*QM7Q`KH BMp`BM+2 M/ /QKBM J`FQp T`QT2`iv
6`QK i?2 TQBMi Q7 pB2r Q7 biiBbiB+H T?vbB+b- i?2 irQ KQbi BKTQ`iMi T`QT@
2`iB2b Q7SLE `2 +QM7Q`KH BMp`BM+2 M/ i?2 /QKBM J`FQp T`QT2`ivX h?2b2 T`QT2`iB2b `2 2tTHBM2/ BM /2iBH BM ST2` *- #mi ?2`2 r2 +M ;Bp2 M BM7Q`KH 2tTHMiBQM 7Q` i?2 +b2 Q7 `/BH SLEX
h?2 `M/QK `/BH GƺrM2` +?BM /2}M2/ #v URXjdV ;2M2`i2b `M/QK MQM@b2H7@+`QbbBM;SLEκ+m`p2 i?i #2;BMb i1M/ i2M/b iQ0X AM 7+i- i?Bb
`M/QK +m`p2 BM/m+2b T`Q##BHBiv K2bm`2 QM i?2 b2i Q7 MQM@b2H7@+`QbbBM;
+m`p2b +QMM2+iBM;1M/0X q2 +M /2MQi2 i?Bb K2bm`2 #vSLEκ(D,1,0)X :Bp2M Mv Qi?2` /QKBMD- T`BK2 2M/a Q7 DM/ M BMi2`BQ` TQBMi b∈D- i?2`2 2tBbib mMB[m2 +QM7Q`KH BbQKQ`T?BbKφ:D→Dbm+? i?i φ(0) =bM/φ(1) =ˆ aX
q2 +M /2}M2 K2bm`2 SLEκ(D, a, b) = SLEκ(φ(D),φ(1), φ(0))ˆ QM i?2 7KBHv Q7 MQM@b2H7@+`QbbBM; +m`p2b HvBM; BM D M/ +QMM2+iBM; a M/b BM +QM7Q`KHHv BMp`BMi rv, B7 i?2 Hr Q7 γ Bb ;Bp2M #vSLEκ(D,1, 0)-i?2M i?2 Hr Q7φ◦γ Bb ;Bp2M #vSLEκ(φ(D),φ(1), φ(0))Xˆ
h?2 /QKBM J`FQp T`QT2`iv K2Mb i?i +QM/BiBQM2/ QM M BMBiBH T`i Q7 i?2 +m`p2 γ[0, t]- i?2 `2KBMBM; +m`p2 ?b i?2 Hr SLEκ(Dt, γ(t), b)X
>2`2- Dt Bb i?2 mMB[m2 +QMM2+i2/ +QKTQM2Mi Q7 D\γ[0, t] +QMiBMBM; b -M/ i?2 K2bm`2 SLEκ(Dt, γ(t), b);`22b rBi? i?2 Q`B;BMH Hr Q7 γ BM +QM7Q`KHHv BMp`BMi rvX
h?2`2 Bb +HQb2 +QMM2+iBQM #2ir22M i?2 /QKBM J`FQp T`QT2`iv M/
i?2 7+i i?i 2[miBQMb /2}MBM; SLE +M #2 `27Q`KmHi2/ b /BzmbBQM 2[miBQMbX 6Q` BMbiM+2- BM i?2 +b2 Q7 i?2 `/BH SLEκ- i?2 +?M;2 Q7 p`B#H2bwt(z) =ft(z)/ei√κ Bt i`Mb7Q`Kb URXjdV BMiQ
dwt(z) =wt(z)1 +wt(z)
1−wt(z)dt−i√
κ wt◦dBt, w0(z) =z, z∈D.
RXj :mbbBM 7`22 }2H/
aiM/`/ `272`2M+2b QM i?2 /Bbi`B#miBQM i?2Q`v `2 (6yj) M/ (ai`yj)X h?2 :mbbBM 7`22 }2H/- i?2 KQbi BKTQ`iMi `M/QK /Bbi`B#miBQM BM irQ /BK2MbBQMb- Bb +Qp2`2/ BM /2iBH BM (a?2yd)- (aaRj) M/ (EJRj)X +QK@
T`2?2MbBp2 i`2iK2Mi Q7 >BH#2`i bT+2b- M BKTQ`iMi iQQH BM i?2 i?2Q`v Q7 i?2 :mbbBM 7`22 }2H/- Bb (CMNd)X
RXjXR a+?r`ix /Bbi`B#miBQMb BM i?2 +QKTH2t THM2
G2i D #2 bBKTHv +QMM2+i2/ /QKBM BM i?2 +QKTH2t THM2X G2i C0∞(D) /2MQi2 i?2 bT+2 Q7 BM}MBi2Hv /Bz2`2MiB#H2 `2H@pHm2/ 7mM+iBQMb rBi? +QK@
T+i bmTTQ`i BM DX b2[m2M+2 Q7 7mM+iBQMb {pn}∞n=1 ⊂C0∞(D)Bb bB/ iQ +QMp2`;2 BM i?2 b2Mb2 Q7 i2bi 7mM+iBQMb iQ i?2 7mM+iBQMp∈C0∞(D)B7
UBV i?2`2 2tBbib +QKT+i bm#b2iK bm+? i?isupp(pn−p)⊆K 7Q` HH n= 1,2, . . .
-k9 AMi`Q/m+iBQM UBBV ∂∂mm+px ∂pypn → ∂m∂m+px ∂pyp- b n → ∞- mMB7Q`KHv QM K 7Q` HH m, p =
1,2, . . .
h?2`2 2tBbib iQTQHQ;v QM C0∞(D) bm+? i?i HBM2` 7mM+iBQMHT QM C0∞(D)Bb +QMiBMmQmb B7 M/ QMHv B7T(pn)→T(p)r?2M2p2`pn→pBM i?2 b2Mb2 Q7 i2bi 7mM+iBQMbX h?2 bT+2 C0∞(D)2M/Qr2/ rBi? i?Bb iQTQHQ;v Bb +HH2/ i?2 bT+2 Q7 i2bi 7mM+iBQMb QMD- M/ Bb /2MQi2/ #vD(D)X
h?2 /mH bT+2 2M/Qr2/ rBi? i?2 r2F@bi` iQTQHQ;v Bb /2MQi2/ #v D(D)- M/ Bib 2H2K2Mib `2 +HH2/ Ua+?r`ixV /Bbi`B#miBQMbX
G2iA/2MQi2 i?2 `2 K2bm`2 QMD- M/ H2i(·,·)/2MQi2 i?2L2(D, A) BMM2` T`Q/m+i- bQ i?i
(p,q) :=
Dp(z)q(z)dA(z), 7Q` `2H@pHm2/ 7mM+iBQMbpM/qX
G2iL1loc(D, A)/2MQi2 i?2 bQ@+HH2/ bT+2 Q7 HQ+HHv BMi2;`#H2 7mM+iBQMb BM DX "v /2}MBiBQM- 7mM+iBQMh #2HQM;b iQ L1loc(D, A) B7h ∈ L1(U) 7Q`
2p2`v QT2M b2iU rBi? Bib +HQbm`2U HvBM; BMDX h?2 bT+2L1loc(D)Bb [mBi2
`B+?- M/ i?2 BM+HmbBQMLp(D, A)⊂L1loc(D, A)?QH/b 7Q` Mvp≥1X G2i h ∈ L1loc(D, A)X 1p2M i?Qm;? Bi Bb MQi i`m2 BM ;2M2`H i?i h ∈ L2(D, A)- i?2 BMM2` T`Q/m+i (h,p) Bb r2HH@/2}M2/ 7Q` Mv i2bi 7mM+iBQM p∈D(D)X JQ`2Qp2`-h/2}M2b +QMiBMmQmb /Bbi`B#miBQM QMD#v
p →(h,p), p∈D(D),
M/ rBi? i?Bb B/2MiB}+iBQM r2 +M r`Bi2 L1loc(D)⊂D(D)X
LQi 2p2`v a+?r`ix /Bbi`B#miBQM QM D+M #2 `2T`2b2Mi2/ #v HQ+HHv BMi2;`#H2 7mM+iBQM b #Qp2X L2p2`i?2H2bb- #v #mb2 Q7 MQiiBQM- r2 H@
rvb /2MQi2 #v (h,p) i?2 `2bmHi Q7 +iBQM Q7 /Bbi`B#miBQM h QM i2bi 7mM+iBQMpX
RXjXk :`22MǶb 7mM+iBQM
"v i?2 _B2KMM KTTBM; i?2Q`2K- ;Bp2M ?vT2`#QHB+ /QKBMD- 7Q` 2+?
w ∈ D i?2`2 2tBbib mMB[m2 +QM7Q`KH BbQKQ`T?BbK fw : D → D- bm+?
i?ifw(w) = 0,M/fw(w)>0X h?2 :`22MǶb 7mM+iBQM Q7D Bb /2}M2/ b GD(z, w) =−log|fw(z)|.
h?2 :`22MǶb 7mM+iBQM Bb bvKK2i`B+ M/ ?`KQMB+ BM #Qi? p`B#H2bX Ai Bb 2bv iQ +?2+F i?i 7Q` i?2 mMBi /BbF- GD(z, w) = log1−z−ww z¯ - M/
7Q` i?2 mTT2` ?H7@THM2GH(z, w) = logz−z−ww¯X
6Q` i2bi 7mM+iBQM p∈D(D)- :`22MǶb i?B`/ B/2MiBiv BKTHB2b
− 1
2π DGD(z, w) Δp(w)dA(w) =p(z). URXj3V q2 +M `2r`Bi2 i?Bb B/2MiBiv mbBM; i?2 7Q`KH HM;m;2 Q7 /Bbi`B#miBQMH Ur2FV /2`BpiBp2b Ub22 (6yj)V BM i?2 7QHHQrBM; 7Q`K
⎧⎨
⎩
−ΔwGD(z, w) = 2πδ(z−w), z∈D,
GD(z, w) = 0, z∈∂D. URXjNV
RXj :mbbBM 7`22 }2H/ k8 PM2 +M mb2 URXjNV b M Hi2`MiBp2 /2}MBiBQM Q7 :`22MǶb 7mM+iBQMX
h?2 GTH+BM Δ Bb M BMD2+iBp2 HBM2` QT2`iQ` BM D(D) U#2+mb2 ker Δ = {0} BM D(D)V- M/ URXj3V vB2H/b i?2 7QHHQrBM; 7Q`KmH 7Q` i?2 BMp2`b2 Q7 Δ
Δ−1p(z) =− 1
2π DGD(z, w)p(w)dA(w). URX9yV RXjXj .B`B+?H2i 2M2`;v M/ 2H2+i`QbiiB+ TQi2MiBH 2M2`;v q2 mb2 i?`22 /Bz2`2Mi BMM2` T`Q/m+ib QM i?2 bT+2 D(D)X PM2 Q7 i?2K-i?2 L2(D, A) BMM2` T`Q/m+i- ?b H`2/v #22M K2MiBQM2/X q2 /2MQi2 Bi
#v(·,·)- Q`- r?2M i?2 /QKBM Bb MQi +H2` 7`QK i?2 +QMi2ti-(·,·)L2(D,A)X 6Q` irQ 7mM+iBQMbp,q∈D
(D)-(p,q) :=
Dp(z)q(z)dA(z).
h?2 .B`B+?H2i BMM2` T`Q/m+i Q7 irQ 7mM+iBQMbp,q∈D(D)Bb /2}M2/ b (p,q)∇:=
D∇p(z)· ∇q(z)dA(z).
u2i MQi?2` BMM2` T`Q/m+i QMD(D)i?i r2 7`2[m2MiHv mb2 Bb (p,q)E(D) :=
D×D2GD(z1, z2)p(z1)q(z2)dA(z1)dA(z2).
h?2 7QHHQrBM; 7Q`KmHb `2HiBM; i?2 BMM2` T`Q/m+ib BMi`Q/m+2/ #Qp2 iQ 2+? Qi?2`- +M #2 2bBHv /2/m+2/ 7`QK :`22MǶb B/2MiBiB2bX AM HH Q7 i?2 7Q`KmHb-p,q∈D(D)X
(p,q) = 1
4π(−Δp,q)E(D), URX9RV
(p,q)∇= (−Δp,q), URX9kV
(p,q)∇= 1
4π(Δp,Δq)E(D). URX9jV h?2 BMM2` T`Q/m+ib BM/m+2 i?2 7QHHQrBM; MQ`Kb QMD(D)X
p:=(p,p), p∇:=(p,p)∇, pE(D):=(p,p)E(D). h?2 MQ`Kp∇ Bb +HH2/ i?2 .B`B+?H2i 2M2`;v Q7 p- M/ i?2 MQ`KpE(D) Bb bQK2iBK2b `272``2/ iQ b i?2 2H2+i`QbiiB+ TQi2MiBH 2M2`;v Q7 pX
RXjX9 .2}MBiBQM Q7 :mbbBM 7`22 }2H/
G2i (Ω,F,P) #2 T`Q##BHBiv bT+2X "v `M/QK /Bbi`B#miBQM UQ`- KQ`2 T`2+Bb2Hv- `M/QK a+?r`ix /Bbi`B#miBQMV QM D r2 K2M K2bm`#H2 KT Φ : Ω→D(D)X
`M/QK /Bbi`B#miBQMΦBb +HH2/ :mbbBM 7`22 }2H/ UGF FV QMDB7 7Q` 2p2`v i2bi 7mM+iBQMp∈D(D)i?2 `M/QK p`B#H2 (Φ,p)Bb +2Mi2`2/
:mbbBM rBi? p`BM+2 p2E(D)X
ke AMi`Q/m+iBQM
"v TQH`BxiBQM- Bi 7QHHQrb i?i i?2 +Qp`BM+2 biBb}2b Cov ((Φ,p),(Φ,q)) = (p,q)E(D).
"2HQr r2 2tTHBM ?Qr QM2 +M +QMbi`m+i :mbbBM 7`22 }2H/b 2tTHB+BiHvX G2iH(D)/2MQi2 i?2 >BH#2`i bT+2 +QKTH2iBQM Q7D(D)rBi? `2bT2+i iQ (·,·)∇X h?2 >BH#2`i bT+2 H(D) Bb b2T`#H2- M/ H(D)⊂ L2(D, A)⊂ L1loc(D)⊂D(D)X
G2i {en}∞n=1 #2 M Q`i?QMQ`KH #bBb BM H(D)- M/ H2i {αn}∞n=1 #2 b2[m2M+2 Q7 BM/2T2M/2Mi biM/`/ :mbbBM `M/QK p`B#H2bX *QMbB/2`
i?2 7Q`KH b2`B2b
h?2 b2`B2b URX99V /Bp2`;2b HKQbi bm`2Hv- #mi M2p2`i?2H2bb 7Q` Mv i2bi 7mM+iBQMp∈D(D)i?2 b2`B2b
>BH#2`i bT+2b (CMNd- 1tKTH2 RXk8)- i?2 b2`B2b URX98V Bb +2Mi2`2/ :mb@
bBM rBi? p`BM+2
B7 i?2 b2`B2b QM i?2 `B;?i ?M/@bB/2 +QMp2`;2bX "mi #v URX9kV- S`b2pHǶb B/2MiBiv M/ URX99V Bb :mbbBM 7`22 }2H/X JQ`2Qp2`- QM2 +M b?Qr i?i 2p2`v :mbbBM 7`22 }2H/ΦQMD+M #2 `2T`2b2Mi2/ bΦ = 2π∞n=1αnen- r?2`2{en}∞n=1 Bb M Q`i?QMQ`KH #bBb 7Q`H(D)- M/{αn}∞n=1 Bb b2[m2M+2 Q7 biM/`/
:mbbBM `M/QK p`B#H2bX
RXjX8 :mbbBM 7`22 }2H/ QM bm#/QKBM
G2i B ⊂ D #2 bBKTHv +QMM2+i2/ bm#/QKBM Q7 DX h?2 bT+2 H(B) Bb +HQb2/ bm#bT+2 Q7H(D)- M/ r2 /2MQi2 Bib Q`i?Q;QMH +QKTH2K2Mi #v Harm(B)- bQ i?iH(D)+M #2 /2+QKTQb2/ Q`i?Q;QMHHv b 7QHHQrb
H(D) =H(B)⊕Harm(B).
q2 /2MQi2 #v PH(B) M/ PHarm(B) i?2 Q`i?Q;QMH T`QD2+iBQMb QMiQ i?2 +Q``2bTQM/BM; +HQb2/ bm#bT+2bX
RXj :mbbBM 7`22 }2H/ kd
"v /2}MBiBQM
f ∈Harm(B)⇔(f, g)∇= 07Q` HHg ∈H(B)
⇔(f,Δg)L2(B,A)= 07Q` HHg∈H(B).
6mM+iBQMb biBb7vBM; i?2 Hbi +QM/BiBQM `2 +HH2/ r2FHv ?`KQMB+ QMBUQ`
?`KQMB+ BM i?2 b2Mb2 Q7 /Bbi`B#miBQMH /2`BpiBp2bV- #mi #v q2vHǶb H2KK (q2v9y)- r2F ?`KQMB+Biv Bb 2[mBpH2Mi iQ ?`KQMB+BivX >2M+2-Harm(B) +QMbBbib T`2+Bb2Hv Q7 i?Qb2 2H2K2Mib Q7 H(D)i?i `2 ?`KQMB+ QMBX
G2iΦB #2 :mbbBM 7`22 }2H/ QMBX Ai Kv #2 r`Bii2M b Q7 BM/2T2M/2Mi biM/`/ :mbbBM p`B#H2bX Ai im`Mb Qmi i?iΦBKv #2 i`2i2/ b `M/QK /Bbi`B#miBQM QM i?2 r?QH2 Q7 D- bm+? i?i 7Q` i2bi
"v 6iQmǶb i?2Q`2K Ub22 (>Q733- *?Ti2` j)V- 2p2`v #QmM/2/ ?`KQMB+
7mM+iBQM +M #2 `2T`2b2Mi2/ #v i?2 SQBbbQM BMi2;`H Q7 #QmM/2/ K2bm`@
#H2 7mM+iBQMh∂QM∂DX AM i?Bb +b2 r2 +M +HHΦˆD i?2 :mbbBM 7`22 }2H/
rBi? .B`B+?H2i #QmM/`v +QM/BiBQMh∂X A7hBb QMHv #QmM/2/ 7`QK #Qp2 Q`
7`QK #2HQr- i?2M i?2 >2`;HQix `2T`2b2MiiBQM i?2Q`2K BKTHB2b i?i Bi +M
#2 `2T`2b2Mi2/ b i?2 SQBbbQM BMi2;`H Q7 bB;M2/ K2bm`2 μhQM ∂DX
k3 AMi`Q/m+iBQM Bb biBHH r2HH@/2}M2/- ?2M+2ΦˆB /2}M2b `M/QK a+?r`ix /Bbi`B#miBQM QM i?2 r?QH2 Q7 DX
RXjXd SmHH#+Fb Q7 /Bbi`B#miBQMb M/ +QM7Q`KH BMp`BM+2 Q7 GF F
G2i φ:D1→D2 #2 +QM7Q`KH BbQKQ`T?BbK- M/ H2iΨ∈ D(D2)X h?2 TmHH#+F Ψ◦φQ7ΨrBi? `2bT2+i iQφBb i?2 /Bbi`B#miBQM QM D1/2}M2/ #v i?2 7Q`KmH
(Ψ◦φ,p) = (Ψ,|(φ−1)|2p(φ−1)), p∈C0∞(D1).
A7ΨBb ML1loc(D2)7mM+iBQM- i?2M i?Bb /2}MBiBQM +Q``2bTQM/b iQ i?2 +?M;2 Q7 p`B#H2b BM i?2 7Q`KmH
D1
Ψ(φ(z))p(z)dA(z) =
D2
Ψ(w)|(φ−1(w))|2p(φ−1(w))dA(w).
h?2 :mbbBM 7`22 }2H/ Bb +QM7Q`KHHv BMp`BMi- K2MBM; i?i i?2 TmHH#+F ΦD2 ◦φQ7 :mbbBM 7`22 }2H/ QM D2 ?b i?2 bK2 Hr bGF F BMD1X RXjX3 SLE M/ GF F
PM2 Q7 i?2 TQbbB#H2 7Q`KmHiBQMb Q7 i?2 +QM7Q`KH }2H/ i?2Q`v U*6hV Bb 7`QK i?2 TQBMi Q7 pB2r Q7 `M/QK }2H/b- BM T`iB+mH`- i?2 :mbbBM 7`22 }2H/X h?2 Ki?2KiB+H bT2+ib Q7 i?Bb 7Q`KmHiBQMb `2 r2HH@mM/2`biQQ/-b22 (EJRj)X
h?2 7+i i?i i?2 :mbbBM 7`22 }2H/ Bb +HQb2Hv `2Hi2/ iQ SLE4 rb }`bi `2HBx2/ BM (aay8)- M/ bBM+2 i?2M HQi Q7 T`Q;`2bb ?b #22M K/2 BM i?Bb /B`2+iBQMX *QMM2+iBQMb #2ir22M GF F M/ SLEκ 7Q` Qi?2` pHm2b Q7 κ?p2 HbQ #22M 2bi#HBb?2/ (.m#yN- a?2Ry)X
"2HQr r2 /2b+`B#2 i?2 bBKTH2bi 2tKTH2 Q7 +QMM2+iBQM #2ir22M SLE M/GF F U+?Q`/H SLE4V- b22 i?2 /2iBHb BM (aayN)- (a?2Ry) M/ (aaRj)X
G2i ΦH :mbbBM 7`22 }2H/ BM i?2 mTT2` ?H7@THM2- M/ +QMbB/2` i?2 7QHHQrBM; KQ/B}+iBQM Q7ΦH,
ΦˆH= ΦH−√ 2 argz.
AM 7+i- ΦˆH Bb :mbbBM 7`22 }2H/ rBi? i?2 #QmM/`v +QM/BiBQM ΦˆH= 0 QM i?2 TQbBiBp2 `2H tBb- M/ΦˆH=πQM i?2 M2;iBp2 `2H tBbX
G2i Bt #2 "`QrMBM KQiBQM BM/2T2M/2Mi Q7ΦˆH- M/ H2i{wt}t≥0 #2 i?2 +Q``2bTQM/BM; +?Q`/HSLE4biQ+?biB+ ~Qr- bQ i?i
dwt(z) = 2
wt(z)dt−2dBt, w0(z) =z, z∈H.
h?2 BMi2`2biBM; +QMM2+iBQM #2ir22M i?2 KQ/B}+iBQM ΦˆHM/ i?2 ~Qr Q7 +?Q`/H SLE4 Bb i?i 7Q` Mv /2i2`KBMBbiB+ iBK2 T > 0- i?2 `M/QK }2H/bΦˆHM/ΦˆH◦wT ?p2 i?2 bK2 HrX
AM ST2` . r2 2tTHBM i?Bb `2HiBQMb?BT BM /2iBH M/ BMp2biB;i2 r?2i?2`
bBKBH` `2bmHi Kv #2 i`m2 7Q` Qi?2` bHBi ?QHQKQ`T?B+ biQ+?biB+ ~QrbX
*?Ti2` k JBM _2bmHib
AM i?Bb +?Ti2` r2 ;Bp2 M Qp2`pB2r Q7 i?2 M2r `2bmHib BMi`Q/m+2/ BM i?Bb i?2bBbX h?2 bi`m+im`2 Bb b 7QHHQrb, a2+iBQMb kXR@kX9 `2 bmKK`B2b Q7 2+?
Q7 i?2 `2b2`+? TT2`b BM+Hm/2/ BM i?2 i?2bBb- M/ BM a2+iBQM kX8 r2 QmiHBM2 TQbbB#H2 /B`2+iBQMb 7Q` 7m`i?2` `2b2`+?X