù❝ S1 (x, y), S2 (x, y), ✈➔ S3 (x, y) t❤ä❛ ♠➣♥ ✷ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✺✮ ✈➔ ✭✸✳✶✻✮✳ ❈ö t❤➸✱ S1 (x, y), S2 (x, y), ✈➔ S3 (x, y) ❝ò♥❣ ♥❤❛✉ t❤ä❛ ♠➣♥✭✸✳✶✺✮✲✭✸✳✶✻✮ t❤❡♦ ❝→❝ ❝➦♣ ✤æ✐ (2j0 , 2n − 2j0 )✲(2k0 , 2n − 2k0 ), (2j0 + 1, 2n − 2j0 + 1)✲(2k0 + 1, 2n − 2k0 + 1), ✈➔ (2j0 , 2n − 2j0 + 1) − (2k0 , 2n − 2k0 + 1), t÷ì♥❣ ù♥❣✳ ✣➸ S ❜✐➸✉ t❤à t➟♣ ❤ñ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ỗ t õ õ số ũ ❞➜✉✳ ❇✐➸✉ t❤à ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ tr➯♥ ❜ð✐ δ ✱ t❤➻ ❦❤✐ ✤â δ = 2n ❝❤♦ ✭✸✳✸✹✮✲✭✸✳✸✺✮✱ ❤♦➦❝ δ = 2n + ❝❤♦ ✭✸✳✸✻✮✳ ❍➺ q✉↔ ✸✳✷✳ ❚♦→♥ tû Tλ ✈ỵ✐ ❤➔♠ ♣❤❛ S(x, y) ∈ S ❧➔ L2 ❜à ❣✐ỵ✐ ❤↕♥ ✈ỵ✐ ✤à♥❤ ♠ù❝ q✉② ✤à♥❤ ữ s ợ j0 < n/2 < k0 ≤ n✱ ❤➔♠ ♣❤❛ ✭✸✳✸✺✮ ✈ỵ✐ ≤ j0 (n − 1)/2 < k0 ≤ n✱ ✈➔ ❤➔♠ ♣❤❛ ✭✸✳✸✻✮ ✈ỵ✐ ≤ j0 < (n + 1)/2 < k0 ≤ n✱ t❛ ❝â ✭✸✳✸✹✮ Tλ ✭✷✮ < = O λ−1/δ ❍➔♠ ♣❤❛ ✭✸✳✸✹✮ ✈ỵ✐ ♠ët tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ ❤♦➦❝ ❧➔ ≤ j0 = n/2 < k0 ≤ n, ❤♦➦❝ ≤ j0 < n/2 = k0 ≤ n; ❤➔♠ ♣❤❛ ✭✸✳✸✺✮ ✈ỵ✐ ♠ët tr♦♥❣ ❤❛✐ tr÷í♥❣ ❤đ♣ ❤♦➦❝ ❧➔ ≤ j0 = (n − 1)/2 < k0 ≤ n, ♦r ≤ j0 < (n − 1)/2 = k0 ≤ n; ❚❛ ❝â Tλ = O λ−1/δ log (λ) ❱➼ ❞ö ✸✳✶✳ ●✐↔ sû S(x, y) t❤ä❛ ♠➣♥ ❣✐↔ ✤à♥❤ ♠➔ t↕✐ ✤â (j1, k1) = (m − 1, 1); (j2 , k2 ) = (1, n − 1)✳ ❚❛ t❤➜② r➡♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❤ä❛ ♠➣♥ ỵ õ ổ tự ✭✸✳✶✼✮ ❝❤♦ β= ✭✸✳✶✺✮✲✭✸✳✶✻✮✱ (j2 − k2 ) − (j1 − k1 ) m+n−4 = , 2(k1 j2 − k2 j1 ) 2(mn − m − n) ✤➙② ❝❤➼♥❤ ❧➔ tè❝ ✤ë ✤÷đ❝ t❤✐➳t ❧➟♣ tr♦♥❣ ❬✺✱ ✻❪✳ ✸✾ ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ỗ ã t t ỡ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣✱ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (Rn ) ✈➔ ♠ët sè ✤à♥❤ q trồ rr ã ợ ❧÷đ♥❣ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❙t❡✐♥✲❲❛✐♥❣❡r t❤ỉ♥❣ q✉❛ ❜➟❝ ❝õ❛ tự ã ữ r t tû ❞❛♦ ✤ë♥❣ tø ❦❤æ♥❣ ❣✐❛♥ Lp (R) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ L2 (R)✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✵ t ỵ tt ❤➔♠ s✉② rë♥❣ ✈➔ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✳ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✭✷✵✶✺✮✳ ❬✷❪ ✣✐♥❤ ❚❤➳ ▲ö❝✱ P❤↕♠ ❍✉② ✣✐➸♥✱ ❚↕ ❉✉② P❤÷đ♥❣✱ ❜✐➳♥✳◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✭✷✵✵✷✮✳ ●✐↔✐ t➼❝❤ ❝→❝ ❤➔♠ ♥❤✐➲✉ ◆♦r♠ ❞❡❝❛② r❛t❡s ♦❢ ♦s❝✐❧❧❛t♦r② ✐♥t❡❣r❛❧s ♦♣❡r❛t♦rs ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ♣❤❛s❡s ❛❝t✐♥❣ ❜❡t✇❡❡♥ Lp ❛♥❞ L2 s♣❛❝❡s✳ Pr❡♣✐♥t✳ ❬✸❪ P✳ ❑✳ ❆♥❤✱ ❱✳ ◆✳ ❍✉② ❛♥❞ ◆✳ ▼✳ ❚✉❛♥✱ ❬✹❪ ■✳ ❘✳P❛r✐ss✐s ✱❆ s❤❛r♣ ❜♦✉♥❞ ❢♦r t❤❡ ❙t❡✐♥ ❲❛✐♥❣❡r ♦s❝✐❧❧❛t♦r② ✐♥t❡❣r❛❧ ✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱❱✳✶✸✻✭✷✵✵✽✮✱♥✳✸✱♣✳✾✻✸✲✾✼✷✳ ❬✺❪ ❉✳ ❍✳ P❤♦♥❣ ❛♥❞ ❊✳ ▼✳ ❙t❡✐♥✱ ▼❛t❤✳ ✶✶✵✱ ✸✾✲✻✷ ✭✶✾✾✷✮✳ ❬✻❪ ❉✳ ❍✳ P❤♦♥❣ ❛♥❞ ❊✳ ▼✳ ❙t❡✐♥✱ ❖s❝✐❧❧❛t♦r② ✐♥t❡❣r❛❧s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ♣❤❛s❡s✳ ■♥✈✳ ▼♦❞❡❧s ♦❢ ❉❡❣❡♥❡r❛t❡ ❋♦✉r✐❡r ■♥t❡❣r❛❧ ❖♣❡r❛t♦rs ❛♥❞ ❘❛❞♦♥ ❚r❛♥s❢♦r♠s✳ ❆♥♥✳ ♦❢ ▼❛t❤✳✱ ✶✹✵✱ ✼✵✸✕✼✷✷ ✭✶✾✾✹✮✳ ✹✶ ... KHOA HỌC TỰ NHIÊN - Nguyễn Thị Xâm MỘT SỐ KẾT QUẢ VỀ TÍNH BỊ CHẶN CỦA TÍCH PHÂN DAO ĐỘNG Chun ngành: Tốn Giải Tích Mã số: 8460101.02 LUẬN VĂN THẠC SĨ KHOA HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC:TS... ①✉➜t ❤✐➺♥ ❝æ♥❣ tr➻♥❤ ❚❤➨♦r✐❡ ❆♥❛❧②t✐q✉❡ ❞❡ ❧❛ ❈❤❛❧❡✉r ❝õ❛ s rr t ỵ tt ữỡ tr r số ỵ tt st ỵ tt số t q✉❛♥❣ ❤å❝✱ ➙♠ ❤å❝✱ ❝ì ❤å❝ ❧÷đ♥❣ tû✱✳✳✳ ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼❝❤... sè ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❜➟❝ d ❝õ❛ ✤❛ t❤ù❝ P (x)✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷ ♥➔② ❞ü❛ tr➯♥ t số ữợ t ỵ d N õ tỗ t số ữỡ c1 ổ ♣❤ö t❤✉ë❝ ✈➔♦ d s❛♦ ❝❤♦ eiP (x) c1 log d ≤ sup p.v P ∈Pd R dx x rữợ