Đánh giá địa phương của hàm đa điều hòa dưới và áp dụng (LV thạc sĩ)

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

❍♦➔♥❣ ❚❤à ❚❤✉ ❍÷ì♥❣

✣⑩◆❍ ●■⑩ ✣➚❆ P❍×❒◆●
❈Õ❆ ❍⑨▼ ✣❆ ✣■➋❯ ❍➪❆ ❉×❰■ ❱⑨ ⑩P ❉Ö◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥✱ ♥➠♠ ✷✵✶✻


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

❍♦➔♥❣ ❚❤à ❚❤✉ ❍÷ì♥❣

✣⑩◆❍ ●■⑩ ✣➚❆ P❍×❒◆●
❈Õ❆ ❍⑨▼ ✣❆ ✣■➋❯ ❍➪❆ ❉×❰■ ❱⑨ ⑩P ❉Ö◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ t➼❝❤
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆❣÷í✐ ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ◗✉❛♥❣ ❉✐➺✉

❚❤→✐ ◆❣✉②➯♥✱ ♥➠♠ ✷✵✶✻

▲í✐ ❝❛♠ ✤♦❛♥

❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ▲✉➟♥ ✈➠♥ ♥➔② ❧➔ ❞♦ ❝❤➼♥❤ t→❝ ❣✐↔ t❤ü❝ ❤✐➺♥ ❞÷î✐ sü
❤÷î♥❣ ❞➝♥ ❝õ❛ ●❙✳ ❚❙❑❍✳ ◆❣✉②➵♥ ◗✉❛♥❣ ❉✐➺✉✳ ▲✉➟♥ ✈➠♥ ❦❤æ♥❣ trò♥❣
❧➦♣ ✈î✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝ ✈➔ ❝→❝ t❤æ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷ñ❝
❝❤➾ rã ♥❣✉ç♥ ❣è❝✳
❚❤→✐ ♥❣✉②➯♥✱ ♥❣➔② ✶✵ t❤→♥❣ ✵✹ ♥➠♠ ✷✵✶✻

❚→❝ ❣✐↔

❍♦➔♥❣ ❚❤à ❚❤✉ ❍÷ì♥❣

✐


ớ ỡ

ữủ t ữợ sỹ ữợ ừ
tọ ỏ t ỡ s s tợ t
ữớ ữợ t t ú ù tr sốt q tr tỹ

ỷ ớ ỡ tợ trữớ ừ
Pỏ ồ ũ t ổ rữớ ồ sữ
ồ rữớ ồ ữ ở
ồ ú ù t
t ỡ ỗ rữớ P
t ú ù ồ t tr q
tr ồ t t

s ổ t t t sỹ tổ s
ở tớ ừ tọ ỏ t ỡ s
s
ỹ ỏ ổ tr ọ ỳ t
sõt rt ữủ sỹ õ õ ỵ ừ t ổ
ữủ t ỡ

t ỡ

t

ồ

ữỡ




▼ö❝ ❧ö❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▲í✐ ❝↔♠ ì♥

✐✐

▼ö❝ ❧ö❝

✐✐✐

▼Ð ✣❺❯

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✷

✶✳✶
✶✳✷
✶✳✸
✶✳✹
✶✳✺

❍➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❉✉♥❣ ❧÷ñ♥❣ t÷ì♥❣ ✤è✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ❝ü❝ trà t÷ì♥❣ ✤è✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
❍➔♠ ●r❡❡♥ ♣❤ù❝ ✈î✐ ❝ü❝ t↕✐ ✈æ ❝ò♥❣

✳
✳
✳
✳
✳

✳
✳
✳

✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳

✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳
✳
✳

✳
✳
✳

✳
✳

✷
✹
✻
✻
✼

✷ ✣→♥❤ ❣✐→ ✤❛ t❤ù❝

✶✺

❑➌❚ ▲❯❾◆

✸✼

❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✸✽

✷✳✶ ✣à♥❤ ❧þ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✷ Ù♥❣ ❞ö♥❣ ❝õ❛ ✤à♥❤ ❧þ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✐✐✐


é
tự ởt ố tữủ q trồ ừ
t õ t t q ỵ rstrss õ r ồ

tử tr t t ợ ừ tự ởt
õ ỵ tỹ t s tr ởt t ợ tổ q
ừ tự tr t ọ ỡ ởt t tự ữ
t tự rst r ừ tự
ự q ừ tự tr ởt t ổ ỹ
tũ ỵ ở ừ tr ởt số t q ừ
ri ữ ỳ ỏ ữợ tr
Cn õ tờ qt ỡ ừ ổ ừ tự õ
ố ử ữ s
ữỡ tr tự ỏ ữợ ỏ
ữợ t q trồ ữủ tữỡ ố
ữỡ ở ừ ỗ ỵ 2.1.2
ở ợ ỏ ữợ t ử tr Cn tọ ởt số
õ õ ỹ tr ở ợ tr ỳ t rt
õ ở s ữỡ t ú tổ ử ỵ 2.1.2
ự ở ợ ừ tự tr t
t tỹ ừ Rn r ú tổ ỏ t ỳ ự ử ừ t
q ổ ừ tự tr t tr t
số t tỹ




ữỡ

tự
r ữỡ ú t tr ởt số ụ ữ
t q tt ữủ sỷ ử ữỡ s

ỏ ữợ

sỷ X ổ tổổ u
[; +)

:

ồ ỷ tử tr tr X ợ ộ R t
X = {x X : u(x) < }

tr X v : X [; +) ồ ỷ tử ữợ tr
X v ỷ tử tr tr X

sỷ

t tr C u :
[; +) ồ ỏ ữợ tr õ ỷ tử tr tr
tọ t tự ữợ tr tr ợ ồ
w tỗ t > 0 s ợ ồ 0 r < t õ


2

1
u(w)
2

u(w + reit ) dt
0

ú ỵ r ợ tr t ỗ t tr ữủ

❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω✳ ❚❛ ❦➼ ❤✐➺✉ t➟♣ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥
Ω ❧➔ SH(Ω)✳
❑➳t q✉↔ s❛✉ ✤➙② ❝❤♦ ♠ët ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ♠ët ❤➔♠ ❦❤↔ ✈✐ ❧➔ ❤➔♠ ✤✐➲✉
❤á❛ ❞÷î✐✳

✣à♥❤ ❧þ ✶✳✶✳✸✳
✈➔ ❝❤➾ ❦❤✐

●✐↔ sû u ∈ C 2 (Ω)✳ ❑❤✐ ✤â u ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω ❦❤✐

u ≥ 0 tr➯♥ Ω✱ ð ✤â

u=

∂2u
∂x2

+

∂2u
∂y 2

❧➔ ▲❛♣❧❛❝❡ ❝õ❛ u✳

●✐↔ sû u ≥ 0 tr➯♥ Ω✳ ▲➜② D ❧➔ ♠✐➲♥ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐
tr♦♥❣ Ω ✈➔ h ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ tr➯♥ ♠✐➲♥ D✱ ❧✐➯♥ tö❝ tr➯♥ D s❛♦ ❝❤♦
lim sup(u − h)(z) ≤ 0 ✤è✐ ✈î✐ ♠å✐ ζ ∈ ∂D✳ ❱î✐ ε > 0✱ ①→❝ ✤à♥❤
❈❤ù♥❣ ♠✐♥❤✳

z→ζ

υε (z) =




u(z) − h(z) + ε |z|2 ♥➳✉ z ∈ D


ε |z|2 ♥➳✉ z ∈ ∂D.

❑❤✐ ✤â υε ♥û❛ ❧✐➯♥ tö❝ tr➯♥ D ♥➯♥ ♥â ✤↕t ❝ü❝ ✤↕✐ tr➯♥ D✳ ❚✉② ♥❤✐➯♥
❞♦ υε = u + 4ε > 0 tr➯♥ D ♥➯♥ υε ✤↕t ❝ü❝ ✤↕✐ tr➯♥ ∂D✳ ❉♦ ✤â
u − h ≤ sup |z|2 tr➯♥ D✳ ❈❤♦ ε → 0 t❛ ✤÷ñ❝ u ≤ h tr➯♥ D ✈➔✱ ❞♦ ✤â✱ u
∂D
✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ D✳
◆❣÷ñ❝ ❧↕✐✱ ❣✐↔ sû u ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω✳ ●✐↔ t❤✐➳t t↕✐ ω ∈ Ω t❛ ❝â
u(ω) < 0✳ ❉♦ ✤â ❝â > 0 s❛♦ ❝❤♦ u ≤ 0 tr➯♥ (ω, )✳ ❚❤❡♦ ✤✐➲✉ ✈ø❛
❝❤ù♥❣ ♠✐♥❤ t❤➻ −u ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ (ω, )✳ ❉♦ ✤â u ❧➔ ❤➔♠ ✤✐➲✉
❤á❛ tr➯♥ (ω, )✳ ❱➟② u(ω) = 0 ✈➔ ❣➦♣ ♠➙✉ t❤✉➝♥✳ ❉♦ ✤â u ≥ 0 ✈➔
✤à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
✣à♥❤ ❧þ s❛✉ ✤➙② ❝❤♦ t❤➜② t➼♥❤ ✤✐➲✉ ❤á❛ ❞÷î✐ ❧➔ ❜➜t ❜✐➳♥ q✉❛ →♥❤ ①↕
❝❤➾♥❤ ❤➻♥❤✳

✣à♥❤ ❧þ ✶✳✶✳✹✳ ●✐↔ sû f : Ω1 → Ω2 ❧➔ →♥❤ ①↕ ❝❤➾♥❤ ❤➻♥❤ ❣✐ú❛ ❤❛✐ t➟♣ ♠ð
✸

tr♦♥❣ C✳ ◆➳✉ u ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω2 t❤➻ u ◦ f ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐
tr➯♥ Ω1 ✳

❱➻ t➼♥❤ ✤✐➲✉ ❤á❛ ❞÷î✐ ❧➔ t➼♥❤ ✤à❛ ♣❤÷ì♥❣ ♥➯♥ ❝❤➾ ❝➛♥ ❝❤ù♥❣
♠✐♥❤ u ◦ f ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ ♠é✐ ♠✐➲♥ ❝♦ ❝♦♠♣❛❝t t÷ì♥❣ ✤è✐ D1 Ω1✳
●✐↔ sû D1 ❧➔ ♠✐➲♥ ♥❤÷ ✈➟②✳ ❑❤✐ ✤â D2 = f (D1) Ω2✳ ❈❤å♥ ❞➣② ❤➔♠
✤✐➲✉ ❤á❛ ❞÷î✐ trì♥ {un} ∈ C ∞(D2) s❛♦ ❝❤♦ un u tr➯♥ D2✳ ❚❤❡♦ ✣à♥❤
❧þ 1.1.3 ❝â un ≥ 0 tr➯♥ D2 ✈î✐ ♠å✐ n ≥ 1✳ ❚❛ ❝â
❈❤ù♥❣ ♠✐♥❤✳

2

(u ◦ f ) = ( (un ) ◦ f ) |f |

tr➯♥ D1.

❉♦ ✤â t❤❡♦ ✣à♥❤ ❧þ 1.1.3 t❛ ❝â u ◦ f ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ D1✳ ◆❤÷♥❣
un ◦ f
u ◦ f tr➯♥ D1 ♥➯♥ u ◦ f ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ D1 ✈➔ ✤à♥❤ ❧þ ✤÷ñ❝
❝❤ù♥❣ ♠✐♥❤✳

✶✳✷ ❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐
✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ ●✐↔ sû Ω ⊂ Cn ❧➔ t➟♣ ♠ð✱ u : Ω → [−∞; +∞) ❧➔

❤➔♠ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✱ ❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❜➡♥❣ −∞ tr➯♥ ♠å✐ t❤➔♥❤ ♣❤➛♥
❧✐➯♥ t❤æ♥❣ ❝õ❛ Ω✳ ❍➔♠ u ❣å✐ ❧➔ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Ω ✭✈✐➳t u ∈ PSH(Ω)✮
♥➳✉ ✈î✐ ♠å✐ a ∈ Ω ✈➔ b ∈ Cn✱ ❤➔♠ λ → u(a + λb) ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ ❤♦➦❝
❜➡♥❣ −∞ tr➯♥ ♠å✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➟♣ {λ ∈ C : a + λb ∈ Ω}✳

✣à♥❤ ❧þ ✶✳✷✳✷✳ ●✐↔ sû u : Ω → [−∞, +∞) ❧➔ ❤➔♠ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✱

❦❤æ♥❣ ✤ç♥❣ ♥❤➜t ❜➡♥❣ −∞ tr➯♥ ♠å✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ Ω ⊂ Cn ✳
❑❤✐ ✤â u ∈ PSH(Ω) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈î✐ ♠å✐ a ∈ Ω, b ∈ Cn s❛♦ ❝❤♦

{a + λb : λ ∈ C, |λ| ≤ 1} ⊂ Ω

✹


t õ
2

1
u(a)
2

u(a + rei b)d := l(u, a, b)



0

s r tứ 1.2.1
ừ sỷ a , b Cn t
ự

U = { C : a + b }

õ U t tr C t () = u(a+b), U ự
() ỏ ữợ tr U ố ự tọ 0 U tỗ
t > 0 s 0 r < t

2

1
(0 )
2

(0 + rei )d
0

ứ a + 0b U õ > 0 s || < t a + 0b + b
ợ 0 r < t õ {a + 0b + rb : || 1} õ tứ tt
2

1
u(a + 0 b)
2

u(a + 0 b + rbei )d
0

(0) 21

2
0

(0 + rei )d

õ ự

õ trữ s ừ t ỏ ữợ

ỵ sỷ Cn t u C 2() õ u
2

PSH() ss Hu (z) = ( zj uzk ) ừ u t z
ữỡ ợ ồ w = (w1 , w2 , ..., wn ) Cn
n

2u
Hu (z)(w, w) =
(z)wj wk 0.
z

z

j
k
j,k=1



❈❤ù♥❣ ♠✐♥❤✳

❙✉② r❛ tø ✤➥♥❣ t❤ù❝✿ ❱î✐ ♠å✐ z ∈ Ω, w ∈ Cn ✈➔ ξ ∈ C t❛ ❝â
1
4

n

ξ u(z

+ ξw)|ξ=0

∂ 2u
=
(z)wj w¯k
∂z
∂
z
¯
j
k
j,k=1

✈➔ ✣à♥❤ ♥❣❤➽❛ 1.2.1 ❝ò♥❣ ✈î✐ ✣à♥❤ ❧þ 1.1.3✳

✶✳✸ ❉✉♥❣ ❧÷ñ♥❣ t÷ì♥❣ ✤è✐
✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ●✐↔ sû Ω ⊂ Cn ❧➔ t➟♣ ♠ð ✈➔ E ⊂ Ω ❧➔ t➟♣ ❇♦r❡❧✳

❉✉♥❣ ❧÷ñ♥❣ t÷ì♥❣ ✤è✐ ❝õ❛ E ✤è✐ ✈î✐ Ω✱ ❦➼ ❤✐➺✉ Cn(E, Ω) ❤❛② ❝â t❤➸ ✈✐➳t
❧➔ Cn(E) ♥➳✉ ❦❤æ♥❣ ❣➦♣ ♣❤↔✐ sü ❤✐➸✉ ♥❤➛♠ ♥➔♦ ❦❤→❝✱ ❧➔ ✤↕✐ ❧÷ì♥❣ ❝❤♦
❜ð✐
(ddc u)n : u ∈ PSH(Ω), −1 ≤ u ≤ 0}.

Cn (E) = Cn (E, Ω) = sup{
E

❇ð✐ ❜➜t ✤➥♥❣ t❤ù❝ ❈❤❡r♥✲ ▲❡✈✐♥❡✲ ◆✐r❡♥❜❡r❣ Cn(E) ❧➔ ❤ú✉ ❤↕♥ ♥➳✉ E

Ω✳

✶✳✹ ❍➔♠ ❝ü❝ trà t÷ì♥❣ ✤è✐
✣à♥❤ ♥❣❤➽❛ ✶✳✹✳✶✳ ●✐↔ sû Ω ⊂ Cn ❧➔ t➟♣ ♠ð ✈➔ E ⊂ Ω✳ ❍➔♠ ❝ü❝ trà
t÷ì♥❣ ✤è✐ ❝õ❛ ❊ ✤è✐ ✈î✐ Ω✱ ✤÷ñ❝ ❦➼ ❤✐➺✉ ❧➔ uE,Ω ✈➔ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣
t❤ù❝
uE,Ω (z) = sup{v(z) : v ∈ PSH(Ω), v ≤ −1

tr➯♥

E, v ≤ 0

tr➯♥

Ω}, z ∈ Ω

❍➔♠ u∗E,Ω ∈ PSH(Ω) ✈➔ −1 ≤ u∗E,Ω ≤ 0, z ∈ Ω, u∗E,Ω(z) = −1 ❦❤✐ z ∈ E ✳
Ð ✤➙②✱ u∗E,Ω ❧➔ ❝❤➼♥❤ q✉② ❤â❛ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ❝õ❛ uE,Ω✳ ❍➔♠ ♥➔② ✤÷ñ❝
✤à♥❤ ♥❣❤➽❛ ❜ð✐
u∗E,Ω (z) = lim sup uE,Ω (ρ).
ρ→z

✻


r ự ợ ỹ t ổ ũ
ởt ợ ỏ q trồ õ ự ử tr t
ự ỵ tt t tở ợ

u PSH() ữủ ồ õ ở t rt

tỗ t số Cu s ợ ồ z Cn
u(z) log z + Cu
2
n
ợ z =
j=1 |zj | , z = (z1 , z2 , ..., zn ).
t ỏ ữợ tr Cn õ ở t rt L(C n)
L ổ õ ữ

L = {u PSH(Cn ) : sup (u(z) log z ) < +}.
zCn

ợ L ữủ ồ ợ ỏ t ợ ừ ợ
L+ ữủ ữ s
L+ = {u PSH(Cn ) : C1 (u), C2 (u)

s ,

C1 (u) + log z u(z) C2 (u) + log z }.

sỷ E t
VE (z) = sup{u(z) : u L, u|E 0}, z Cn

ồ r ự ừ t E ợ ỹ t

ử

sỷ E = B(a, r) = {z Cn :

z a r}

VB(a,r) = log+

õ

za
, z Cn
r



✳ ❚r♦♥❣ ✤â✱ log+ z−a
= max(0, log z−a
r
r )✳
❚❤➟t ✈➟②✱ ✈➳ ♣❤↔✐ t❤✉ë❝ L ✈➔ ≤ 0 ❦❤✐ z ∈ E ✳ ❱➟② t❤❡♦ ✤à♥❤ ♥❣❤➽❛✱
VB(a,r) ≥ log+

z−a
, z ∈ Cn .
r

●✐↔ sû u ∈ L, u|E ≤ 0✳ ▲➜② w ∈ Cn\E ✈➔ ①→❝ ✤à♥❤ ❤➔♠
1
ω−a
υ(t) = u(a + (w − a)) − log+
,
t
|t| r
(0, w−a

ð ✤â✱ t ∈ (0, w−a
r )\{0}✳ ❍➔♠ υ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ t❤❡♦ t ∈
r )\{0}
✈➔ ❞♦ u ∈ L ♥➯♥ υ(t) ≤ c ❦❤✐ t → 0✳ ❱➟② υ t❤→❝ tr✐➸♥ tî✐ ❤➔♠ ✤✐➲✉ ❤á❛
w−a
❞÷î✐ υ˜ tr➯♥ t ∈ (0, w−a
t❤➻ υ˜(t) ≤ 0✳ ❱➟② t❤❡♦
r )✳ ❑❤✐ |t| =
r
♥❣✉②➯♥ ❧þ ❝ü❝ ✤↕✐✱ υ˜ ≤ 0 tr➯♥ (0, w−a
˜(1) =
r )✳ ✣➦❝ ❜✐➺t υ(1) = υ
≤ 0✳ ❚ø ✤â u(w) ≤ log+ w−a
❦❤✐ w ∈ Cn\E ✳ ◆➳✉
u(w) − log+ w−a
r
r
w ∈ E t❤➻ u(w) ≤ 0 = log+ w−a
r ✳
✈î✐ ♠å✐ w ∈ Cn ✈➔ ✤➥♥❣ t❤ù❝
❉♦ ✤â u(w) ≤ log+ w−a
r
VB(a,r) (z) = log+

z−a
r

✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✸✳ ●✐↔ sû P ❧➔ ✤❛ t❤ù❝ tr➯♥ Cn ✈➔ B(a, r) ❧➔ ❤➻♥❤ ❝➛✉ t➙♠

❜→♥ ❦➼♥❤ r tr♦♥❣ Cn✳ ❍➔♠ u(z) = deg1 P log
❱➟②✱ ✈➼ ❞ö tr➯♥ ❝❤♦ t❛

a

|P (z)|
P B(a,r)

1
|P (z)|
z−a
log
≤ max 0, log
deg P
P B(a,r)
r

∈L

✈➔ u|B(a,r) ≤ 0.

, ∀z ∈ Cn .

❉♦ ✤â t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ♠➔ ✤÷ñ❝ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❇❡r♥st❡✐♥✲
❲❛❧s❤✿
deg P
|P (z)| ≤ P

B(a,r)

max 1,

z−a
r

.

❉÷î✐ ✤➙② t❛ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè t➼♥❤ ❝❤➜t ✤è✐ ✈î✐ ❤➔♠ VE ✳
✽


ỵ

E1 E2 t VE1 VE2

E t ỹ VE +.
E ổ t ỹ t VE L.
E ổ t ỹ t VE L ỹ tr Cn \ E.

VEF
= VE F t ỹ

Ej

E t VEj

VE .

ứ ừ VE ú t õ

E1 E2 t VE VE sỷ E t ỹ õ u L
ợ E {u = } õ ợ ồ M > 0, u+M VE õ VE = +
tr {u > } s r VE +. ữủ VE + õ
{uj }j L, uj |E 0 supj uj (z) = + ỡ tr Cn
ự tọ ồ U = {uj : j 1} ổ tr ữỡ t
tr t ợ ồ a Cn õ r > 0 số C s

ự

1

2

z B(a, r), j : uj (z) C.

tr ợ supj uj (z) = + ỡ tr Cn ữ tỗ
t B(a, r) Cn ừ {uj } t õ t {uj } s
Mj = sup uj B j ợ ồ j ứ ử r ự õ ỹ
t õ
uj (z) Mj + log+

za
, z Cn .
r

ự õ z0 Cn s
lim sup exp(uj (z0 ) Mj ) > 0.
j

●✐↔ sû ♥❣÷ñ❝ ❧↕✐✱ lim sup exp(uj (z) − Mj ) ≤ 0 ✈î✐ ♠å✐ z ∈ Cn✳ ❚❛ ❝â✱
j

1
lim sup exp(uj (z) − Mj ) ≤ , ∀z ∈ B(a, r)
2
j

✈➔ j ✤õ ❧î♥.

✣✐➲✉ ✤â tr→✐ ✈î✐ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ Mj ✳ ✣➦t
δ = lim sup exp(uj (z0 ) − Mj ).
j

❈❤å♥ ❞➣② {uj }k≥1 s❛♦ ❝❤♦
k

lim sup exp(ujk (z0 ) − Mjk ) = δ
j

❳➨t ❤➔♠

✈➔

Mjk ≥ 2k

✈î✐ ♠å✐ k.

∞

2−k (ujk (z) − Mjk ), z ∈ Cn .

ω(z) =
k=1

❚ø ✭✶✳✷✮ t❛ ❝â

2−k (ujk (z) − Mjk ) − 2−k log+

✣➦t

R
≤ 0, ∀z ∈ B(a, R), R > r.
r

∞

2−k (ujk (z) − Mjk ) − 2−k log+

ωk (z) =
k=1

✭✶✳✸✮

R
, z ∈ Cn .

r

t❤➻ ωk ∈ PSH(B(a, R)) ✈➔ ωk ≤ 0✳ ❱➟② ❤➔♠ ∞k=1 ωk = ω − log+ Rr ❤♦➦❝
❧➔ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ B(a, R) ❤♦➦❝ ❜➡♥❣ −∞✳ ❉♦ R > 0 ✤÷ñ❝
❝❤å♥ tò② þ ♥➯♥ ω ❧➔ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ❤♦➦❝ ❜➡♥❣ −∞ tr➯♥ Cn✳ ◆❤÷♥❣
ω(z0 ) > −∞ ♥➯♥ ω ❧➔ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ Cn ✳ ❍ì♥ ♥ú❛ tø ✭✶✳✷✮
t❛ s✉② r❛ ω ∈ L✳ ◆➳✉ z ∈ E t❤➻ uj (z) ≤ 0✳ ❱➟②
k

∞

∞
−k

ω(z) ≤ −

2 Mjk ≤
k=1

−1 = −∞.
k=1

❉♦ ✤â E ❧➔ ✤❛ ❝ü❝✳
✸✳ ❚ø ❝❤ù♥❣ ♠✐♥❤ ♣❤➛♥ ✷ t❛ ❝â ♣❤➛♥ ✸✳
✹✳ ❉♦ ✸✳ t❛ ❝â VE∗ ❧➔ ❜à ❝❤➦♥ ✤à❛ ♣❤÷ì♥❣✳ ❚❛ ❝â ❞➣② {uj } ⊂ L, uj |E
✶✵

≤

ỡ tr Cn sỷ B Cn \ E ởt
õ õ t t uj uj ỏ ữợ ỹ tr B
uj
VE tr B VE ỹ tr B õ VE ỹ tr Cn

E E F VE VEF
sỷ u L, u|E 0 ứ tt t
õ L, F { = } E t õ t 0 tr E
u + L u + |EF 0 ợ ồ > 0 õ
0, uj

VE

u + VEF

tr Cn.


u(z) VEF t ồ z { > } ứ õ u(z) VEF
ợ ồ

z Cn VE VEF
t ữủ ự
ứ Ej E s r VE VE VE VE tợ
ỏ ữợ u VE õ t tt E ổ ỹ õ u L
t P = {VE < VE } ỹ õ u = 0 tr E \ P
j

VE
u VE\P VE\P = VE VE

j

j

j

j

j

j

K1 K2 ... t t
ừ Cn K =



Kj t lim VKj (z) = VK (z) ợ ồ z Cn
j

j=1

ứ t õ VK (z) VK (z) ... VK (z) ợ
z Cn tỗ t lim VK (z) VK (z) ợ ồ z Cn
j
sỷ u L, u 0 tr K > 0 {z Cn : u(z) < }
t ự K õ j0 s Kj {z Cn : u(z) < } ứ õ
u VK lim VK tr Cn ữ VK (z) lim VK (z)
j
j

ữủ ự

ự

1

2

j

0

j0

q

j

j

K Cn t t t VK ỷ tử ữợ

tr Cn




●✐↔ sû u ∈ L, u|K ≤ 0✳ ❑❤✐ ✤â u ∗ χ ∈ L ✈➔ u ∗ χ u ❦❤✐
0 tr➯♥ Cn ✳ ❱➟② ✈î✐ δ > 0 ✈➔ ✈î✐ ♠é✐ x ∈ K ❝â sè x > 0 s❛♦ ❝❤♦

❈❤ù♥❣ ♠✐♥❤✳

u(x) ≤ u ∗ χ x (x) < u(x) + δ ≤ δ.

❉♦ ✤â tç♥ t↕✐ ❧➙♥ ❝➟♥ Vx ❝õ❛ x s❛♦ ❝❤♦ ✈î✐ ♠å✐ t ∈ Vx : u ∗ χ (t) < δ✳
❍å {Vx}x∈K ❧➔ ♠ët ♣❤õ ♠ð ❝õ❛ K ✳ ❱➟② ❝â {Vx , ..., Vx } ♣❤õ K ✳ ✣➦t
= min{ x , ..., x }✳ ❑❤✐ ✤â
x

1

1

r

r

u ∗ χ (z) ≤ u ∗ χ j (z), ∀j = 1, 2, ..., r, ∀z ∈ Cn .

◆❤÷ ✈➟② ✈î✐ ♠å✐ z ∈ K : u ∗ χ (z) < δ✳ ❉♦ ✤â u ∗ χ (z) − δ < 0 tr➯♥ K ✳
❱➟②
u ∗ χ (z) − δ ≤ VK (z), ∀z ∈ Cn .

❚ø ✤➙② t❛ ✤÷ñ❝
VK (z) = sup{u ∗ χ (z) − δ : > 0, δ > 0}

✈➔ ❤➔♠ VK ❧➔ ♥û❛ ❧✐➯♥ tö❝ ❞÷î✐✳

❍➺ q✉↔ ✶✳✺✳✼✳

◆➳✉ K ⊂ Cn ❧➔ t➟♣ ❝♦♠♣❛❝t ✈➔ VK∗ |K ≡ 0 t❤➻ VK ❧➔ ❤➔♠

❧✐➯♥ tö❝ tr➯♥ Cn ✱ ð ✤â VK∗ ❝❤➼♥❤ q✉② ❤â❛ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ❝õ❛ VK .

▲➜② a ∈ K ✳ ❉♦ VK∗ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ t↕✐ a ✈➔ VK∗ (a) = 0
♥➯♥ ❝â r > 0 s❛♦ ❝❤♦
❈❤ù♥❣ ♠✐♥❤✳

∀z ∈ B(a, r) : VK∗ (z) < 1.
+ z−a
∗
❉♦ ✤â VK∗ (z) − 1 ≤ log+ z−a
+ 1 ✈î✐ ♠å✐ z ∈ Cn
r ✳ ❱➟② VK (z) ≤ log
r
✈➔ ❤➔♠ VK∗ ∈ L✳ ❚ø ❣✐↔ t❤✐➳t t❛ ❝â VK∗ (z) ≤ VK (z) ✈î✐ ♠å✐ z ∈ Cn✳ ❱➟②
VK∗ = VK ✈➔ ❦➳t ❧✉➟♥ s✉② r❛ tø ❍➺ q✉↔ 1.5.6

✶✷


❍➺ q✉↔ ✶✳✺✳✽✳

●✐↔ sû K ⊂ Cn ❧➔ t➟♣ ❝♦♠♣❛❝t ✈➔

K = {z ∈ Cn : d(z, K) ≤ }, > 0.
❑❤✐ ✤â VK ❧✐➯♥ tö❝ tr➯♥ Cn ✈➔ ✈î✐ ♠å✐ z ∈ Cn

lim VK (z) = VK (z).
→0

❚➟♣ K ❧➔ t➟♣ ❝♦♠♣❛❝t✱ VK∗ |K ≥ 0 ✈➔ K = B(a, )✳
a∈K
∗
∗
❚❛ ❝❤ù♥❣ ♠✐♥❤ VK |K ≤ 0 ✈➔ ❞♦ ✤â VK |K = 0✳ ❍➺ q✉↔ 1.5.6 ❝❤♦ t❛
❦➳t q✉↔ VK ❧➔ ❤➔♠ ❧✐➯♥ tö❝ ✈➔ ❞➵ t❤➜② VK VK tr➯♥ Cn ❦❤✐
0✳
▲➜② t ∈ K ✳ ❑❤✐ ✤â ❝â a ∈ K s❛♦ ❝❤♦ t ∈ B(a, )✳ ❚ø K ⊃ B(a, ) ♥➯♥
VK (z) ≤ log+ z−a ✈î✐ ♠å✐ z ∈ Cn ✳ ❉♦ ✤â VK∗ (z) ≤ log+ z−a ✈î✐ ♠å✐
z ∈ Cn ✳ ✣➦❝ ❜✐➺t VK∗ (t) ≤ log+ t−a = 0✳ ❉♦ ✤â VK∗ |K ≤ 0 ✈➔ ❤➺ q✉↔
✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❈❤ù♥❣ ♠✐♥❤✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✾✳ ❚➟♣ ❝♦♥ E ⊂ Cn ❣å✐ ❧➔ L✲ ❝❤➼♥❤ q✉② t↕✐ a ∈ E ♥➳✉

❤➔♠ VE ❧✐➯♥ tö❝ t↕✐ a✳ ◆➳✉ E ❧➔ L✲ ❝❤➼♥❤ q✉② t↕✐ ♠å✐ a ∈ E t❤➻ E ✤÷ñ❝
❣å✐ ❧➔ t➟♣ L✲ ❝❤➼♥❤ q✉②✳
❚r÷í♥❣ ❤ñ♣ E ❧➔ t➟♣ ❝♦♠♣❛❝t tr♦♥❣ Cn t❤➻ tø ❍➺ q✉↔ 1.5.7 ❝â t❤➸ ❝❤♦
❦➳t q✉↔ s❛✉✿ ❚➟♣ E ❧➔ L✲ ❝❤➼♥❤ q✉② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ VE∗|E ≡ 0.
❙❛✉ ✤➙② ❧➔ ❦❤→✐ ♥✐➺♠ ❤➔♠ ❝ü❝ trà ❙✐❝✐❛❦ ✈➔ ♠ët sè ❦➳t q✉↔ ✈➲ ♠è✐ ❧✐➯♥ ❤➺
❣✐ú❛ ❤➔♠ ●r❡❡♥ ✈î✐ ❝ü❝ trà t↕✐ ∞ ✈➔ ❤➔♠ ❙✐❝✐❛❦✳
●✐↔ sû K ⊂ Cn ❧➔ t➟♣ ❝♦♠♣❛❝t ✈➔
PK = {p : Cn → C : p

❧➔ ✤❛ t❤ù❝✱

p

K

≤ 1, deg p ≥ 1}.

✣➦t
1

ΦK (z) = sup{|p(z)| deg p : p ∈ PK }, z ∈ Cn .

❍➔♠ ΦK (z) ✤÷ñ❝ ❣å✐ ❧➔ ❤➔♠ ❝ü❝ trà ❙✐❝✐❛❦✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉✳
✶✸


✣à♥❤ ❧þ ✶✳✺✳✶✵✳ ◆➳✉ K ⊂ Cn ❧➔ t➟♣ ❝♦♠♣❛❝t t❤➻ VK (z) = log ΦK (z), z ∈
ˆ ❧➔ ❜❛♦ ❧ç✐ ✤❛ t❤ù❝
Cn . ❍ì♥ ♥ú❛ VK (z) = VKˆ (z) ✈î✐ ♠å✐ z ∈ Cn ✱ ð ✤â K
❝õ❛ K ✳
❈❤ù♥❣ ♠✐♥❤✳

❱➟②

❱î✐ ♠é✐ p ∈ PK ✳ ❍➔♠ u(z) = deg1 p log |p(z)| ∈ L ✈➔ u|K ≤ 0✳
1
log |p(z)| = u(z) ≤ VK (z), z ∈ Cn .
deg p

❚ø ✤â log ΦK (z) ≤ VK (z) ✈î✐ ♠å✐ z ∈ Cn.
●✐↔ sû δ ✈➔ > 0✳ ✣➦t Kδ = {z ∈ Cn : d(z, K) ≤ δ✳ ❑❤✐ ✤â ❍➺ q✉↔ 1.5.8
❝❤♦ t❤➜② VK VK ❦❤✐ δ 0 ✈➔ VK ❧✐➯♥ tö❝✳ ❱➟② ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤
VK ≤ log ΦK ✳ ✣➦t u = VK ✳ ❱î✐ j ≥ 1 ✈➔ z ∈ Cn ✱ ✤➦t
δ

δ

δ

δ

hj (t, z) =




|t| (j −1 + exp u(t−1 z))1−j −1 + j −1 (t, z) , t ∈ C \ {0}


j −1 z ,

t = 0.

❑❤✐ ✤â✱ hj ❧➔ ❤➔♠ ❧✐➯♥ tö❝✱ t❤✉➛♥ ♥❤➜t ✭♥❣❤➽❛ ❧➔ h(λt, λz) = |λ| h(t, z)✱
✈î✐ ♠å✐ z ∈ C✮✱ h−1
j (0) = {0} ✈➔ lim hj (t, z) = exp u(z)✳ ❍ì♥ ♥ú❛ log hj ∈
j→∞
n+1
L(C )✳ ✣➦t Vj (z) = hj (1, z), z ∈ Cn ✳ ❚ø t➟♣ ♠ð U = {z ∈ Cn :
exp u(z) < 1 + } ❝❤÷❛ K t❛ ❝â Vj ≤ 1 + 2 tr➯♥ K ✈î✐ j ✤õ ❧î♥✳ ❱➟②
Vj ≤ (1 + 2 )ΦK tr➯♥ Cn ✈î✐ j ✤õ ❧î♥✳ ❱➟② ❝❤♦ j → ∞ t❛ ❝â exp u(z) ≤
(1+2 )ΦK (z) ✈î✐ ♠å✐ z ∈ Cn ✳ ❈❤♦
0 t❛ ✤÷ñ❝ ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳
❚ø ❜✐➸✉ ❞✐➵♥ tr➯♥ t❛ ❝â

VK (z) = VKˆ (z), ∀z ∈ Cn .

✶✹


❈❤÷ì♥❣ ✷

✣→♥❤ ❣✐→ ✤❛ t❤ù❝
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ t❛ tr➻♥❤ ❜➔② ✤→♥❤ ❣✐→ ✤ë ❧î♥ ❝→❝ ❤➔♠ ✤❛ ✤✐➲✉
❤á❛ ❞÷î✐ t♦➔♥ ❝ö❝ tr➯♥ Cn ✭t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥ ❝❤✉➞♥ ❤â❛ ♥➔♦ ✤â✮
❞ü❛ tr➯♥ ✤ë ❧î♥ tr➯♥ ♥❤ú♥❣ t➟♣ r➜t ❜➨ ❝❤➾ ❝➛♥ ❝â ✤ë ✤♦ ▲❡❜❡s❣✉❡ ❞÷ì♥❣✳
⑩♣ ❞ö♥❣ ✈✐➺❝ ✤→♥❤ ❣✐→ ✤ë ❧î♥ ❝→❝ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ✈➔♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉
✤→♥❤ ❣✐→ ✤ë ❧î♥ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ tr➯♥ ❝→❝ t➟♣ ❝♦♥ ❣✐↔✐ t➼❝❤ t❤ü❝ ❝õ❛ Rn ✈➔
♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❝õ❛ ❦➳t q✉↔ ♥➔② ✈➔♦ ✤→♥❤ ❣✐→ ♠æ✤✉♥ ❝õ❛ ✤❛ t❤ù❝ tr➯♥
❝→❝ t➟♣ ♥➡♠ tr♦♥❣ ❝→❝ t➟♣ ✤↕✐ sè ❤❛② t❤➟♠ ❝❤➼ ❧➔ ❣✐↔✐ t➼❝❤ t❤ü❝✳

✷✳✶ ✣à♥❤ ❧þ ❝❤➼♥❤
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ❍➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ f
♥➳✉ ♥â t❤ä❛ ♠➣♥
f = 0❀

: Cn −→ R

t❤✉ë❝ ❧î♣

Fr (r > 1)

✭✐✮ sup
B (0,r)
✭✐✐✮ sup

c

f ≥ −1.

Bc (0,1)

❉÷î✐ ✤➙②✱ B(x, ρ) ✈➔ Bc(x, ρ) ❧➔ ❝→❝ ❤➻♥❤ ❝➛✉ ❒❝❧✐t ✈î✐ t➙♠ x ✈➔ ❜→♥
❦➼♥❤ ρ tr♦♥❣ Rn ✈➔ Cn t÷ì♥❣ ù♥❣✳

✶✺


❈❤♦ ❤➻♥❤ ❝➛✉ B(x, t) t❤ä❛ ♠➣♥
B(x, t) ⊂ Bc (x, at) ⊂ Bc (0, 1)

✭✷✳✶✮

ð ✤â a > 1✱ ❧➔ ♠ët ❤➡♥❣ sè ❝è ✤à♥❤✳

✣à♥❤ ❧þ ✷✳✶✳✷✳

❚ç♥ t↕✐ ❝→❝ ❤➡♥❣ sè c = c(a, r) > 0 ✈➔ d = d(n) > 0 s❛♦

❝❤♦ ❜➜t ✤➥♥❣ t❤ù❝

sup f ≤ c log
B(x,t)

d|B(x, t)|
+ sup f

|ω|
ω

✭✷✳✷✮

✤ó♥❣ ✈î✐ ♠é✐ f ∈ Fr ✈➔ ♠å✐ t➟♣ ❝♦♥ ✤♦ ✤÷ñ❝ ω ⊂ B(x, t)✱ ð ✤➙② |ω| ❧➔ ✤ë
✤♦ ▲❡❜❡s❣✉❡ ❝õ❛ ω ✳

✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧þ ♥➔② ❝❤ó♥❣ t❛ ❝➛♥ ♠ët sè ❦➳t q✉↔
♣❤ö trñ ❝❤♦ ❤➔♠ ✤❛ ✤✐➲✉ ❤á❛ ❞÷î✐ ✈➔ ✤✐➲✉ ❤á❛ ❞÷î✐✳
❳➨t ❤å Ar ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ❧✐➯♥ tö❝ ❦❤æ♥❣ ❞÷ì♥❣ f : D → R s❛♦
❝❤♦
❈❤ù♥❣ ♠✐♥❤✳

−1 ≤ sup f.
Dr

✭✷✳✸✮

Ð ✤➙② Dr := {z ∈ C; |z| < r}, D := D1 ✈➔ r ❧➔ ♠ët sè ❝è ✤à♥❤✱ 0 < r < 1✳
❑➳t q✉↔ ❜ê trñ ✤➛✉ t✐➯♥ ❝õ❛ ❝❤ó♥❣ tæ✐✳

▼➺♥❤ ✤➲ ✷✳✶✳✸✳

❱î✐ ♠é✐ f ∈ Ar tç♥ t↕✐ ♠ët ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ hf :

C −→ R ✈➔ ♠ët ❤➡♥❣ sè cf > 0 s❛♦ ❝❤♦

✭✐✮ hf /cf ∈ L(C)❀
✭✐✐✮ f = hf tr➯♥ Dr ❀

✭✐✐✐✮ sup cf < ∞.
f ∈Ar

≤ |z| ≤ 1+r
❈❤♦ R := { 1+3r
4
2 } ❧➔ ❤➻♥❤ ✈➔♥❤ ❦❤✉②➯♥ tr♦♥❣
D \ Dr ✱ ✈➔ ❦➼ ❤✐➺✉ X ❧➔ ❤å ❝→❝ ✤÷í♥❣ trá♥ ✤ç♥❣ t➙♠ 0 ✈➔ ❜→♥ ❦➼♥❤ ♥➡♠
❈❤ù♥❣ ♠✐♥❤✳

✶✻


tr♦♥❣ R✳
✣➸ ❝❤ù♥❣ ♠✐♥❤ ▼➺♥❤ ✤➲ tr➯♥ t❛ ❝➛♥ ❦➳t q✉↔ s❛✉✳

❇ê ✤➲ ✷✳✶✳✹✳

❈❤♦ f ∈ Ar ✈➔

t(f ) := sup inf f (z).
S∈X z∈S

❚❤➻

C(r) := inf t(f ) > −∞.
f ∈Ar

❈❤ù♥❣ ♠✐♥❤✳

✭✷✳✹✮

▲➜② {fi}i≥1 ⊂ Ar s❛♦ ❝❤♦
lim t(fi ) = C(r).

i→∞

❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ ❞➣② ♥➔② ❦❤æ♥❣
❝❤ù❛ ❤➔♠ ❦❤æ♥❣✳ ❱î✐ ♠é✐ S ⊂ X ❝❤ó♥❣ t❛ ✤➦t
Si := {z ∈ S; fi (z) = min fi }
S

✈➔
Si .

Ki :=
s∈X

❉♦ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ fi ♥➯♥ Ki ❧➔ t➟♣ ❝♦♠♣❛❝t✳ ❚➟♣ ❝→❝ ❣✐→ trà ❜→♥ ❦➼♥❤ ❝õ❛
1+r
1−r
❝→❝ ✤✐➸♠ t❤✉ë❝ Ki ❧➜♣ ✤➛② ✤♦↕♥ t❤➥♥❣ Ir := [ 1+3r
4 , 2 ] ✤ë ❞➔✐ w(r) := 4 ✳
❉♦ ✤â ✤÷í♥❣ ❦➼♥❤ s✐➯✉ ❤↕♥ δ(Ki) ❝õ❛ Ki t❤ä❛ ♠➣♥
δ(Ki ) ≥ δ(Ir ) =

w(r)
.
4

✭✷✳✺✮

❇➙② ❣✐í ❝❤ó♥❣ t❛ ✤➦t
mi := max fi
Ki

✈➔

gi :=

fi
.
|mi |

✭✷✳✻✮

Ð ✤➙② mi < 0✱ ♥➳✉ ❦❤æ♥❣ t❤➻ fi ✤ç♥❣ ♥❤➜t ❜➡♥❣ ✵✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛
♣❤↔✐ ÷î❝ ❧÷ñ♥❣ |mi| ❜➡♥❣ ♠ët ❤➡♥❣ sè ❦❤æ♥❣ ♣❤ö t❤✉ë❝ i ≥ 1✳ ✣➸ ❦➳t t❤ó❝
✶✼


❝❤ó♥❣ t❛ s➩ s♦ s→♥❤ gi ✈î✐ ❤➔♠ ❝ü❝ trà t÷ì♥❣ ✤è✐ uK ,D ❝õ❛ ❝➦♣ (Ki, D)✳
❈❤ó þ r➡♥❣✱ uK ,D ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐
i

i

✭✷✳✼✮

uKi ,D(z) := sup{v(z) : v ∈ SH(D), v|Ki ≤ −1, v ≤ 0}

✈î✐ z ∈ D✳ Ð ✤➙② SH(D) = PSH(D) ✈î✐ n = 1✳ ❚ø gi ≤ −1 tr➯♥ Ki✱ t❤❡♦
✤à♥❤ ♥❣❤➽❛✱ ❝❤ó♥❣ t❛ ❝â
✭✷✳✽✮

gi ≤ uKi ,D .

❑➼ ❤✐➺✉ (uK ,D)∗ ❧➔ ❝❤➼♥❤ q✉② ❤â❛ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ❝õ❛ uK ,D✳ ❉♦ ✤â ❤➔♠
♥➔② ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ tr♦♥❣ D✳ ❉♦ gi ≤ 0 ✈➔ uK ,D ≤ 0 ♥➯♥ t❤❡♦ ❜➜t
✤➥♥❣ t❤ù❝ ✭✷✳✽✮ ❝❤ó♥❣ t❛ ❝â
i

i

i

|(uKi ,D )∗ | ≤ |gi | =

|fi |
.
|mi |

❚ø ✤â✱ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣ ð ♠ët ✤✐➸♠ ♥❤➜t ✤à♥❤ z0 ∈ D✱ t❛ ❝â
|mi | ≤

|fi (z0 )|
.
|(uKi ,D )∗ (z0 )|

✭✷✳✾✮

✣➸ ❝❤å♥ z0 ✈➔ ÷î❝ ❧÷ñ♥❣ ♠➝✉ tr♦♥❣ ✭✷✳✾✮ ❝❤ó♥❣ t❛ sû ❞ö♥❣ ♠è✐ q✉❛♥ ❤➺
❣✐ú❛ ❤➔♠ ❝ü❝ trà t÷ì♥❣ ✤è✐ ✈➔ ❞✉♥❣ ❧÷ñ♥❣ cap(Ki, D)✱ ✈î✐
(uKi ,D )∗ dxdy.

cap(Ki , D) :=
D

❉♦ (uK ,D)∗ ❧➔ ❤➔♠ ✤✐➲✉ ❤á❛ ♥❣♦➔✐ tr➯♥ ♣❤➛♥ ❜ò ❝õ❛ Ki ❝❤ó♥❣ t❛ ❝â t❤➸
✈✐➳t ❧↕✐ ✈➳ ♣❤↔✐ ♥❤÷ s❛✉✳
▲➜② R ⊂ D ❧➔ ❤➻♥❤ ✈➔♥❤ ❦❤✉②➯♥ tò② þ ❝❤ù❛ ❤➻♥❤ trá♥ conv(R) = {z; |z| ≤
1+r
2 } ✈➔ ρ ❧➔ ❤➔♠ trì♥ ✈î✐ ❣✐→ tr♦♥❣ ❝♦♥✈ ✭❘✬✮ ✈➔ ❜➡♥❣ ✶ tr♦♥❣ conv(R )\R ✳
❚❤❡♦ ❝æ♥❣ t❤ù❝ ●r❡❡♥✬s t❛ ❝â
i

ρ (uKi ,D )∗ dxdy = |

cap(Ki , D) =
D

R

(uKi ,D )∗ ρdxdy| ≤ C max |(uKi ,D )∗ |.

❚ø ❤➔♠ (uK ,D)∗ ❦❤æ♥❣ ❞÷ì♥❣ ✈➔ ✤✐➲✉ ❤á❛ tr♦♥❣ D \ Ki✱ t❛ ❝â
i

max |(uKi ,D )∗ | ≤ C |(uKi ,D )∗ (z0 )|
R

✶✽

R


ợ số C ử tở r ộ z0 R
t ủ ợ t tự s ú t t r t
tự
|mi |

C |fi (z0 )|
cap(Ki , D)

ú ợ ộ z0 R ữ t ừ Ar ừ
fi t õ 0 > max fi 1 ồ z0 ởt õ số t ỹ
R
t ú t õ
mi

C
.
cap(Ki , D)

ử ỵ s s ởt r r tr
t tự s ợ ữớ s ừ Ki õ trũ ợ
ữủ ừ Ki
(Ki ) exp(

2

).
cap(Ki , D)

t ủ ợ t tự tr t tự ú t õ
4
=: C (r)
w(r)

|mi | C log

ợ ồ i 1 ứ ừ fi s r
inf t(f ) = lim t(fi ) inf |mi | C (r) > .

f Ar

i

i

ờ ữủ ự
ớ ú t t ự 2.1.3 f Ar
ờ tỗ t ởt ữớ trỏ Sf X s
inf f C(r) > .
Sf

Sf

ừ trỏ Dr(f ) õ
1 + 3r
1+r

r(f )
.
4
2



❇➙② ❣✐í ❝❤ó♥❣ t❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ❝➛♥ t➻♠ hf (z) : C −→ R
❜ð✐

hf (z) :=






















(z ∈ Dr(f ) )

f (z)
max f (z),

4|z|
2C(r) log 3+r

4|z|
2C(r) log 3+r

)
log 4r(f
3+r

(z ∈ D Dr(f ) )
(z ∈ C D).

)
log 4r(f
3+r

⑩♣ ❞ö♥❣ ❜ê ✤➲ ❞→♥ ❝→❝ ❤➔♠ ✤✐➲✉ ❤á❛ ❞÷î✐ ✈➔ ❞♦ tr♦♥❣ ❝æ♥❣ t❤ù❝ t❤ù
❜❛ ❧➔ ♥❤ä ❤ì♥ C(r) < 0 tr➯♥ Sf ✈➔ ❧î♥ ❤ì♥ ✵ tr➯♥ ∂ D ✈➔ ✈➻ f ❧➔ ❧✐➯♥ tö❝✱
t❛ ❝â hf ❧➔ ✤✐➲✉ ❤á❛ ❞÷î✐ tr➯♥ C✳ ❍ì♥ t❤➳ ♥ú❛✱ t❤❡♦ ✣à♥❤ ♥❣❤➽❛ 1.5.1✱
4r(f )
3 + r h ∈ L(C).
f

2C(r)

log

❚❛ ✤➦t

4r(f )
3+r.
2C(r)

log
cf :=

❑❤✐ ✤â

1 + 3r
3 + r < ∞.
2C(r)

log
cf ≤

❑➳t ❤ñ♣ ❧↕✐ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳

▲➜② k ❧➔ ♠ët ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ❜→♥ ❦➼♥❤ tr➯♥ Cn
✈➔ t❤ä❛ ♠➣♥
k(x)dxdy = 1,
Cn

sup ♣(k) ⊂ Bc (0, 1),

✭✷✳✶✵✮

ð ✤â✱ z = x+iy ✈î✐ x, y ∈ Rn✳ ❈❤♦ Ω ⊂ Cn ❧➔ ♠ët ♠✐➲♥✳ ❱î✐ f ∈ PSH(Ω)✱
❝❤ó♥❣ t❛ ❦➼ ❤✐➺✉ fε ❧➔ ❤➔♠ ①→❝ ✤à♥❤ ❜ð✐
k(z)f (w − εz)dxdy,

fε (w) :=
Cn

✷✵

✭✷✳✶✶✮