j j ( ϕ j ) p (dd c ϕ j ) n < (dd c ϕ j ) n <.
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- Shavonne Bond
- 6 years ago
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1 ÆÆÄË ÈÇÄÇÆÁÁ ÅÌÀÅÌÁÁ ½º¾ ¾¼¼µ ÓÒÖÒÒ Ø ÒÖÝ Ð E p ÓÖ 0 < p < 1 Ý ÈÖ ËÙÒ ÚÐе Ê ÞÝ ÃÖÛµ Ò È º Ñ ÀÓÒ À º Ô ÀÒÓµ ØÖغ Ì ÒÖÝ Ð E p ØÙ ÓÖ 0 < p < 1º ÖØÖÞØÓÒ Ó Ö¹ ØÒ ÓÙÒ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ Ò ØÖÑ Ó F p Ò Ø ÔÐÙÖÓÑÔÐÜ p¹òöý ÔÖÓÚº ½º ÁÒØÖÓÙØÓÒº ÄØ C n ÓÙÒ ÝÔÖÓÒÚÜ ÓÑÒ ºº ØÖ Ü Ø ÓÙÒ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ ϕ : (, 0) Ù ØØ Ø ÐÓ ÙÖ Ó Ø Ø {z : ϕ(z) < c} ÓÑÔØ Ò ÓÖ ÚÖÝ c (, 0)º ÁÒ Ø ÖØÐ ÓÙÖ Ð Ó Ø Ø ÙÒØÓÒ ÛÐÐ Ø ÓÒÚÜ ÓÒ E 0 (= E 0 ()) ÓÒ ØÒ Ó ÐÐ ÓÙÒ ÔÐÙÖ ÙÖÑÓÒ Ì ÙÒØÓÒ ϕ Ò ÓÒ Ù ØØ lim z ξ ϕ(z) = 0 ÓÖ ÚÖÝ ξ Ò (ddc ϕ) n < ÛÖ (dd c ) n Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖº ÙÑ ØØ u ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ Ò ÓÒ Ò [ϕ j ] ϕ j E 0 Ö Ò ÕÙÒ Û ÓÒÚÖ ÔÓÒØÛ ØÓ u ÓÒ j º Á ØÖ Ò ÒÓ Ñ ÒØÖÔÖØØÓÒ ÕÙÒ [ ] ÛÐÐ ÒÓØ Ý [ ]º ÓÖ p > 0 Ü ÓÒ Ö Ø ÓÐÐÓÛÒ ÖØÓÒ ½µ sup j ¾µ sup j ( ϕ j ) p (dd c ϕ j ) n < (dd c ϕ j ) n <. Á Ø ÕÙÒ [ϕ j ] Ò Ó Ò Ù ØØ (1) ÓÐ ØÒ Û Ý ØØ u ÐÓÒ ØÓ E p Ò (2) ÓÐ ØÒ u ÐÓÒ ØÓ Fº ÒÐÐÝ ÓØ (1) Ò (2) Ö Ø ØÒ u F p º ÓÖ p = 0 Û Ý Ý ÓÒÚÒØÓÒ ØØ u Fº Ì ÒÖÝ Ð F p Ò E p Ö ØÛÓ Ó Ø Ó ÐÐ ÖÐÐ Ð º ÓÖ p 1 Ø Ð F p Ò E p ÛÖ ÒØÖÓÙ Ò ÜØÒ ÚÐÝ ØÙ Ò Ò Ö Û ÛÐÐ ØÙÝ ØÑ ÓÖ 0 < p < 1º ÓÖ ÙÖØÖ ÒÓÖÑØÓÒ ÓÙØ Ø ÖÐÐ Ð ºº Ò Ø ÖÖÒ ØÖÒº ÁØ ÓÐÐÓÛ ÖÓÑ ¾¼¼¼ ÅØÑØ ËÙØ Ð ØÓÒ ÈÖÑÖÝ ¾Í½ ËÓÒÖÝ ¾Ï¾¼º ÃÝ ÛÓÖ Ò ÔÖ ÖÐÐ Ð ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÖÐØ ÔÖÓÐÑ ÔÐÙÖÓÑÔÐÜ ÒÖÝ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ º ½½ ÁÒ ØÝØÙØ ÅØÑØÝÞÒÝ ÈÆ ¾¼¼
2 ½¾¼ Ⱥ Ø Ðº ØØ ÒÝ ÙÒØÓÒ Ò E p Ò E Ò Ò Ý Ø ÓÔÖØÓÖ (dd c ) n ÛÐÐ Ò ÓÒ E p p 0 ÓÖ Ø ÒØÓÒ Ó Eµº ÆÓÛ ÐØ e p (u) Ò Ý e p (u) = ( u) p (dd c u) n ÓÖ p > 0º Ì ÒØÖÐ e p (u) Ø ÔÐÙÖÓÑÔÐÜ Ô¹ÒÖÝ Ó Ø ÙÒØÓÒ uº Ò ½½ Ø ÔÐÙÖÓÑÔÐÜ p¹òöý ÛÐÐ Ù ØÓ ØÙÝ E p º ÁÒ ½½ ÈÖ ÓÒ ÔÖÓÚ ØØ p 1 Ò u 0, u 1,...,u n E 0 ØÒ ( u 0 ) p dd c u 1 dd c u n D n,p e p (u 0 ) p/(p+n) e p (u 1 ) 1/(p+n) e p (u n ) 1/(p+n) Ð Ó µ ÛÖ D n,p ÓÒ ØÒØ ÔÒÒ ÓÒÐÝ ÓÒ n Ò pº Ì ÀÐÖ ØÝÔ ÒÕÙÐØÝ ÙÒÑÒØÐ ØÓÓÐ Ò º ÁÒ ËØÓÒ ¾ Û ÛÐÐ ÜØÒ Ø ØÑØ ØÓ p > 0 ÖØ ÓÒ ÕÙÒ Ø ÓÐÐÓÛ ØØ F p Ò E p Ö ÓÒÚÜ ÓÒ ÓÖÓÐÐÖÝ ¾ºµº Ì Ñ Ó Ø ÖØÐ ØÓ ÔÖÓÚ Ø ÓÐÐÓÛÒ ÖØÖÞØÓÒ Ó Ø ÖÐØ ÔÖÓÐÑ ÄØ n 1 p > 0 Ò µ ÒÓÒ¹ÒØÚ Ñ ÙÖ ÒÓØ Ò ÖÐÝ Ó ÒØ ØÓØÐ Ñ µº ÌÒ ØÖ Ü Ø ÙÒÕÙ ÙÒØÓÒ u E p Ù ØØ (dd c u) n = µ Ò ÓÒÐÝ ØÖ Ü Ø ÓÒ ØÒØ A > 0 Ù ØØ ( ϕ) p dµ Ae p (ϕ) p/(n+p) ÓÖ ÚÖÝ ϕ E 0 ÌÓÖÑ ºµº ÓÖ p 1 Ø Û ÔÖÓÚ Ò º ÖÐØ ÖÐØ ÔÖÓÐÑ ÓÖ Ø p = 0 Û ÓÒ Ö Ò º ÁÒ ËØÓÒ Û ÛÐÐ ÔÖÓÚ Ò ÔÔÐØÓÒ Ó Ø ÖÑÛÓÖ ÒÙ Ý Ø ÒÖÝ Ð ØØ u E 0 Ò ÓÒÐÝ ½µ u F p ÓÖ ÚÖÝ p 0 ¾µ lim u(z) = 0 ÓÖ ÚÖÝ ξ z ξ µ supe p (u) 1/p < º p>0 Ï Ò Ø ÖØÐ Ý ÓÒ ØÖÙØÒ ØÛÓ ÜÑÔÐ Û ÑÓØÚØ Ø Ö¹ ØÖÞØÓÒº Ì ÙØÓÖ ÛÓÙÐ Ð ØÓ ØÒ ÍÖÒ ÖÐÐ ÆÙÝÒ ÎÒ Ã٠˹ ÛÓÑÖ ÃÓÓÞ Ò Ñ Ö ÓÖ ÚÐÙÐ ÐÔ ÛØ Ø ÑÒÙ ÖÔغ ¾º ÀÐÖ ØÝÔ ÒÕÙÐØݺ Ï ÛÐÐ ÔÖÓ Ò ½½ Ý Ù Ò ÄÑÑ ¾º½ ÐÓÛ ÓÙÒØÖÔÖØ Ó ÄÑÑ º½ Ò ½½ º ÄÑÑ ¾º½º ÄØ u, v PSH() L () lim z ξ u(z) =lim z ξ v(z) =0 ÓÖ ÚÖÝ ξ Ò T ÔÓ ØÚ ÐÓ ÙÖÖÒØ Ó Ö (n 1, n 1)º
3 ÒÖÝ Ð E p ÓÖ 0 < p < 1 ½¾½ ÓÖ 0 < p < 1 ( u) p dd c v T p 1 1 p ( ( u) p dd c u T ) ( p ) 1 p+1 ( v) p dd c p+1 v T. ÈÖÓÓº ÄØ 0 < p < 1 Ò w = ( v) p º ÌÒ w PSH() L () Ò lim z ξ w(z) = 0 ÓÖ ÚÖÝ ξ º Ï Ú (2.1) ( u) p dd c v T = = 1 p 1 p 1 p p 2 ÀÐÖ³ ÒÕÙÐØÝ ÝÐ ¾º¾µ ( u) p (dd c ( w) 1/p ) T ( u) p ( w) 1/p 1 dd c ( w) T ( u) p dd c v T 1 p ( u) p ( w) 1/p 2 d( w) d c ( w) T ( u) p ( w) 1/p 1 dd c w T = 1 p = 1 p = 1 p [ [ [ ( u)dd c w T ( w)dd c u T ( v) p dd c u T ] p [ ] p [ ] p [ Ý ÓÑÒÒ ÒÕÙÐØ ¾º½µ Ò ¾º¾µ Û Ø ( u) p dd c v T 1 p [ [ p 1+p 1 ( v) p dd c u T [ ] p [ ( u) p dd c v T ( v) p dd c v T] 1 p. ÌÙ Ø Ö ÒÕÙÐØÝ Úº ( u) p ( v) 1 p dd c w T. ( v)dd c w T ( w)dd c v T ] 1 p ] 1 p ( v) p dd c v T] 1 p. ( v) p dd c v T ] p 2[ ] 1 p ( u) p dd c u T ] p(1 p) ÌÓÖÑ ¾º¾º ÄØ u 0, u 1,...,u n E 0 Ò p > 0º ÙÑ ØØ X ÒÓÒ¹ÑÔØÝ Ø n 1 Ò ÒØÖ Ò ØØ F : X n+1 R ÙÒØÓÒ Û ÝÑÑØÖ Ò Ø Ð Ø n ÚÖÐ º Á ØÖ Ü Ø ÓÒ ØÒØ C > 0 Ù ØØ F(u 0, u 1,...,u n ) CF(u 0, u 0, u 2,...,u n ) p+1f(u p 1, u 1, u 2,...,u n ) p+1, 1
4 ½¾¾ Ⱥ Ø Ðº ØÒ F(u 0, u 1,...,u n ) C α(n,p) F(u 0,...,u 0 ) p p+1f(u 1,...,u 1 ) 1 p+1 F(u n,...,u n ) 1 p+1, ÛÖ α(n, p) ÚÒ Ý α(1, p) = 1, α(n, p) = α(n 1, p) + ( (p + 1)(p + n 1) 1 + p(p + n) ÅÓÖÓÚÖ C 1 ØÒ ( ) p + 1 n 1 α(n, p) = (p + 2) (p + 1). p ÈÖÓÓº º ÌÓÖÑ º½ Ò ½½ º ) α(n 1, p). p + 1 ÄØ p > 0º Ì ÑÙØÙÐ ÔÐÙÖÓÑÔÐÜ Ô¹ÒÖÝ (u 0,...,u n ) p Ò Ý (u 0,...,u n ) p = ( u 0 ) p dd c u 1 dd c u n. ÓÖ p 1 ÌÓÖÑ ¾º ÐÓÛ Û ÔÖÓÚ Ò ½½ º Á p = 0 ØÒ ¾º µ Ò ÒØÖÔÖØ ÓÖÓÐÐÖÝ º Ò º ÌÓÖÑ ¾º º ÄØ p > 0 Ò u 0, u 1,...,u n E 0 º ÌÒ ¾º µ (u 0,..., u n ) p D n,p e p (u 0 ) p/(p+n) e p (u 1 ) 1/(p+n) e p (u n ) 1/(p+n), ÛÖ p α(n,p)/(1 p) 0 < p < 1, D n,p = 1 p = 1, p pα(n,p)/(p 1) p > 1, Ò α(n, p) = (p + 2) ( p+1) n 1 p (p + 1)º ÈÖÓÓº ÄØ 0 < p < 1 Ò (u 0, u 1,...,u n ) p = F(u 0, u 1,...,u n ) Ò ÌÓ¹ ÖÑ ¾º¾º Ì ÔÖÓÓ ØÒ ÓÐÐÓÛ ÖÓÑ ÄÑÑ ¾º½ Ò ÌÓÖÑ ¾º¾º ÓÖÓÐÐÖÝ ¾ºº ÓÖ ÒÝ p 0 Ø Ð F p Ò E p Ö ÓÒÚÜ ÓÒ º ÈÖÓÓº Ì ÓÐÐÓÛ Ò Ý Ù Ò ÌÓÖÑ ¾º º Á q > p > 0 ØÒ F q F p Ý ÀÐÖ³ ÒÕÙÐØݺ Ï ÛÐÐ Ò Ø ØÓÒ Ý ÜÔÐÒÒ ÛÝ ÑÐÖ Ö ÙÐØ ÓÖ E p ÒÓØ ÔÓ Ðº ÄØ q > p > 0 ܺ ÌÒ Ø ÓÐÐÓÛ ÖÓÑ ÜÑÔÐ ¾º Ò ØØ E p \ E q ÒÓÒ¹ÑÔØݺ ÜÑÔÐ ¾º ÐÓÛ ÓÛ ØØ E q \ E p ÒÓÒ¹ÑÔØÝ ÛÐк Ö Ø ÒÓØ ØØ u 1,...,u k E 0 ØÒ k ¾ºµ e p (u u k ) e p (u j ).
5 ÒÖÝ Ð E p ÓÖ 0 < p < 1 ½¾ Ï ÛÐÐ Ð Ó Ò Ø ÓÐÐÓÛÒ ÐÑѺ ÄÑÑ ¾ºº ÄØ p>0 Ò u, v E 0 º ÌÒ e p (u+v) e p (u) e p (v) 0º ÈÖÓÓº ÄØ 0 < p < 1º ÀÐÖ³ ÒÕÙÐØÝ ØÓØÖ ÛØ ¾º µ Ò Ø Ø ØØ ( u v) p ( u) p + ( v) p ÝÐ n ¾ºµ e p (u) e p (u + v) e p (u) + C e p (u) p+ne j p (v) p+n j p+n Ò Ø 0 < p < 1 ÔÖÓÚº ÙÑ ÒÓÛ ØØ p 1º Í Ò ÅÒÓÛ ³ ÒÕÙÐØÝ Û Ø ¾ºµ e p (u + v) 1/p ÑÔÐÓÝÒ ¾º µ ØÓ ØÑØ Ò j=0 [ ( u) p (dd c (u + v)) n] 1/p [ + ( u) p (dd c u) n j (dd c v) j ( v) p (dd c u) n j (dd c v) j ØÓØÖ ÛØ ¾ºµ ÓÑÔÐØ Ø ÔÖÓÓº ( v) p (dd c (u + v)) n] 1/p. ÓÖ j = 1,...,n ÓÖ j = 0,...,n ÊÑÖº ÁØ ÓÐÐÓÛ ÖÓÑ Ø ØÑØ ¾ºµ Ò ÜÑÔÐ º½½ Ò ØØ ( p>0 E p) \ F º ÜÑÔÐ ¾ºº ÄØ q > p > 0 Ò g = g(z, z 0 ) Ø ÔÐÙÖÓÑÔÐÜ ÖÒ ÙÒØÓÒ ÛØ ÔÓÐ z 0 º Ò v j = j p max(g, 1/j p+n ) E 0 º ÌÒ e p (v j ) = (2π) n Ò e q (v j ) = (2π) n j n(p q) Ò lim j e q (v j ) = 0º ÌÖÓÖ ÄÑÑ ¾º ÑÔÐ ØØ ØÖ Ü Ø ÒØÖ s j Ù ØØ Ø Ö Ò ÕÙÒ Ò Ý u k = v s1 + + v sk ÓÒÚÖ ÔÓÒØÛ ØÓ ÙÒØÓÒ u E q º ÁÒÕÙÐØÝ ¾ºµ ÑÔÐ ØØ e p (u k ) k(2π) n º ÌÙ u / E p º º Ì ÖÐØ ÔÖÓÐÑ ÄÑÑ º½º ÄØ p 0 Ò K {F p, E p }º Á u K Ò v PSH() v 0 ØÒ max(u, v) Kº ÈÖÓÓº ÓÖ Ø p = 0 º Ò ÓÖ Ø p 1 º ÄØ 0 < p < 1 Ò u E p º ÌÒ Ý ÒØÓÒ ØÖ Ü Ø Ö Ò ÕÙÒ [u j ] u j E 0 Û ÓÒÚÖ ÔÓÒØÛ ØÓ u j Ò sup j e p (u j ) < º ËØ w j = max(u j, v)º ÌÒ [w j ] w j E 0 Ö Ò ÕÙÒ Û ÓÒÚÖ ÔÓÒØÛ ØÓ max(u, v) j Ò sup j e p (w j ) sup j e p (u j ) < Ò Ì max(u, v) E p º Á u F p ØÒ Û ØÓÒÐÐÝ Ò Ì ØÓ ÔÖÓÚ ØØ sup j (ddc w j ) n < º ÙØ u j w j Û ÑÔÐ ØØ sup j (ddc w j ) n sup j Ì (ddc u j ) n < º
6 ½¾ Ⱥ Ø Ðº ÓÖ p 1 ÄÑÑ º¾ ÐÓÛ Û ÔÖÓÚ Ò º Ý Ù Ò ÌÓÖÑ ¾º ØÓØÖ ÛØ ÄÑÑ º½ Ø ÔÖÓÓ Ó ÄÑÑ º Ò Ð Ó ÚÐ ÓÖ Ø 0 < p < 1º ÄÑÑ º¾º ÄØ p > 0º Á ψ PSH() C() ψ < 0 Ò u E p ØÒ χ A (dd c u) n = χ A (dd c max(u, ψ)) n, ÛÖ χ A Ø ÖØÖ Ø ÙÒØÓÒ Ó Ø Ø A = {z : u > ψ}º ÄÑÑ º º ÄØ p 0º Á u, v E p Ö Ù ØØ u v ØÒ ( ϕ)(dd c v) n ÓÖ ÚÖÝ ϕ PSH() ÛØ ϕ 0º ( ϕ)(dd c u) n ÈÖÓÓº Ö Ø ÙÑ ØØ ϕ E 0 º ÌÒ ÒØÖØÓÒ Ý ÔÖØ µ ÑÔÐ ØØ º½µ ( ϕ)(dd c u) n = ÙØ u v Ý ÙÑÔØÓÒ Ò ØÖÓÖ º¾µ ( u)(dd c ϕ) (dd c u) n 1 ( u)(dd c ϕ) (dd c u) n 1 ; Ý Ù Ò ÒØÖØÓÒ Ý ÔÖØ ÓÒ Ò Û Ø º µ ( v)(dd c ϕ) (dd c u) n 1 = ( v)(dd c ϕ) (dd c u) n 1. ( ϕ)(dd c v) (dd c u) n 1 Ò ØÖÓÖ Ì ( ϕ)(ddc u) n Ì ( ϕ)(ddc v) (dd c u) n 1 Ý º½µ º µº ÓÒØÒÙÒ Ò ÑÐÖ ÑÒÒÖ Ù Ò ÒØÖØÓÒ Ý ÔÖØ Ò Ø ÙÑÔ¹ ØÓÒ u v ÝÐ Ø Ö ÒÕÙÐØÝ ÛÒ ϕ E 0 º Ì ÒÖÐ ØÒ ÓÐÐÓÛ ÖÓÑ ÌÓÖÑ ¾º½ Ò ØÓØÖ ÛØ Ø ÑÓÒÓØÓÒ ÓÒÚÖÒ ØÓÖѺ ÓÖ p = 0 ÌÓÖÑ º ÐÓÛ Û ÔÖÓÚ Ò ÌÓÖÑ º½µ Ò ÓÖ p 1 Ø ÓÐÐÓÛ ÖÓÑ Ø ÔÖÓÓ Ó ÌÓÖÑ º¾ Ò º ÀÖ Û ÛÐÐ Ù Ø ÑØÓ Ó ØÓ Ú Ø Ö ÙÐØ ÓÖ 0 < p < 1º ÌÓÖÑ ºº ÄØ p 0º Á u E Ò v E p Ö Ù ØØ (dd c v) n (dd c u) n ØÒ u vº ÈÖÓÓº ÙÑ ØØ 0 < p < 1 Ò ÐØ h E 0 C() ÒÓØ ÒØÐÐÝ 0º ÓÖ m 1 ÄÑÑ º½ Ò º¾ ÑÔÐÝ ØØ (dd c max(v, mh)) n = χ {v>mh} (dd c v) n + χ {v mh} (dd c max(v, mh)) n. ÃÓÓÞ³ ØÓÖÑ ½¼ Ò Ð Ó ÈÖÓÔÓ ØÓÒ º½ Ò µ ÑÔÐ ØØ ØÖ Ü Ø g m E 0 Ù ØØ (dd c g m ) n = χ {v mh} (dd c max(v, mh)) n º
7 ÒÖÝ Ð E p ÓÖ 0 < p < 1 ½¾ ÌÙ (dd c (u + g m )) n (dd c max(v, mh)) n º ÌÓÖÑ º½ Ò ÓÛ ØØ max(v, mh) u + g m Ò ºµ v = lim sup m max(v, mh) u + lim supg m. m ÄØ w m = sup j m g j º ÌÒ w m E 0 ÛÖ w ÒÓØ Ø ÙÔÔÖ ÑÓÒ¹ ØÒÙÓÙ ÖÙÐÖÞØÓÒ Ó Ø ÙÒØÓÒ wº ÅÓÖÓÚÖ [w m ] Ö Ò ¹ ÕÙÒ Û ÓÒÚÖ ÔÓÒØÛ ØÓ lim sup m g m m º Ü m 1 Ò ÐØ j mº ÄÑÑ º Ò Ø Ø ØØ max(v, jh) g j w m ÑÔÐÝ ØØ e p (wm) m p ( h) p (dd c wj) n m p ( h) p (dd c g j ) n ( ) m p = ( jh) p χ j {v jh} (dd c max(v, jh)) n ( ) m p sup e p (max(v, jh)) < j j m Ò ØÖÓÖ w m = 0º ÀÒ lim sup m g m = lim m w m = 0 ÐÑÓ Ø ÚÖÝÛÖ Ò Ý ÒÕÙÐØÝ ºµ Ø ÓÐÐÓÛ ØØ v uº Ì ÒÜØ ÓÖÓÐÐÖÝ Û ÔÖÓÚ Ò ½ ÓÖ p 1 Ò p = 0º Í Ò ÜØÐÝ Ø Ñ ÑØÓ ØÓØÖ ÛØ ÌÓÖÑ º ÝÐ Ø Ö Ø ØØÑÒغ Ì ÓÒ ØØÑÒØ ÓÐÐÓÛ ÖÓÑ ÜÑÔÐ º Ò ½ º ÓÖÓÐÐÖÝ ºº Á u p 0 E p ØÒ lim sup z ξ u(z) = 0 ÓÖ ÚÖÝ ξ º ÅÓÖÓÚÖ ÓÖ p 0 ØÖ Ü Ø ÙÒØÓÒ v E p Ù ØØ lim inf z ξ v(z) = ÓÖ ÚÖÝ ξ º Ï ÒÓÛ ÔÖÓÚ ÖØÖÞØÓÒ Ó Ø ÖÐØ ÔÖÓÐÑ Ò E p ÓÖ p > 0º ÓÖ p 1 Ø Û ÔÖÓÚ Ò ÌÓÖÑ º¾ º ÌÓÖÑ ºº ÄØ p > 0 Ò µ ÒÓÒ¹ÒØÚ Ñ ÙÖº ÌÒ ØÖ Ü Ø ÙÒÕÙ ÙÒØÓÒ u E p Ù ØØ (dd c u) n = µ Ò ÓÒÐÝ ØÖ Ü Ø ÓÒ ØÒØ A > 0 Ù ØØ ºµ ÓÖ ÚÖÝ ϕ E 0 º ( ϕ) p dµ Ae p (ϕ) p/(n+p) ÈÖÓÓº ÄØ 0 < p < 1º ÙÑ ØØ ØÖ Ü Ø ÙÒÕÙ u E p Ù ØØ (dd c u) n = µº ÌÖ Ü Ø ÕÙÒ [u j ] u j E 0 Û ÓÒÚÖ ÔÓÒØÛ ÓÒ ØÓ u j Ò lim j e p (u j ) = e p (u) < ÄÑÑ ¾º½ Ò µº ÄØ ϕ E 0 º ÌÒ ÌÓÖÑ ¾º ÑÔÐ ØØ ØÖ Ü Ø ÓÒ ØÒØ C > 0
8 ½¾ Ⱥ Ø Ðº Ù ØØ Ì ( ϕ)(ddc u j ) n Ce p (ϕ) p/(p+n) e p (u j ) 1/(p+n) Ò ØÖÓÖ ( ϕ) p dµ lim inf j Ae p (ϕ) p/(n+p). ( ϕ) p (dd c u j ) n Ce p (u) 1/(p+n) e p (ϕ) p/(p+n) ÓÖ Ø ÓÒÚÖ ÙÑ ØØ ØÖ Ü Ø ÓÒ ØÒØ A > 0 Ù ØØ ºµ ÓÐ º ÁÒ ÔÖØÙÐÖ Ø ÙÑÔØÓÒ ÑÔÐ ØØ µ ÚÒ ÓÒ ÔÐÙÖÔÓÐÖ Ø Ò Ó ÌÓÖÑ º½½ Ò ÓÛ ØØ ØÖ Ü Ø ÙÒØÓÒ φ E 0 Ò 0 f L 1 loc ((ddc φ) n ) Ù ØØ µ = f(dd c φ) n º ÃÓÓÞ³ ØÓÖÑ ½¼ ÈÖÓÔÓ ØÓÒ º½ µ ÑÔÐ ØØ ØÖ Ü Ø u j E 0 Ù ØØ (dd c u j ) n = min(f, j)(dd c φ) n º ÀÒ sup j e p (u j ) < A (n+p)/p < Ò ØÖÓÖ ØÖ Ü Ø u E p Ù ØØ (dd c u) n = µº ÍÒÕÙÒ ÓÐÐÓÛ ÖÓÑ ÌÓÖÑ ºº Í Ò ÌÓÖÑ º ØÓØÖ ÛØ Ø ÑØÓ Ó ¾ Û ÓØÒ ÓÖÓÐÐÖÝ ºº ÄØ n 1 Ò ψ PSH() ÛØ lim z ξ ψ(z) = 0 ÓÖ ÚÖÝ ξ Ò ϕ L q ((dd c ψ) n ) ϕ 0 1 < q < º ÌÒ ØÖ Ü Ø ÙÒÕÙ ÙÒØÓÒ u E n(q 1) Ù ØØ (dd c u) n = ϕ(dd c ψ) n º ÅÓÖÓÚÖ Ì (ddc ψ) n < ØÒ u F n(q 1) º º ÖØÖÞØÓÒ Ó ÓÙÒ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ º Ì ÓÐÐÓÛÒ ÛÐйÒÓÛÒ ÐÑÑ Ò ÐÑÒØÖÝ ÜÖ Ò L p ¹ØÓÖݺ ÄÑÑ º½º ÄØ q > 1 Ò ÙÑ ØØ u Ò E q ÒÓØ ÒØÐÐÝ 0º ÌÒ { [ lim e p(u) 1/p = inf α R : χ { u>α} (dd c u) n] } = 0. p ÈÖÓÓº ËØ M = inf{α R : Ì χ { u>α}(dd c u) n = 0}º ÏØÓÙØ ÐÓ Ó ÒÖÐØÝ Û Ò ÙÑ Ì ØØ M > 0º Ì 0 < M < Mº Á A = {z : u(z) > M} Ò C 1 = χ A(dd c u) n ØÒ C 1 > 0 Ò > C 2 = ( u) q (dd c u) n A ( u) q (dd c u) n M q C 1. Ì ÓÖ p > q Ø ØÒ ÓÐÐÓÛ ØØ e p (u) 1/p ( A ( u)p (dd c u) n ) 1/p ÌÙ º½µ lim inf p e p(u) 1/p M, Ò 0 < M < M Û Ó Ò ÖØÖÖÐݺ ÅÓÖÓÚÖ ÓÖ p > q Û Ú e p (u) 1/p = ( ( u) q ( u) p q (dd c u) n) 1/p 1/p C 2 M 1 q/p. MC 1/p 1 º
9 ÒÖÝ Ð E p ÓÖ 0 < p < 1 ½¾ ÀÒ º¾µ lim supe p (u) 1/p M. p ÁÒÕÙÐØ º½µ Ò º¾µ ÓÑÔÐØ Ø ÔÖÓÓº ÌÓÖÑ º¾º ÙÒØÓÒ u ÐÓÒ ØÓ E 0 Ò ÓÒÐÝ ½µ u F p ÓÖ ÚÖÝ p 0 ¾µ lim u(z) = 0 ÓÖ ÚÖÝ ξ z ξ µ supe p (u) 1/p < º p>0 ÈÖÓÓº ÏØÓÙØ ÐÓ Ó ÒÖÐØÝ ÙÑ ØØ u(z) < 0 ÓÖ z º ÄØ u E 0 º ÌÒ ÔÖÓÔÖØ ½µ Ò ¾µ ÓÐÐÓÛ ÖÓÑ Ø ÒØÓÒ Ó E 0 Ò Ì F p º Ì ÙÒØÓÒ u ÓÙÒ Ý ÙÑÔØÓÒ Ò ØÖÓÖ e p (u) 1/p C 1 ( (ddc u) n ) Ì 1/p ÛÖ C 1 0 ÓÒ ØÒغ ÌÙ sup p>0 e p (u) 1/p < Ò lim p ( (ddc u) n ) 1/p = 1º ÓÖ Ø ÓÒÚÖ ÙÑ ØØ u ÙÒØÓÒ Ø ÝÒ ½µ µº ÄØ M Ò ÄÑÑ º½º ÌÒ M < Ý µº ÅÓÖÓÚÖ M > 0 Ò u < 0 Ý ÙÑÔØÓÒº ÄØ A = {z : u(z) < M}º Ì Ø A ÓÔÒ Ò u ÙÔÔÖ ÑÓÒØÒÙÓÙ ÌA (ddc u) n = 0 Ò u M ÓÒ \ Aº ÆÓÛ ÙÑ ØØ u ÙÒÓÙÒ Ò ÐØ ε > 0 Ù ØØ ε z 2 < M ÓÒ º ËØ v(z) = max(u(z), ε z 2 2M)º ÌÒ v F p L () ÓÖ p 0º u ÙÒÓÙÒ Ø Ø {u < v} Ì = {u < ε z 2 2M} Ì ÒÓÒ¹ÑÔØÝ Ò ÓÔÒº ÄÑÑ º Ò ÑÔÐ ØØ {u<v} (ddc v) n {u<v} (ddc u) n Ì A (ddc u) n = 0 Ò {u < v} A ÙØ {u<v} (dd c v) n = {u<v} (dd c (ε z 2 2M)) n = Cλ({u < v}) > 0, ÛÖ λ Ø Ä Ù Ñ ÙÖ Ò C ÓÒ ØÒØ ÔÒÒ ÓÒÐÝ ÓÒ n Ò εº Ì ÓÒØÖØÓÒ Û ÑÔÐ ØØ u ÓÙÒº ÌÙ u E 0 º ÜÑÔÐ º º ÄØ = (0, 1) Ø ÙÒØ ÐÐ Ò C n Ò [a k ] ÕÙÒ Ò Ù ØØ a k ζ ÓÖ ÓÑ ζ º ÄØ T ak = T k Ø ÙØÓÑÓÖÔ Ñ Ó Û ÑÔ a k ØÓ 0 ºº T k (z) = T ak (z) = 1 a k 2 1 ak 2 ( z, a k a k a k 2 z) + a k ( a k 2 z, a k ), 1 z, a k ÛÖ x, y = n x jȳ j Ø Ù ÙÐ ÒÒÖ ÔÖÓÙØ Ò C n º Ì ÖÐ ÂÓÒ Ó T k Ø z ÚÒ Ý T k (z) 2 = F(z, a k ) 1 z, a k 4n,
10 ½¾ Ⱥ Ø Ðº ÛÖ F ÓÙÒ ÙÒØÓÒº ÅÓÖÓÚÖ ÓÖ ÐÐ ÓÑÔØ Ù Ø K Û Ú max z K T k (z) 2 C 1 ÛÖ C 1 ÓÒ ØÒØ ÒÓØ ÔÒÒ ÓÒ kº Ò ϕ j (z) = 2 j max(log T j (z), 1)º ÌÒ ϕ j PSH() L () lim z ξ ϕ j (z) = 0 ÓÖ ÚÖÝ ξ Ò ( (dd c ϕ j ) n = dd c 1 ) n 2 j max(log T j, 1) = 1 2 jn (dd c max(log T j, 2 j )) n = 1 2 jn T k 2 (dd c max(log z, 2 j )) n 1 2 jn (2π)n max T 1 k 2 C 2 (0,e 1 ) 2 jn, ÛÖ C 2 ÓÒ ØÒØ ÒÓØ ÔÒÒ ÓÒ jº ËØ ( k ) 1 u k (z) = max 2 j log T j(z), 1. ÌÒ u k PSH() L () lim z ξ u k (z) = 0 ÓÖ ÚÖÝ ξ Ò u k k ϕ jº Ì ÓÑÔÖ ÓÒ ÔÖÒÔÐ ºº µ ØÓØÖ ÛØ ÄÑÑ ¾º Ò ÓÛ ØØ u k E 0 º Ì ÙÒØÓÒ u Ò Ý ( ) 1 u(z) = max 2 j log T j(z), 1 ÐÓÒ ØÓ F L () Ò ØÖÓÖ ØÓ F p ÓÖ ÐÐ p 0º ÙØ u E 0 Ò lim inf z ζ u(z) lim j u(a j ) = 1º ÜÑÔÐ ºº ÄØ = (0, 1) C 2 Ò ÐØ [a j ] Ò [b j ] 0 < a j, b j < 1 Ö Ò ÕÙÒ Û ÓÒÚÖ ØÓ 0 j º ÓÖ j N Ò ϕ j : R { } Ý ϕ j (z) = a j max(log z, log b j )º ÌÒ ϕ j PSH() L () Ò lim z ξ ϕ j (z) = 0 ÓÖ ÚÖÝ ξ º ÅÓÖÓÚÖ { (2π) 2 dd c ϕ j dd c a 2 j ϕ k = dσ b j j = k, (2π) 2 a j a k dσ max(bj,b k ) ÓØÖÛ ÛÖ dσ r Ø ÒÓÖÑÐÞ Ä Ù Ñ ÙÖ ÓÒ (0, r) Ò ϕ j E 0 Ò ØÖÓÖ Ø ÙÒØÓÒ u k : R Ò Ý u k = k ϕ j Ò E 0 º Ì ÙÒØÓÒ u k Ö ÖÐÐÝ ÝÑÑØÖ ºº u k ( z ) = u k (z) Ò
11 (4.3) = k j,l=1 ( u k ) p (dd c u k ) 2 = ÒÖÝ Ð E p ÓÖ 0 < p < 1 ( u k ) p dd c ϕ j dd c ϕ l = ( u k ) p( k ) 2 dd c ϕ j k ( u k (max(b j, b l ))) p (2π) 2 a j a l j,l=1 k ( u k (b j )) p/2 ( u k (b l )) p/2 (2π) 2 a j a l = (2π) 2( k ) 2. ( u k (b j )) p/2 a j j,l=1 ÄØ z Ù ØØ z = b j º ÌÒ { ak log b k k j, ϕ k (z) = a k log b j ÓØÖÛ Ò ØÖÓÖ k=1 ϕ k(z) = j k=1 a k log b k +log b j k=j+1 a k = c j º ÙÑ ÒÓÛ ØØ Ø ÕÙÒ [a j ] Ò [b j ] Ö Ó Ò Ù ØØ ½µ ¾µ µ a j <, a j log b j = ( c j ) p/2 a j < º ÄØ u : R { } Ò Ý u = lim k u k º ÌÒ u ÔÐÙÖ Ù¹ ÖÑÓÒ Ò Ø Ø ÐÑØ Ó Ö Ò ÕÙÒ Ó ÔÐÙÖ ÙÖÑÓÒ Ì ÙÒØÓÒ Ò u(1/2, 0) > º ÙÑÔØÓÒ ½µ ÑÔÐ ØØ (ddc u) 2 < Ò ÖÓÑ ÒÕÙÐØÝ º µ Ò ÙÑÔØÓÒ µ Ø ÓÐÐÓÛ ØØ sup k ( u k ) p (dd c u k ) 2 <. ÀÒ u F p ÓÖ p 0º ÙØ ÙÑÔØÓÒ ¾µ ÝÐ u(0) = º ÄØ ÒÓÛ Ø ÕÙÒ [a j ] Ò [b j ] Ò Ý a j = 1/2 j Ò b j = e 2j /j º Ì ÕÙÒ Ö ØÓ 0 j Ò Ý ØÖØÓÖÛÖ ÐÙÐØÓÒ ØÝ Ø Ý ÙÑÔØÓÒ ½µ µº ÀÒ Ø ÙÒØÓÒ Ò ÓÒ Ý u(z) = 1 2 j max(log z, log e 2j /j ) = ( ) 1 max 2 j log z, 1 j ÐÓÒ ØÓ F p ÓÖ ÚÖÝ p 0 Ò lim z ξ u(z) = 0 ÓÖ ÚÖÝ ξ º ÙØ u E 0 Ò u ÙÒÓÙÒº ½¾
12 ½ ¼ Ⱥ Ø Ðº ÊÖÒ ½ ¾ ½¼ ½½ Ⱥ Ì ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÓÒ ÓÙÒ ÝÔÖÓÒÚÜ ÓÑÒ Èºº Ø ÍÑ ÍÒÚº ¾¼¼¾º Ⱥ Ò Êº ÞÝ ÇÒ Ø ÖÐØ ÔÖÓÐÑ Ò Ø ÖÐÐ Ð ÒÒº ÈÓÐÓÒº Åغ ¾¼¼µ ¾ ¾º º ÓÖ Ò º º ÌÝÐÓÖ ÒÛ ÔØÝ ÓÖ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ Ø Åغ ½ ½¾µ ½¼º ͺ ÖÐÐ ÈÐÙÖÓÑÔÐÜ ÒÖÝ º ½¼ ½µ ½¾½º ÌÛÓ ÜÑÔÐ Ò ÔÐÙÖÔÓØÒØÐ ØÓÖÝ Å ËÛÒ ÍÒÚº Ö Ö ÖÔÓÖØ ½ ¾¼¼¼µº Ì ÒÖÐ ÒØÓÒ Ó Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÒÒº ÁÒ Øº ÓÙÖÖ ÖÒÓе ¾¼¼µ ½½º ͺ ÖÐР˺ ÃÓÓÞ Ò º Ö ËÙÜØÒ ÓÒ Ó ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ ÛØ Û ÒÙÐÖØ Åغ º ¾¼ ¾¼¼µ ¾¾º ͺ ÖÐÐ Ò Äº ÈÖ ÓÒ Ò ÒÖÝ ØÑØ ÓÖ Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔ¹ ÖØÓÖ ÒÒº ÈÓÐÓÒº Åغ ½µ ½¼¾º ͺ ÖÐÐ Ò Âº ÏÐÙÒ ÅÓÒÑÔÖ ÒÓÖÑ ÓÖ ÐعÔÐÙÖ ÙÖÑÓÒ ÙÒ¹ ØÓÒ Åغ ËÒº ¾¼¼µ ¾¼½¾½º ˺ ÃÓÓÞ Ì ÖÒ Ó Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÁÁ ÁÒÒ ÍÒÚº Åغ º ½µ ¾º ĺ ÈÖ ÓÒ ÖÐØ ÔÖÒÔÐ ÓÖ Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ Öº Åغ ½µ º ÔÖØÑÒØ Ó ÅØÑØ Å ËÛÒ ÍÒÚÖ ØÝ Ë¹½ ¼ ËÙÒ ÚÐÐ ËÛÒ ¹ÑÐ ÔÖºÑÙÒº ÔÖØÑÒØ Ó ÅØÑØ ÍÒÚÖ ØÝ Ó ÙØÓÒ ÀÓ ËÙ ÈÑ ÀÒÓµ Ù Ý ÌÙÐÑ ÀÒÓ ÎØÒÑ ¹ÑÐ ÔÔÚÒÝÓÓºÓÑ ÔÖØÑÒØ Ó ÅØÑØ ÂÐÐÓÒÒ ÍÒÚÖ ØÝ ÊÝÑÓÒØ ¼¹¼ ÃÖÛ ÈÓÐÒ ¹ÑÐ ÊкÞÝÞѺٺٺÔÐ ÊÚ ¾¾º½º¾¼¼ ½½¾µ
dz k dz j. ω n = 1. supφ 1.
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