❇❐ ●■⑩❖ ❉Ư❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❇Ò■ ❚❍➚ ◆❍⑨■ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T) ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❍➔ ◆ë✐ ✲ ✷✵✶✽ ❇❐ ●■⑩❖ ❉Ö❈ ❱⑨ ✣⑨❖ ❚❸❖ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ❍⑨ ◆❐■ ✷ ❇Ị■ ❚❍➚ ◆❍⑨■ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T) ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽✹✻✵✶✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ữớ ữợ ề ì ị ❍➴❈ R ❚➟♣ t➜t ❝↔ ❝→ sè t❤ü❝ R+ ❚➟♣ ❤đ♣ ❝→❝ sè t❤ü❝ ❦❤ỉ♥❣ ➙♠ Rd ❑❤ỉ♥❣ ❣✐❛♥ ✈❡❝tì d ổ ữợ ỳ tỷ tr ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ·, · T = R/2π Z ❍➻♥❤ ①✉②➳♥ ♠ët ❝❤✐➲✉ C(T) ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tư❝✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T C α (T) ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ ❍☎♦❧❞❡r✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T Lp (T) ❑❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❜➟❝ p ❦❤↔ t➼❝❤✱ t✉➛♥ ❤♦➔♥ ❝❤✉ ❦ý 2π ✱ ♥❤➟♥ ❣✐→ trà ♣❤ù❝ tr➯♥ T f T f Lp (T) ❈❤✉➞♥ ❝õ❛ ❤➔♠ f tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp (T) ❚➼❝❤ ♣❤➙♥ ❧➜② tr➯♥ ✤♦↕♥ ❜➜② ❦ý ❝â ✤ë ❞➔✐ 2π ▲✐➯♥ ❤ñ♣ ❝õ❛ f ▲❮■ ❈❷▼ ❒◆ ✣➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ✈ỵ✐ ✤➲ t➔✐ ✧❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ ổ Lp (T) trữợ t tổ ữủ tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ❝ò♥❣ ❝→❝ t❤➛② ❝ỉ ♣❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷ ✤➣ ✤ë♥❣ ✈✐➯♥ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ q✉❛✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ t❤➛② ❣✐→♦ ❚❙✳ ❇ò✐ ữớ ữớ trỹ t ữợ õ õ ỵ qỵ tổ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❜➔✐ ▲✉➟♥ ✈➠♥ ♥➔②✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝õ❛ ❜↔♥ t❤➙♥ ♥➯♥ ❝❤➢❝ ❝❤➢♥ ✤➲ t➔✐ ♥➔② ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟②✱ tỉ✐ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❝↔♠ t❤ỉ♥❣ ỳ ỵ õ õ t ổ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ❝õ❛ tỉ✐ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✽ ❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ❇ò✐ ❚❤à ◆❤➔✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❦➳t q✉↔ ❝õ❛ tæ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝ò♥❣ ✈ỵ✐ sü ❣✐ó♣ ✤ï ❝õ❛ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ❝→❝ t❤➛② ❝æ tr♦♥❣ ❦❤♦❛ t♦→♥✱ t sỹ ữợ t t t ❣✐→♦ ❚❙✳ ❇ò✐ ❑✐➯♥ ❈÷í♥❣✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❧➔♠ ▲✉➟♥ ✈➠♥✱ tæ✐ ❝â t❤❛♠ ❦❤↔♦ ♥❤ú♥❣ t➔✐ ❧✐➺✉ ❝â ❧✐➯♥ q✉❛♥ ✤➣ ✤÷đ❝ ❤➺ t❤è♥❣ tr♦♥❣ ♠ư❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ▲✉➟♥ ✈➠♥ ✧❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp (T)✧ ❦❤ỉ♥❣ ❝â trò♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ▲✉➟♥ ✈➠♥ ❦❤→❝✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✼ ♥➠♠ ✷✵✶✽ ❍å❝ ✈✐➯♥ ❇ò✐ ❚❤à ◆❤➔✐ ✸ ▼ö❝ ❧ö❝ ▲❮■ ❈⑩▼ ❒◆ ▲❮■ ❈❆▼ ✣❖❆◆ ✸ ▼Ð ✣❺❯ ✻ ✶ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✶✶ ✶✳✶ ❑❤æ♥❣ ❣✐❛♥ ✤ë ✤♦✱ ❤➔♠ ❝❤➾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❑❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ Lp (T)✱ (Z) ✈➔ W 1,1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷ ❈❍❯➱■ ❋❖❯❘■❊❘ ✶✻ ✷✳✶ ❍➺ sè ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✷ ❍➺ sè ❋♦✉r✐❡r tr♦♥❣ L1 (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❍➺ sè ❋♦✉r✐❡r tr♦♥❣ L2 (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ❍➔♠ ❝ü❝ ✤↕✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T) ✸✻ ✸✳✶ ❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ L2 (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✷ P❤➙♥ t➼❝❤ ❈❛❧❞❡râ♥✲❩②❣♠✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✸ ❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ Lp (T) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹ ▼Ö❈ ▲Ö❈ ❑➌❚ ▲❯❾◆ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ é ỵ t ❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt t❤✉ë❝ ❧➽♥❤ ✈ü❝ ❣✐↔✐ t➼❝❤ ✤✐➲✉ ❤á❛✱ ①✉➜t ❤✐➺♥ ❧➛♥ ✤➛✉ t✐➯♥ ❦❤✐ ❉✳ ❍✐❧❜❡rt ♥❣❤✐➯♥ ❝ù✉ t rt t ữợ ♥❤↔②✑ ❝õ❛ ♥â ❦❤✐ ✤✐ q✉❛ ♠ët ✤÷í♥❣ ❝♦♥❣✳ ❙❛✉ ✤â✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ q t t õ ỗ ỵ tt t ý P ❈❛❧❞❡râ♥ ❝ơ♥❣ ♥❤÷ ❙✳●✳ ▼✐❦❤❧✐♥ ✈➔ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝ ①➙② ❞ü♥❣ ❧➯♥✳ ❱➲ ♠➦t ✤à♥❤ ♥❣❤➽❛✱ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt ✤ì♥ ❣✐↔♥ ❧➔ t➼❝❤ ♣❤➙♥ ✏s✉② rë♥❣✑ ♥❤÷ s❛✉ ∞ f (y) dy, −∞ x − y H(f )(x) = π ð ✤â f : R → C ❧➔ ♠ët ❤➔♠ ✤õ tèt✳ ❈â t❤➸ t❤➜② t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ♥➔② t❤❡♦ ❤❛✐ ♥❣❤➽❛✿ ❝â ✤✐➸♠ ❦➻ ❞à t↕✐ x ✈➔ ♠✐➲♥ ❧➜② t➼❝❤ ♣❤➙♥ ❧➔ ✈æ t tr ữ ợ ❤↕♥ ❝õ❛ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ t❤ỉ♥❣ t❤÷í♥❣ t❤➻ s➩ ❝â r➜t ➼t ❤➔♠ ✤➸ t➼❝❤ ♣❤➙♥ ♥➔② ❤ë✐ tö✳ ❈❤➥♥❣ ❤↕♥ ❤➔♠ f (x) = e−x ❧➔ ❤➔♠ ❦❤→ tèt ♥❤÷♥❣ ❦❤✐ t❤❛② ✈➔♦ t➼❝❤ ♣❤➙♥ tr➯♥ t❛ ❝❤➾ ✤÷đ❝ ♠ët t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❦❤ỉ♥❣ ❤ë✐ tư t↕✐ ❜➜t ❝ù ✤✐➸♠ x ♥➔♦✳ ◆➳✉ ❤✐➸✉ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❡♦ ❦✐➸✉ ❣✐→ trà ❝❤➼♥❤ ✭♣r✐♥❝✐♣❛❧ ✈❛❧✉❡✮✱ ♥❣❤➽❛ ✻ ▼Ð ✣❺❯ ❧➔ H(f )(x) = lim+ →0 M →+∞ π s❛♦ ❝❤♦ |{t ∈ T : |(Hf )(t)| > λ}| ≤ A f λ L1 (T) ✈ỵ✐ ♠å✐ λ > ✈➔ f ∈ L2(T)✳ ❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ λ ≤ f L1 (T) t❤➻ A = 2π t❤ä❛ ♠➣♥✳ ●✐↔ sû λ > f L1 (T) ỵ rõ t õ f = g+b✱ ✈ỵ✐ g ∈ L∞ (T) ✈➔ ❞♦ ✈➟② g ∈ L2 (T) s✉② r❛ Hg ∈ L2 (T)✳ ❚ø ✤â✱ t❛ ❝ô♥❣ ❝â b = f −g ∈ L2 (T) ✈➻ ✈➟② Hb ∈ L2 (T)✳ ❍ì♥ ♥ú❛✱ bl ∈ L2 (T) ✈➔ b = l bl ❤ë✐ tư tr♦♥❣ L2 (T)✱ ❞♦ t➼♥❤ ❦❤ỉ♥❣ ❣✐❛♦ ♥❤❛✉ ❝õ❛ ❣✐→ ❝õ❛ ❝→❝ bl ✳ ❱➻ ✈➟② Hb = tr♦♥❣ L2 (T)✳ ✹✺ l Hbl ❤ë✐ tư ❈❤÷ì♥❣ ✸✳ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T) ▼➦t ❦❤→❝✱ ✈➻ Hf = Hg + Hb✱ t❛ s✉② r❛ |{t ∈ T : |(Hf )(t)| > λ}| ≤ t ∈ T : |(Hg)(t)| > λ + t ∈ T : |(Hb)(t)| > λ = γ + β ❉♦ g ∈ L2 ♥➯♥ |(Hg)(t)|2 γ≤ λ 2 T dt |g(t)|2 dt ✈➻ H L2 (T)→L2 (T) = λ T |g(t)|dt ✈➻ g L∞ (T) ≤ 2λ ≤ λ T · 2π ≤ f L1 (T) ✈➻ f L1 (T) ≤ f L1 (T) λ ≤ ❚❛ ❧↕✐ ❝â b ∈ L1 ♥➯♥ β≤| 2I(l)| + t∈T\ l ≤ 2I(l) : |(Hb)(t)| > l 4π f λ L1 (T) |(Hb)(t)| + λ T\∪l 2I(l) dt t ỵ rõ ≤ 4π f λ ✈➻ |Hb| ≤ L1 (T) + λ |(Hbl )(t)|dt l T\2I(l) |Hbl | ❤➛✉ ❦❤➢♣ ♥ì✐✳ ❈✉è✐ ❝ò♥❣✱ ❝❤ó♥❣ t❛ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ l |(Hbl )(t)|dt ≤ (const ) f l T\2I(l) ✹✻ L1 (T) ✭✸✳✻✮ ❈❤÷ì♥❣ ✸✳ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T) ⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✸✳✹ tr➯♥ ❦❤♦↔♥❣ T \ 2I(l)✱ ❝❤ó♥❣ t❛ ❝â |Hbl (t)|dt = T\2I(l) T\2I(l) 2π bl (τ ) cot I(l) (t − τ ) dτ td ❞♦ t − τ ❜à ❝❤➦♥ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ ✵✱ ✈➻ τ ∈ I(l) ✈➔ t ∈ / 2I(l), = T\2I(l) 2π bl (τ ) cot I(l) 1 (t − τ ) − cot (t − cl ) 2 ð ✤â cl ❧➔ t➙♠ ❝õ❛ I(l), ✈➔ = T\2I(l) 2π bl (τ ) I(l) ≤ (const ) |bl (τ )| I(l) sin sin 12 (t − dτ dt bl (τ )dτ = 0, I(l) (τ − cl ) τ ) sin 21 (t − cl ) dτ dt |I(l)| dtdτ R\2I(l) |t − τ t − cl | ▼➔ |t − cl | ≤ |t − τ | + |τ − cl | ≤ |t − τ | + |I(l)| ❦❤✐ τ ∈ I(l) ❦❤✐ t ∈ R \ 2I(l), ≤ 2|t − τ | ♥➯♥ |I(l)| dt ≤ R\2I(l) |t − τ t − cl | |I(l)| dt R\2I(l) |t − cl | ∞ 2r =4 dt ð ✤â 2r = |I(l)| 2r t = ✹✼ ❈❤÷ì♥❣ ✸✳ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ Lp(T) t ỵ s r❛ ✈➳ tr→✐ ❝õ❛ ✭✸✳✻✮ ≤ (const ) |bl (τ )|dτ l I(l) = (const ) b L1 (T) ≤ (const ) f L1 (T) ❙✉② r❛ ✭✸✳✻✮ ✤÷đ❝ ự õ ỵ ữủ ự q✉↔ ✸✳✶✸ ✭❬✶❪✮✳ ❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt ❧➔ (p, p) ♠↕♥❤ ✈ỵ✐ < p < ∞✱ ✈➔ (Hf )(n) = −i sign(n)f (n) ✈ỵ✐ ♠å✐ f ∈ Lp(T), n ∈ Z✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ t❛ ❝â H ❧➔ (2, 2) t t t ỵ H ❝á♥ ❧➔ (1, 1) ②➳✉ tr➯♥ L2 (T)✱ ❞♦ ✤â ♥â ❧➔ ❤➔♠ ✤ì♥ ❣✐↔♥ tr➯♥ T✳ ❱➻ ✈➟② H ❧➔ (p, p) ♠↕♥❤ ✤è✐ ✈ỵ✐ < p < ỵ s r r❛ H : Lp (T) → Lp (T) ❜à ❝❤➦♥ ✈➔ t✉②➳♥ t➼♥❤ ✈ỵ✐ < p < 2✳ ❱ỵ✐ < p < ∞ ❝❤ó♥❣ t❛ s➩ sû ❞ư♥❣ t➼♥❤ ❝❤➜t ✤è✐ ♥❣➝✉ ✈➔ ♣❤↔♥✲tü ❧✐➯♥ ❤ñ♣ H ∗ = −H tr➯♥ L2 (T) ✭①❡♠ ❇ê ✤➲ ✸✳✸✮ ✤➸ ✤÷❛ ✈➲ tr÷í♥❣ ❤đ♣ < p < 2✳ ●å✐ p ❧➔ sè t❤ä❛ ♠➣♥ Hf p 2π = sup{ 2π p + p = 1✳ ◆➳✉ f ∈ Lp ∩ L2 (T) t❤➻ (Hf )gdt| : g ∈ Lp (T) ❝â ❝❤✉➞♥ ❜➡♥❣ ✶} = sup{| T (Hf )gdt : g ∈ Lp (T) ∩ L2 (T) ❝â ❝❤✉➞♥ ❜➡♥❣ ✶} T t❤❡♦ t➼♥❤ trò ♠➟t ❝õ❛ Lp ∩ L2 tr♦♥❣ Lp = sup{ 2π f (Hg)dt : g ∈ Lp (T) ∩ L2 (T) ❝â ❝❤✉➞♥ ❜➡♥❣ ✶} T ✈➻ H ∗ = −H tr➯♥ L2 (T) ✹✽ ❈❤÷ì♥❣ ✸✳ ❇■➌◆ ✣✃■ ❍■▲❇❊❘❚ ❚❘➊◆ ❑❍➷◆● ●■❆◆ Lp(T) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎ ♦❧❞❡r✱ t❛ s✉② r❛ Hf p ≤ f Lp (T) sup{ ≤ (const )p f Hg Lp (T) : g ∈ Lp (T) ∩ L2 (T) ✈ỵ✐ ❝❤✉➞♥ ✶} Lp (T) , ✈➻ t➼♥❤ ❜à ❝❤➦♥ (p , p ) ♠↕♥❤ ✭✈ỵ✐ < p < 2✮ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ð tr➯♥✳ ❱➻ ✈➟② H ❜à ❝❤➦♥ ✈➔ t✉②➳♥ t➼♥❤ tr➯♥ t➟♣ ❝♦♥ trò ♠➟t Lp ∩ L2 (T) ❝õ❛ Lp (T)✳ ❱➻ ✈➟② H ❝â t❤➸ ✤÷đ❝ t❤→❝ tr✐➸♥ t✉②➳♥ t➼♥❤ ❜à ❝❤➦♥ tr➯♥ t♦➔♥ Lp (T)✳ ❈✉è✐ ❝ò♥❣✱ ❝❤♦ f ∈ Lp (T)✱ < p < ∞✱ ❧➜② fm ∈ Lp ∩L2 (T) ✈ỵ✐ fm → f tr♦♥❣ Lp (T)✳ ❉♦ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ H tr➯♥ Lp s✉② r❛ Hfm → Hf tr♦♥❣ Lp ✳ ❱➻ ✈➟② fm → f ✈➔ Hfm → Hf tr♦♥❣ L1 (T)✳ ❈❤✉②➸♥ q✉❛ ❣✐ỵ✐ ❤↕♥✱ t❛ s✉② r❛ (Hfm )(n) = −i sign(n)fm (n) ❤❛② (Hf )(n) = −i sign(n)f (n)✱ ✤➙② ❧➔ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ✹✾ ❑➌❚ ▲❯❾◆ r q tr t tổ ữợ ✤➛✉ ❧➔♠ q✉❡♥ ✈ỵ✐ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝ ✈➔ ❧➔♠ ✈✐➺❝ ♥❣❤✐➯♠ tó❝✳ ◗✉❛ ✤â✱ tỉ✐ ❝â ♥➨t ❤➻♥❤ ❞✉♥❣ ✤➛✉ t✐➯♥ ✈➲ t♦→♥ ❤å❝ ❤✐➺♥ ✤↕✐✱ ❝❤✉②➯♥ ♥❣➔♥❤ ●✐↔✐ t ỗ tớ t ữủ sỹ ú t ❤å❝✱ ♠➔ ✤➦❝ ❜✐➺t tr♦♥❣ ▲✉➟♥ ✈➠♥ ♥➔②✱ tæ✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♠ët ❝→❝❤ ❦❤→✐ q✉→t ✈➲ ❤➺ sè ❋♦✉r✐❡r✱ ✈➔ ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp (T)✱ ❝â t❤➸ ①❡♠ ♥❤÷ ❧➔ ♠ët t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ tèt ❝❤♦ ♥❤ú♥❣ ♥❣÷í✐ q✉❛♥ t➙♠ ✈➲ ✈➜♥ ✤➲ ♥➔②✳ ✣➲ t➔✐ ✏❇✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ ❦❤æ♥❣ ❣✐❛♥ Lp (T) ✤➣ tr➻♥❤ ❜➔② ❝→❝ ✈➜♥ ✤➲ s❛✉ ✤➙②✿ ✶✳ ❚r➻♥❤ ❜➔② ♠ët sè ✤à♥❤ ❧➼ ✈➲ t➼❝❤ ❝❤➜t ❝õ❛ ❤➺ sè ❋♦✉r✐❡r✳ ❚➼♥❤ ❝❤➜t ❤ë✐ tö t❤❡♦ ❝❤✉➞♥ ✈➔ t❤❡♦ ✤✐➸♠ ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r✱ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❝ü❝ ✤↕✐ ❍❛r❞②✲▲✐tt❧❡✇♦♦❞ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♥â❀ ✷✳ ❚r➻♥❤ ❜➔② ♣❤➙♥ t➼❝❤ ❈❛❧❞❡râ♥✲❩②❣♠✉♥❞ t❤æ♥❣ q✉❛ ❞➣② ❤➔♠ tr✉♥❣ ❜➻♥❤ ♥❤à ♥❣✉②➯♥ ❝❤♦ ❝↔ tr÷í♥❣ ❤đ♣ t♦➔♥ ❦❤ỉ♥❣ ❣✐❛♥ ✈➔ ❤➻♥❤ ①✉②➳♥ T ✈➔ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t (1, 1) ②➳✉ ✈➔ (p, p) ♠↕♥❤ ✈ỵ✐ < p < ∞ ❝õ❛ ❜✐➳♥ ✤ê✐ ❍✐❧❜❡rt tr➯♥ ❦❤æ♥❣ ❣✐❛♥ Lp (T)✳ ✺✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❘✐❝❤❛r❞ ❙✳ ▲❛✉❣❡s❡♥ ✭✷✵✵✾✮✱ ❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s ▲❡❝t✉r❡ ◆♦t❡s ✱ ❯♥✐✲ ✈❡rs✐t② ♦❢ ■❧❧✐♥♦✐s ❛t ❯r❜❛♥❛✕❈❤❛♠♣❛✐❣♥✱ ❯❙❆✳ ❬✷❪ ❏✳ ❏✳ ❇❡♥❡❞❡tt♦ ✭✷✵✵✶✮✱ ❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ❈❘❈ Pr❡ss✳ ❬✸❪ ❏✳ ❉✉♦❛♥❞✐❦♦❡t①❡❛ ✭✷✵✵✶✮✱ ❋♦✉r✐❡r ❆♥❛❧②s✐s✱ ❚r❛♥s❧✳ ❉✳ ❈r✉③❯r✐❜❡✱ ●r❛❞✉❛t❡ ❙t✉❞✐❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✷✾✱ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐✲ ❡t②✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✳ ❬✹❪ ❘✳ ❊✳ ❊❞✇❛r❞s✱ ●✳ ■✳ ●❛✉❞r② ✭✶✾✼✼✮✱ ▲✐tt❧❡✇♦♦❞✲P❛❧❡② ❛♥❞ ▼✉❧t✐♣❧✐❡r ❚❤❡♦r②✱ ❊r❣❡❜♥✐ss❡ ❞❡r ▼❛t❤❡♠❛t✐❦ ✉♥❞ ✐❤r❡r ●r❡♥③❣❡❜✐❡t❡✱ ❇❛♥❞ ✾✵✱ ❙♣r✐♥❣❡r✕❱❡r❧❛❣✱ ❇❡r❧✐♥✕◆❡✇ ❨♦r❦✳ ❬✺❪ ❨✳ ❑❛t③♥❡❧s♦♥ ✭✷✵✵✷✮✱ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s✱ ❙❡❝♦♥❞ ❝♦rr❡❝t❡❞ ❡❞✐t✐♦♥✱ ❉♦✈❡r P✉❜❧✐❝❛t✐♦♥s✱ ■♥❝✳✱ ◆❡✇ ❨♦r❦✳ ❬✻❪ ❊✳ ▼✳ ❙t❡✐♥ ✭✶✾✼✵✮✱ ❙✐♥❣✉❧❛r ■♥t❡❣r❛❧s ❛♥❞ ❉✐❢❢❡r❡♥t✐❛❜✐❧✐t② Pr♦♣❡r✲ t✐❡s ♦❢ ❋✉♥❝t✐♦♥s✱ Pr✐♥❝❡t♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙❡r✐❡s✱ ◆♦✳ ✸✵✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆✳❏✳✳ ✺✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✼❪ ❆✳ ❩②q♠✉♥❞ ✭✷✵✵✷✮✱ ❚r✐❣♦♥♦♠❡tr✐❝ ❙❡r✐❡s✱ ❱♦❧✳ ■✱ ■■✳ ❚❤✐r❞ ❡❞✐t✐♦♥✱ ❋♦r❡✇♦r❞ ❜② ❘✳ ❆✳ ❋❡❢❢❡r♠❛♥✱ ❈❛♠❜r✐❞❣❡ ▼❛t❤❡♠❛t✐❝❛❧ ▲✐❜r❛r②✱ ❈❛♠✲ ❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ✺✷ ... (f ∗ g)(t) ❦❤✐ δ → ữỡ ỵ ❦❤↔ tê♥❣ tr♦♥❣ Lp(T) ✈➔ C(T)✮✳ ◆➳✉ {kn} ❧➔ ♠ët ❤↕❝❤ ❦❤↔ tê♥❣✱ ♥❣❤➽❛ ❧➔ ∞ n=1 kn < ∞✱ ❤➔♠ f ∈ Lp(T), ≤ p < ∞ t❤➻ tr♦♥❣ Lp(T), ❦❤✐ n → ∞ kn ∗ f → f ❚÷ì♥❣ tü✱ ♥➳✉ f... t➼♥❤ ❝❤➜t ❝õ❛ ❤➺ sè ❋♦✉r✐❡r ❝õ❛ t➼❝❤ ❝❤➟♣ ✭①❡♠ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✮✿ ✣à♥❤ ỵ f Lp(T), p ∞ ✈➔ g ∈ L1(T) t❤➻ f ∗ g ∈ Lp(T) ✈ỵ✐ f ∗g Lp (T) ≤ f Lp (T) g L1 (T) ✭✷✳✹✮ ✈➔ f ∗ g(n) = f (n)g(n), n ∈... A ❈❤÷ì♥❣ ✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ❝❤➾ ❝õ❛ t➟♣ A✳ ✶✳✷ ❑❤→✐ ♥✐➺♠ ❦❤æ♥❣ ❣✐❛♥ Lp(T) (Z) ✈➔ W 1,1 ❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ♥❤➢❝ ❧↕✐ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❝ì ❜↔♥ ✤÷đ❝ sû ❞ö♥❣ tr♦♥❣