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✐ ✱ ✈➻ sü ❣✐ó♣ ✤ï✱ ❝❤➾ ❜↔♦ t➟♥ t➻♥❤✱ ũ ỳ ổ ũ ỵ ❚❤➛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❤ö ❦✐➳♥ t❤ù❝ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❈❛♦ ❤å❝✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥✱ ❦❤✉②➳♥ ❦❤➼❝❤✱ ❣✐ó♣ ✤ï tỉ✐ r➜t ♥❤✐➲✉ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤å❝ t➟♣✳ ▼➦❝ ❞ò ✤➣ ❝è ❣➢♥❣ r➜t ♥❤✐➲✉ ✈➔ ♥❣❤✐➯♠ tó❝ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ữ ợ q ợ ổ t ự ❦❤♦❛ ❤å❝ ✈➔ ❝á♥ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ t❤ü❝ ❤✐➺♥ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sõt ữủ ỵ õ ❣â♣ ❝õ❛ ❝→❝ t❤➛② ❝æ ✈➔ ❝→❝ ❜↕♥ ✤➸ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❍➔ ◆ë✐✱ ♥➠♠ ✷✵✶✾ ✷ ▼ð ✤➛✉ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ t❤✉ ❤ót ♥❤✐➲✉ sü q✉❛♥ t➙♠ ❝õ❛ ❝→❝ ♥❤➔ ❚♦→♥ ❤å❝ ✈➔ ❝→❝ t ỵ tứ t ổ tr r ❆♥❛❧②t✐q✉❡ ❞❡ ❧❛ ❈❤❛❧❡✉r ❝õ❛ ❏♦s❡♣❤ ❋♦✉r✐❡r ✈➔♦ ♥➠♠ ✶✽✷✷✳ t ỵ tt ữỡ tr r số ỵ tt st ỵ tt sè❀ ❝→❝ ❜➔✐ t♦→♥ ✈➲ q✉❛♥❣ ❤å❝✱ ➙♠ ❤å❝✱ ❝ì ❤å❝ ❧÷đ♥❣ tû✱✳✳✳ ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣✳ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✤➣ ✈➔ ✤❛♥❣ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ♥❤✐➲✉ ù♥❣ ❞ư♥❣ ❦❤→❝ ♥❤❛✉ ✈➔ t❤✉ ❤ót ✤÷đ❝ ♥❤✐➲✉ sü q✉❛♥ t➙♠ tø ❝→❝ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ❬✸✲✻❪✳ ◆❤✐➲✉ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ rt ộ ỹ ữợ t trỹ t trà t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ✈➔ tè❝ ✤ë s✉② ❣✐↔♠ ❝õ❛ ❝❤✉➞♥ ❝õ❛ ❚➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ✭①❡♠ ❬✸✱ ✺✱ ✻❪ ✮✳ ◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ❜❛ ❝❤÷ì♥❣✿ ✳ ❈❤÷ì♥❣ ♥➔② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣ ✈➔ ♠ët sè ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (R ) ✈➔ L (R )✳ ✳ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ✈➲ ✈✐➺❝ ✤→♥❤ ❣✐→ ❝➟♥ tr➯♥ ✈➔ ữợ t ý I() = dx , x e R ữợ ữủ tr ữợ tổ q ✤❛ t❤ù❝ P (x)✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✹❪✳ ✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ t➻♠ ❤✐➸✉ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣ ❋♦✉r✐❡r ❞↕♥❣✿ (T φ)(x) = e ψ(x, y)φ(y)dy, 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➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✳ Ω Ω {(Ω , ϕ )} R R ϕ R Ω {Ω } Ω, Ω ⊂ U ≤ ϕ (x) ≤ 1, x ∈ Ω, j = 1, 2, , Ω ϕ (x) = 1, x ∈ Ω ϕ ∈ C (R ), supp ϕ ⊂ Ω , j = 1, 2, , {ϕ } {Ω } Ω ❚❛ õ ỵ s ỡ R {U } { } {U } rữợ ự ỵ t t : R R ❧➔ ❤➔♠ ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✿ ρ(x) := Ce 0, , ♥➳✉ x < ♥➳✉ x ≥ tr♦♥❣ ✤â✱ C ❧➔ ❤➡♥❣ sè s❛♦ ❝❤♦ ρ(x)dx = R ✹ ❍➔♠ ρ ❝â ❝→❝ t➼♥❤ ❝❤➜t ✿ ρ ∈ C (R ), s✉♣♣ρ = B[0, 1] = x ∈ R x ≤ , ρ(x) ≥ 0, ρ(x)dx = 1, R ✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ x ✳ ❱ỵ✐ ♠é✐ > 0✱ t❛ ①➨t ❤➔♠ ρ ♥❤÷ s❛✉ ρ = ρ x ❍➔♠ ρ ❝ơ♥❣ ❝â ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ ρ, ❝ư t❤➸ ❧➔ ρ ∈ C (R ), s✉♣♣ρ = B[0, ] = x ∈ R x ≤ , ρ (x) ≥ 0, ρ (x)dx = 1, R ✈➔ ρ ❧➔ ❤➔♠ ❝❤➾ ♣❤ư t❤✉ë❝ ✈➔♦ x ✳ ❱ỵ✐ ♠é✐ ❤➔♠ f ∈ L (R )✱ ✤➦t f (x) = (f ∗ ρ ) (x) = f (y)ρ (x − y)dy R ❱✐➺❝ ✤➦t ♥➔② ❝â ♥❣❤➽❛ ✈➻ f (y)ρ (x − y)dy = R f (x − y)ρ (y)dy = f (y)ρ (x − y)dy R f ∈ L (R ) f ∈ C (R ) supp f = K ⊂ R f ∈ C (R ) supp f ⊂ K K = K + B[0, ] = x ∈ R d(x, K) ≤ f ∈ C(R ), lim sup |f (x) − f (x)| = 0, K ⊂ R ✭✐✮ ❉➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ tø ✤➥♥❣ t❤ù❝ s❛✉ f (y)ρ (x − y)dy D f (y)D ρ (x − y)dy = R R ✭✐✐✮ ❉♦ supp f = K ♥➯♥ f (y)ρ (x − y)dy = f (x) R f (y)ρ (x − y)dy R ❱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 R ✺ (f (x − y) − f (x)) p(y)dy = ♥➯♥ |f (x) − f (x)| ≤ sup |f (x − y) − f (x)| ▼➔ f ∈ C(R ) ♥➯♥ f ❧✐➯♥ tö❝ ✤➲✉ tr➯♥ tø♥❣ t➟♣ ❝♦♠♣❛❝t K ⊂ R ✳ ❉♦ ✤â lim sup |f (x) − f (x)| = 0, K ⊂ R ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ K⊂R ≤ ϕ(x) ≤ ∀x ∈ R supp ϕ ⊂ K ϕ(x) = 1, ∀x ∈ K ❳➨t χ(x) ❧➔ ❤➔♠ ✤➦❝ tr÷♥❣ ❝õ❛ t➟♣ K χ(x) := 1, ♥➳✉ x ∈ K 0, ♥➳✉ x ∈ /K ❈â χ ∈ L (R ) ⊂ L (R ), supp χ = K χ∗ρ ϕ ∈ C (R ) >0 ∈ C (R ), supp(λ ∗ ρ ✱ tù❝ ❧➔ , ✱ ♥➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶ ❝â ) ⊂ K , ≤ (χ ∗ ρ )(x) ∀x ∈ R ▼➔ (χ ∗ ρ )(x) = χ(x − y)ρ (y)dy ∀x ∈ R , ♥➯♥ (χ ∗ ρ )(x) ≤ ρ (y)dy = (χ ∗ ρ )(x) = ρ (y)dy = 1, x ∈ K ✈➔ ◆❤÷ ✈➟② ❤➔♠ ❝➛♥ t➻♠ ❧➔ ϕ(x) = χ ∗ ρ (x) ✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❚ø ❣✐↔ t❤✐➳t K ❧➔ t➟♣ ❝♦♠♣❛❝t✱ {U } K t❛ ❝â W := K \ ∪ ✻ U ⊂U ởt tỗ t > s❛♦ ❝❤♦ W ⊂ W + B(0, ) ⊂ U ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ❤➔♠ ψ ∈ C (R ; [0; 1]) s❛♦ ❝❤♦ V := W + B(0, ▲↕✐ ❝â✱ W := K \ ∪ ) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U , ψ (x) = 1, x ∈ V ⊂ V ♠➔ V ❧➔ t➟♣ ♠ð ♥➯♥ U W := K \ V õ tỗ t ∈C U ⊂U > s❛♦ ❝❤♦ W ⊂ W + B(0, ) ⊂ U ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷✱ ❝â ♠ët ❤➔♠ (R ; [0; 1]) s❛♦ ❝❤♦ V := W + B(0, ) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U , ψ (x) = 1, x ∈ V ❈ù ♥❤÷ t❤➳ t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❞➣② ❝→❝ ❤➔♠ {ψj} ψ ∈C (R ; [0; 1]) , V := W + B(0, ψ (x) = 1, x ∈ V , ✈➔ ❝→❝ t➟♣ {V , W } ) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U ψ (x) > 0, x ∈ ∪ V (⊃ K) , ✈➔ ψ (x) < N + 1, x ∈ R ❈â K V tỗ t số > s ❝❤♦ K ⊂ K + B(0, ) ⊂ ∪ V ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷ ❝â ❤➔♠ ❦❤æ♥❣ ➙♠ φ t❤ä❛ ♠➣♥ φ ∈ C (R ), K ⊂ K + B(0, /2) ⊂ s✉♣♣φ ⊂ K + B(0, ) ⊂ ∪ V, ✈➔ ≤ φ(x) ≤ 1, x ∈ R , φ(x) = 1, x ∈ K + B(0, /2) ✣➦t ψ (x) ϕ (x) := φ(x) ψ (x) + (1 − φ(x)) N + − ✼ t❤ä❛ ♠➣♥ ψ (x) ❝â ≤ ϕ (x) ≤ 1, x ∈ K, j = 1, 2, , N, ϕ ∈ C (R ), supp ϕ ⊂ U , j = 1, 2, , N, ✈➔ ϕ (x) = 1, x ∈ K ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ◆➳✉ f, g ∈ L (R ) t❛ ✤à♥❤ ♥❣❤➽❛ f ∗ g(x) = f (x − y)g(y)dy = R f (y)g(x − y)dy R ①→❝ ✤à♥❤ ✈ỵ✐ ♠å✐ x ∈ R ✳ ❚❛ ❣å✐ f ∗ g ❧➔ t➼❝❤ ❝❤➟♣ ❝õ❛ ❤➔♠ f t❤❡♦ ❤➔♠ g ✳ ❘ã r➔♥❣✱ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔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 R R |f (x − y)| dx dy ≤ f |g(y)| ≤ R g R R R ♥➯♥ f ∗ g ∈ L (R ) ✈➔ f ∗g R ≤ f R g R ❚ê♥❣ q✉→t✱ ✈ỵ✐ f ∈ L (R ), g ∈ L (R )(1 ≤ p ≤ ∞) t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❨♦✉♥❣ ♥❤÷ s❛✉ f ∗g ≤ f g S Rn S (R ) S (R ) = {ϕ ∈ C (R ) : sup x D ϕ (x) < ∞ R ✽ ∀α, β ∈ Z } ❚❛ ❦➼ ❤✐➺✉ M := sup{|x| + |y| (x, y) ∈ s✉♣♣ψ}, C := sup{x : }, C := sup{x : |x| ≤ λ ≤ |x| ≤ 2M} λ D = {y ∈ M : |x − y| < |x + y|}; D = {y ∈ M : |x − y| ≥ |x + y|} B := {y ∈ M : |x ± y| < λ ; B := {y : |x ± y| ≥ λ } ✭✸✳✹✮ ❚❛ t❤➜② |K(x, y)| dy = |K(x, y)| dy + M D |K(x, y)| dy := I + I D ❚r♦♥❣ ❝→❝ ✤→♥❤ ❣✐→ s❛✉✱ ♠ët ❤➡♥❣ sè ❝❤✉♥❣ C s➩ ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ t➜t ❝↔ ❝→❝ ÷ỵ❝ t➼♥❤ ❝õ❛ K(x, y)✳ ❚❛ ❝â ❜ê ✤➲ s❛✉✳ x, y ∈ R ✭✶✮ ✭✷✮ j∈N j ≤ |x − y | |x − y| ✭✸✳✺✮ ≤ |x − y | |x − y | ✭✸✳✻✮ j ✣➛✉ t✐➯♥✱ t❛ ❝â ✿ − (x x y +y )= = = x x y +y x y = x x y +y x = (x + y) x y + x y x y +2 ≥ ✷✼ y x y + 2xy x y + 2xy x y ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ≥ (x x y ❉➵ ❞➔♥❣ t❤➜② x +y +y |x| + |y| ≥ ✭✸✳✼✮ ) ❉♦ ✤â ✭✸✳✺✮ • |x |x| + |y| ≥ x y ✭✸✳✽✮ j := 2m + ❙û ❞ö♥❣ ✭✸✳✽✮✱ t❛ ❝â −y | = |x − y| ≥ |x − y| ≥ |x − y| x y x−y =2 x +y x−y ✭✸✳✾✮ , t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥✳ ✭✸✳✻✮ • |x j := 2m + ❙û ❞ö♥❣ ✭✸✳✽✮✱ t❛ t❤➜② −y | = |(x ) − (y ) = |x − y | x ≥ |x − y | ≥2 +y x +y x −y | = |x − y | + x y x y ≥ |x − y | x +y ✭✸✳✶✵✮ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❇➜t ✤➥♥❣ t❤ù❝ ✭✸✳✺✮ ❤♦➦❝ ✭✸✳✻✮ trð t❤➔♥❤ ✤➥♥❣ t❤ù❝ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y ✱ ❤♦➦❝ x = −y, x, y ≤ |x − y | C C , |x − y| C , |x − y | ✷✽ j j ●✐↔ sû ❝â ♠ët sè ❞÷ì♥❣ δ s❛♦ ❝❤♦ |x| ≥ δ ✈➔ |y| ≥ δ ✳ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✾✮✱ ✭✸✳✶✵✮ t❛ ❝â✿ |x −y |x −y +y ≥ δ |x − y|; x +y | ≥ |x − y | ≥ δ |x − y | | ≥ |x − y| x t ự ữủ ữợ t C = δ ✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❇ê ✤➲ t✐➳♣ t❤❡♦ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❦➳t q✉↔ ❝❤➼♥❤ ð ❜➔✐ ✈✐➳t ♥➔②✳ ≤ p ≤ r = 2q = C |K(x, y)| dy) = max{(sup R |K(x, y)| dx) , (sup R R } < ∞ R φ ∈ L (R) ≤ T T φ T φ ≤C φ T T φ L (R) ✭✸✳✶✶✮ L (R) C T L (R) (C ) ❚ø ❣✐↔ t❤✐➳t✱ t❛ s✉② r❛ r➡♥❣ = + L (R) ≤ q, r ≤ ∞, (1/r) + (1/p) = 1, ✈➔ − ❳➨t φ ∈ L (R)✳ ❇➡♥❣ ❝→❝❤ ✤ê✐ ❜✐➳♥ sè tr♦♥❣ t➼❝❤ ♣❤➙♥ ✈➔ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍♦❧❞❡r ❝❤♦ (1/r) + (1/p) = 1, t❛ ❝â T φ(x)T φ(x)dx = e R R ψ(x, y)φ(y)dy T φ(x)dx R = e R ψ(x, y)T φ(x)dx φ(y)dy R T T φ(y)φ(y)dy ≤ φ = T T φ R ❉♦ ✤â ✱ T φ ≤ T T φ ✷✾ φ ✭✸✳✶✷✮ ❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ t♦→♥ tû T ✈➔ T t❛ t❤➜② (T T φ)(x) = e ψ(z, x)T φ(z)dz e ψ(z, x) R = e R φ(y) = ψ(z, y)φ(y)dy dz R e R ψ(z, x)ψ(z, y)dz dy R ❚ø ✤â✱ t♦→♥ tû T T ❧➔ ởt t tỷ t ợ K(x, y) ữủ ✤à♥❤ ♥❤÷ s❛✉ K(x, y) = e ✭✸✳✶✸✮ ψ(z, x)ψ(z, y)dz R ✈➔ (T T φ)(x) = φ(y)K(x, y)dy R ❑➳t ❤đ♣ ✭✸✳✸✮✱ ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ ❨♦✉♥❣ tr♦♥❣ ❝❤÷ì♥❣ ✶ ✈➔ ≤C T T φ = + − t❛ ✤÷đ❝ ✭✸✳✶✹✮ φ ❑➳t ❤đ♣ ✭✸✳✶✷✮ ✈➔ ✭✸✳✶✹✮ ✤÷đ❝ t❤♦↔ ♠➣♥ ✭✸✳✶✶✮✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ❚❛ ✤➦t Q (z) := S(x, z) − S(y, z), ✤÷đ❝ ①❡♠ ữ ởt z M ợ t❤❛♠ sè x, y ∈ M✳ ❚❛ q✉❛♥ t➙♠ ✤➳♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ tr♦♥❣ ❤➔♠ sè Q d Q dz (z) ≥ C |x − y | ✈ỵ✐ ❤➡♥❣ sè C > 0, ✈➔ t❤➯♠ ✈➔♦ ✤â Q d Q dz (z)✳ (z) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ ❦❤✐ k = (z) ≥ C |x − y | ✈ỵ✐ ❤➡♥❣ sè C > 0, ✈➔ t❤➯♠ ✈➔♦ ✤â Q ✭✸✳✶✺✮ ✭✸✳✶✻✮ (z) ❧➔ ❤➔♠ ✤ì♥ ✤✐➺✉ ❦❤✐ k = ❚❛ ①→❝ ✤à♥❤ β t❤æ♥❣ q✉❛ ❝→❝ sè k , j , k , j ♥❤÷ s❛✉ −j )−( ( β := ❤♦➦❝ ❧➔ −j ) 2(j k − j k ) 1 β := max ; 2k 2k ✸✵ ; ✭✸✳✶✼✮ ✭✸✳✶✽✮ ≤ p ≤ 2, r = 2q = ✭✶✮ ✭✸✳✶✺✮ j < =O λ T ✭✷✮ ✭✸✳✶✺✮ ✭✸✳✶✻✮ j = , j = =O λ T log λ , ✭✸✳✶✽✮ β ✭✸✳✶✺✮ ✭✸✳✶✻✮ − j )( ( −j ) γ ✱ ❤♦➦❝ j > γ ✈➔ j < γ ✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ tr÷í♥❣ ❤đ♣ ❦❤✐ j < γ ✈➔ j > γ ✱ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❝â t❤➸ ❝♦✐ ❧➔ t÷ì♥❣ tü✳ ❚❤❡♦ ❣✐↔ ✤à♥❤ ✭✸✳✶✺✮✲✭✸✳✶✻✮✿ |Q (z)| ≥ C |x − y |; |Q (z)| ≥ C |x − y |, ♥➯♥ →♣ ❞ö♥❣ ❇ê ✤➲ ❱❛♥❞❡r ❈♦r♣✉t✱ t❛ ♥❤➟♥ ✤÷đ❝ |K(x, y)| ≤ C λ |x − y | ,λ |x − y | |K(x, y)| ≤ C λ |x − y | ,λ |x − y | ❉♦ ✤â ✭✸✳✷✺✮ ❇➙② ❣✐í t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ |K(x, y)| dy ≤ Cλ ✭✸✳✷✻✮ M ❜➡♥❣ ❝→❝❤ ❝❤✐❛ ❧➔♠ ✹ tr÷í♥❣ ❤đ♣✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ♥➔②✱ t❛ sû ❞ö♥❣ ❇ê ✤➲ ✸✳✶ ✤➸ ❝â ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ s❛✉ |x ± y| |K(x, y)| ≤ C λ |x ± y| ,λ ✭✸✳✷✼✮ , ✈ỵ✐ ♠å✐ x ∈ M✱ ✈➔ ✈ỵ✐ ♠å✐ y tr♦♥❣ D ✈➔ D , ❞ü❛ tr➯♥ t➼♥❤ ❝❤➤♥ ❤♦➦❝ ❧➫ j , j t ữ s ❑➳t ❤ñ♣ ✭✸✳✷✺✮ ✈➔ ❇ê ✤➲ ✸✳✶ t❛ ❝â j ,j |x − y| |K(x, y)| ≤ C λ |x − y| ;λ ✭✸✳✷✽✮ ◆➯♥ t❛ ❝â |K(x, y)| dy = M |K(x, y)|dy + B |K(x, y)|dy B |x − y| ≤C λ |x − y| dy + λ B ≤C λ B |t| ≤C λ λ |t| dt + λ C C +λ ✸✹ λ dt dy ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ |K(x, y)| dy ≤ C λ ✭✸✳✷✾✮ +λ M ❚❛ ✤→♥❤ ❣✐→ tr♦♥❣ ✭✸✳✷✾✮ ❜➡♥❣ ❝→❝❤ ❝❤å♥ α t❤ä❛ ♠➣♥ λ ❉♦ ✤â α= =λ j γ −j γ ; γ −γ α+γ −j α+γ −j = α.γ α.γ 2β := ❱➻ ✈➟②✱ |K(x, y)| dy ≤ Cλ M • ❙û ❞ö♥❣ ❇ê ✤➲ ❱❛♥❞❡r ❈♦r♣✉t ✈➔ ❇ê ✤➲ ✸✳✶ t❛ ❝â j ,j |K(x, y)| ≤ C λ |x − y | ;λ |x − y | ;λ |x − y| ✭✸✳✸✵✮ , ✈ỵ✐ ♠å✐ x, y ∈ M ❱ỵ✐ y ∈ D t❛ ❝â ✤→♥❤ ❣✐→ |x − y| |K(x, y)| ≤ C λ ❑❤✐ ✤â |K(x, y)| dy = D |K(x, y)|dy + D B |K(x, y)|dy D ≤C λ |x − y| B |x − y| dy + λ B |t| ≤C λ |t| dt + λ C ≤C λ dy B dt C λ +λ λ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ |K(x, y)| dy ≤ C λ +λ B ❚è✐ ÷✉ ❤â❛ ✤→♥❤ ❣✐→ tr➯♥✱ t❛ ✤÷đ❝ I ≤ Cλ ✸✺ ✭✸✳✸✶✮ ❱ỵ✐ y ∈ D t❛ t❤➜② |K(x, y)| ≤ C min{λ |x + y| |x + y| ;λ } ❑❤✐ ✤â |K(x, y)| dy = |K(x, y)|dy + D D B |K(x, y)|dy D |x + y| ≤C λ B |x + y| dy + λ B ≤C λ |t| |t| dt + λ C ≤C λ dy B dt C λ +λ λ ✣✐➲✉ ♥➔② ❞➝♥ ✤➳♥ |K(x, y)| dy ≤ C λ +λ ✭✸✳✸✷✮ B ❚è✐ ÷✉ ❤â❛ ✤→♥❤ ❣✐→ tr➯♥ t❛ ✤÷ñ❝ I ≤ Cλ ❉♦ ✤â |K(x, y)| dy ≤ I + I ≤ Cλ M • j ❚ø ❇ê ✤➲ ✸✳✶ t❛ ❝â j x −y ≥ C|x − y| |x + y| , ✈ỵ✐ ♠å✐ x, y ∈ N✱ ✈ỵ✐ y ∈ D t❛ ❝â |K(x, y)| ≤ C λ |x − y| ;λ |x − y| ❚÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✷✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ I ≤ Cλ ❱ỵ✐ y ∈ D ✱ t❛ ❝â |K(x, y)| ≤ C λ |x − y| ✸✻ ;λ |x + y| ❚÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✷ t❛ ❝â I ≤ Cλ ❉♦ ✈➟② |K(x, y)| dy ≤ I + I ≤ Cλ M • j ❇➡♥❣ ✈✐➺❝ ❤♦→♥ ✤ê✐ ✈❛✐ trá ❝õ❛ j ✈➔ j j t❛ ❝â t❤➸ tr❛♥❤ ❝ù ❝❤♦ tr÷í♥❣ ❤đ♣ ♥➔② t÷ì♥❣ tü ♥❤÷ ❚r÷í♥❣ ❤đ♣ ✸✳ õ ợ tt trữớ ủ j , j t❛ ❝â |K(x, y)| dy ≤ I + I ≤ Cλ M ❚÷ì♥❣ tü ✤→♥❤ ❣✐→ ♥➔② ❝ơ♥❣ ✤ó♥❣ ❦❤✐ t❛ t❤❛② |K(x, y)| dy ❜ð✐ M M |K(x, y)| dx✱ ✈➔ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✷ t❛ s✉② r❛ C ≤ T λ ✭✸✮ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❝❤➾ ❝❤ù♥❣ ♠✐♥❤ tr÷í♥❣ ❤đ♣ j < γ ✈➔ j = γ ✈➔ ❝→❝ tr÷í♥❣ ❤đ♣ ỏ õ t ữủ ự tữỡ tỹ ❑❤✐ ✤â j ,j |K(x, y)| ≤ C λ |x − y| |x − y| ;λ ✣✐➲✉ ♥➔② ❝❤♦ t❛ |x − y| |K(x, y)| dy ≤ C λ M ≤C λ |x − y| dy + λ B B |x − y| +λ dy B =C λ ✈➻ +λ log λ ≤ Cλ α+γ −j = 2β αγ ❉♦ ✤â |K(x, y)| dy ≤ Cλ M ✸✼ log λ log(λ), dy ❚÷ì♥❣ tü ✤→♥❤ ❣✐→ ♥➔② ❝ơ♥❣ ✤ó♥❣ ❦❤✐ t❛ t❤❛② M |K(x, y)| dy ❜ð✐ M |K(x, y)| dx✱ ✈➔ →♣ ❞ö♥❣ ❇ê ✤➲ ✸✳✷ t❛ s✉② r❛ C ≤ T log λ λ ❦➳t ❤ñ♣ ợ t ự ữủ trữớ ủ trữớ ủ ỏ õ t ự ♠✐♥❤ ✈➔ t✐➳♥ ❤➔♥❤ j ,j t÷ì♥❣ tü t❤❡♦ ❝→❝ tr÷í♥❣ ❤đ♣ ð ✭✸✮✱ ♥❤÷♥❣ ❝→❝ ②➳✉ tè ❧♦❣❛r✐t ♥➯♥ ữủ t tr ữợ t ỵ ữủ ❝❤ù♥❣ ♠✐♥❤ ①♦♥❣✳ ❈❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ✭✸✳✶✺✮ ✭✸✳✶✻✮ ( β− − j )( − j ) ≤ 2k β− 1 ; 2k 2k 2k − j )( (k − k ) ( = −j ) 4k k (j k − j k ) ≤ β ≤ max 1 ; 2k 2k ≤ ≤ ✭✸✳✸✸✮ ✭✸✳✶✺✮ ✭✸✳✶✻✮ ❚❛ t tự ỗ t ữợ S (x, y) = a x S (x, y) = a S (x, y) = a x Ð ✤➙② ❝â t❤➸ ❣✐↔ ✤à♥❤ r➡♥❣ a S(x, y) ❝ò♥❣ ❞➜✉✱ tù❝ ❧➔✱ a a a x = 0, ✈➔ a ≥ ✈ỵ✐ t➜t ❝↔ ✸✽ ✭✸✳✸✹✮ y ; y y a ; ✭✸✳✸✺✮ ✭✸✳✸✻✮ = ●✐↔ sû ♠å✐ ❤➺ sè tr♦♥❣ = j , ❚❛ ❝â t❤➸ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ r➡♥❣ tø♥❣ ✤❛ t❤ù❝ S (x, y), S (x, y), ✈➔ S (x, y) t❤ä❛ ♠➣♥ ✷ ✤✐➲✉ ❦✐➺♥ ✭✸✳✶✺✮ ✈➔ ✭✸✳✶✻✮✳ ❈ö t❤➸✱ S (x, y), S (x, y), ✈➔ S (x, y) ❝ò♥❣ ♥❤❛✉ t❤ä❛ ♠➣♥✭✸✳✶✺✮✲✭✸✳✶✻✮ t❤❡♦ ❝→❝ ❝➦♣ ✤ỉ✐ (2j , 2n − 2j )✲(2k , 2n − 2k ), (2j + 1, 2n − 2j + 1)✲(2k + 1, 2n − 2k + 1), ✈➔ (2j , 2n − 2j + 1) − (2k , 2n − 2k + 1), t÷ì♥❣ ù♥❣✳ ✣➸ S ❜✐➸✉ t❤à t➟♣ ❤đ♣ t➜t ❝↔ tự ỗ t õ õ ❤➺ sè ❝ò♥❣ ❞➜✉✳ ❇✐➸✉ t❤à ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ tr➯♥ ❜ð✐ δ ✱ t❤➻ ❦❤✐ ✤â δ = 2n ❝❤♦ ✭✸✳✸✹✮✲✭✸✳✸✺✮✱ ❤♦➦❝ δ = 2n + ❝❤♦ ✭✸✳✸✻✮✳ S(x, y) ∈ S T ✭✶✮ ✭✸✳✸✹✮ ✭✸✳✸✺✮ ≤ j < n/2 < k ≤ n ✭✸✳✸✻✮ (n − 1)/2 < k ≤ n T ✭✷✮ L ≤ j < ≤ j < (n + 1)/2 < k ≤ n =O λ ✭✸✳✸✹✮ ≤ j = n/2 < k ≤ n, ≤ j < n/2 = k ≤ n; ✭✸✳✸✺✮ ≤ j = (n − 1)/2 < k ≤ n, T =O λ ≤ j < (n − 1)/2 = k ≤ n; log (λ) ✭✸✳✶✺✮ ✭✸✳✶✻✮ S(x, y) (m − 1, 1); (j , k ) = (1, n − 1) ✭✸✳✶✼✮ β= (j − k ) − (j − k ) m+n−4 = , 2(k j − k j ) 2(mn − m − n) ✸✾ (j , k ) = ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ t➼❝❤ ♣❤➙♥ ❞❛♦ ✤ë♥❣✳ ◆ë✐ ❞✉♥❣ ỗ t ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♣❤➙♥ ❤♦↕❝❤ ✤ì♥ ✈à✱ t➼❝❤ ❝❤➟♣✱ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❣✐↔♠ ♥❤❛♥❤ S (R ) ✈➔ ♠ët sè ✤à♥❤ ❧➼ q✉❛♥ trå♥❣ ❝õ❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ rr ợ ữủ t tr tổ q tự ữ r ❝❤✉➞♥ ❝õ❛ t♦→♥ tû ❞❛♦ ✤ë♥❣ tø ❦❤æ♥❣ ❣✐❛♥ L (R) ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ L (R)✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ✹✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ✣➦♥❣ ❆♥❤ ❚✉➜♥✱ ✳ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✭✷✵✶✺✮✳ ❬✷❪ ✣✐♥❤ ❚❤➳ ▲ư❝✱ P❤↕♠ ❍✉② ✣✐➸♥✱ ❚↕ ❉✉② P❤÷đ♥❣✱ ✳◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐ ✭✷✵✵✷✮✳ ❬✸❪ P✳ ❑✳ ❆♥❤✱ ❱✳ ◆✳ ❍✉② ❛♥❞ ◆✳ ▼✳ ❚✉❛♥✱ L ❬✹❪ ■✳ ❘✳P❛r✐ss✐s ✱ L ✳ Pr❡♣✐♥t✳ s❤❛r♣ ❜♦✉♥❞ ❢♦r t❤❡ ❙t❡✐♥ ❲❛✐♥❣❡r ♦s❝✐❧❧❛t♦r② ✐♥t❡❣r❛❧ ✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱❱✳✶✸✻✭✷✵✵✽✮✱♥✳✸✱♣✳✾✻✸✲✾✼✷✳ ❬✺❪ ❉✳ ❍✳ P❤♦♥❣ ❛♥❞ ❊✳ ▼✳ ❙t❡✐♥✱ ▼❛t❤✳ ✳ ■♥✈✳ ✱ ✸✾✲✻✷ ✭✶✾✾✷✮✳ ❬✻❪ ❉✳ ❍✳ P❤♦♥❣ ❛♥❞ ❊✳ ▼✳ ❙t❡✐♥✱ ✳ ❆♥♥✳ ♦❢ ▼❛t❤✳✱ ✹✶ ✱ ✼✵✸✕✼✷✷ ✭✶✾✾✹✮✳ ... 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... t ổ tr r tq ❞❡ ❧❛ ❈❤❛❧❡✉r ❝õ❛ ❏♦s❡♣❤ ❋♦✉r✐❡r ✈➔♦ ♥➠♠ ✶✽✷✷✳ ◆❤✐➲✉ t ỵ tt ữỡ tr r số ỵ tt st ỵ tt số ❝→❝ ❜➔✐ t♦→♥ ✈➲ q✉❛♥❣ ❤å❝✱ ➙♠ ❤å❝✱ ❝ì ❤å❝ ❧÷đ♥❣ tû✱✳✳✳ ✤➲✉ ❝â t❤➸ ✤÷❛ ✈➲ ✈✐➺❝ ♥❣❤✐➯♥... ) ⊂ supp ψ ⊂ W + B(0, ) ⊂ U ψ (x) > 0, x ∈ ∪ V (⊃ K) , ✈➔ ψ (x) < N + 1, x ∈ R ❈â K ⊂ V tỗ t số > s K ⊂ K + B(0, ) ⊂ ∪ V ❚❤❡♦ ♠➺♥❤ ✤➲ ✶✳✷ ❝â ❤➔♠ ❦❤æ♥❣ ➙♠ φ t❤ä❛ ♠➣♥ φ ∈ C (R ), K ⊂ K + B(0,