dz k dz j. ω n = 1. supφ 1.
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- Marion Lynch
- 6 years ago
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1 ÆÆÄË ÈÇÄÇÆÁÁ ÅÌÀÅÌÁÁ ½º¾ ¾¼¼µ ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ Ý ËÛÓÑÖ ÒÛ ÃÖÛµ ØÖغ Ï ØÙÝ ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ º ÇÙÖ Ö ÙÐØ Ò¹ ÖÐÞ ÓÑ ØÓÖÑ Ó Ù Ò Ö ÖÓÑ Ø ØØÒ Ó ÙÖ ØÓ ÖØÖÖÝ ÑÒÓÐ µ Ò Ò ÛÖ ÓÑ ÓÔÒ ÕÙ ØÓÒ ÔÓ Ý ØѺ ½º ÁÒØÖÓÙØÓÒº ËÒ Ø ÓÖÒÖ ØÓÒ Ö ÙÐØ Ó ÓÖ Ò ÌÝÐÓÖ Ì½ Ò Ì¾ µ ÔÐÙÖÔÓØÒØÐ ØÓÖÝ Ò ÓÑÒ Ó C n ÓÑ ÙØ Ó ÚÖÝ ÒØÒ Ú ØÙ º ÊÒØÐÝ Ò Ã½ ½ Ò ¾ ÔÐÙÖÔÓØÒØÐ ØÓÖÝ Ò Ø ØØÒ Ó ÓÑÔØ ÃÐÖ ÑÒÓÐ Ò ÚÐÓÔº ËÙ ØÓÖÝ ÒØÖ¹ ØÒ ÔÔÐØÓÒ Ò ÓÑÔÐÜ ÝÒÑ ÖÒØÐ Ò ÐÖ ÓÑØÖÝ Ò Ð Ó Ò ÔÖÓÐÑ Ò Ø ØÓÖÝ Ý Ø Û ÑÒ ÔÐÙÖÔÓØÒØРع ÓÖÝ Ò ÝÔÖÓÒÚÜ ÓÑÒ Ò C n µº Ï ÖÖ ØÓ ½ ÛÖ ÓÑ ÒØÖ¹ ØÓÒ ØÛÒ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ Ò C n ÛØ ÐÓÖØÑ ÖÓÛØ Ò Ø PSH(P n, ω FS ) ÙÒØÓÒ ÓÒ Ø ÓÑÔÐÜ ÔÖÓØÚ Ô P n ÕÙÔÔ ÛØ Ø ÙÒËØÙÝ ÑØÖ ω FS Ö ÓÛÒº ÁÒ ¾ Ø ÙØÓÖ Ò Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ Ò ÔÖÓÚ ÚÖÓÙ Ö ÙÐØ ÓÒÖÒÒ Øº ÌÝ ÐÑ ØØ ØÖ Ö ÙÐØ ØÐÐ ÓÐ Ò ÖØÖÖÝ ÑÒ ÓÒ ÙØ ØÝ Ö ØÖØ ØÑ ÐÚ ØÓ Ø ÙÖ Ò Ø ÒØÓÒ ÑÙ ÑÔÐÖ Ò ØØ º ÀÖ Ò ËØÓÒ ¾ Û Ú Ø ÒÖÐ ÒØÓÒ Ó Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ Ò ÔÝ Ø ÓÑÒ Ó ÒØÓÒº ÆÜØ Û ÒØÖÓÙ ÖÐÐ Ð Ó PSH(, ω) ÙÒØÓÒ Ò ÔÖÓÚ ÚÖÐ ÔÖÓÔÖØ ÒÖÐÞÒ ÓÑ Ö ÙÐØ Ò ¾ º ËÓÑ ÔÖÓÓ ÖÓÑ Ø ØÓÒ ÖÐÝ ÚÐÝ ÓÒ ØÖ Ø ÒÐÓÙ ÈÖÓÔÓ ØÓÒ ¾º½ Ò ¾º ÌÓÖÑ ¾ºµº Ï ÐÐ Ð Ó ÓØÒ ÖÖ ØÓ ¾ ÛÒ Ò n¹ñò ÓÒÐ Ö ÙÐØ ÓÐÐÓÛ ÖØÐÝ ÖÓÑ Ø ÙÖ Ò ÐÐ ÓÙ ÓÒÐÝ ÓÒ ØÓ ÔÓÒØ ÛÖ ÒÐÓ Ö Ð ÐÖº Ì Ö ÙÐØ ÖÓÑ Ø ØÓÒ ÛÐÐ Ù ØÓ ÔÖÓÚ ÓÙÖ ÑÒ ØÓÖѺ ¾¼¼¼ ÅØÑØ ËÙØ Ð ØÓÒ ¾Í¼ ¾Í¼ º ÃÝ ÛÓÖ Ò ÔÖ ÔÐÙÖÔÓØÒØÐ ØÓÖÝ ÃÐÖ ÑÒÓÐ ÖÐÐ Ð º ½ ÁÒ ØÝØÙØ ÅØÑØÝÞÒÝ ÈÆ ¾¼¼
2 ½¼ ˺ ÒÛ ÌÓÖÑ ½º½ ÅÒ Ö ÙÐصº ÚÖÝ PSH(, ω) ÙÒØÓÒ ÛØ ÓÙÒ p¹òöý ÐÑØ Ó Ö Ò ÕÙÒ Ó ÙÒØÓÒ j Û ÐÓÒ ØÓ L () PSH(, ω) Ò ÛÓ p¹òö ØÒ ØÓ Ø p¹òöý Ó Ë¹ ØÓÒ 3µº ÆÓØ ØØ Ò ÒÐÓÓÙ Ö ÙÐØ ØÖÙ Ò Ø ØÓÖÝ ½ µ ÙØ Ø ÔÖÓÓ ÖÐ ÓÒ ÚÖÐ ÖØÖ ÒÓÒØÖÚÐ Ö ÙÐØ ºº ÖÐÐ ÓÑÔÓ ØÓÒ Ü ØÒ Ö ÙÐØ ÓÖ Ø ÖÐØ ÔÖÓÐÑ Ò ÐÓÐ ÓÑÔÖ ÓÒ ÔÖÒ¹ Ôеº ÌØ ÔÖÓÓ ÒÒÓØ ÖÔØ Ò Ø ÃÐÖ ÑÒÓÐ ØØÒ ÑÒÐÝ Ù ØÖ ÒÓ ÒÐÓÙ Ó Ø ÐÓÐ ÓÑÔÖ ÓÒ ÔÖÒÔÐ Ò ÐÐ ÅÓÒÑÔÖ Ñ ÙÖ Ö ÔÖÓÐØÝ Ñ ÙÖ Ò Ø ÃÐÖ µº ÇÙÖ ÔÖÓÓ ÓÛÚÖ Ò ÔÔÐ Ò ÓØ ØÙØÓÒ Ó ØØ ÝÔÖÓÙØ Û ÓØÒ ÖÒØ ÔÖÓÓ Ó Ø Ö ÙÐØ Ò Ø Ø ÑÓÖ ØÒÐ ÙØ ÒÓØ ÖÕÙÖÒ ÚÝ ÑÒÖݵº ÁÒ ËØÓÒ Û ÒÖÐÞ Ø ÐÓе ÓÑÔÖ ÓÒ ÔÖÒÔÐ ÖÓÑ Ã½ ØÓ ÖÐÐ Ð º Ï ÖÖ ØÓ ½ Ò ¾ ÓÖ ÐÐ ÒÓØÓÒ Ù Ò Ø ÔÔÖº ÅÓÖ ÖÓÙÒ Ò ÔÐÙÖÔÓØÒØÐ ØÓÖÝ Ò ÓÙÒ Ò ÃРþ º ¾º ÒØÓÒ º ÄØ ÓÑÔØ n¹ñò ÓÒÐ ÃÐÖ ÑÒÓÐ ÕÙÔÔ ÛØ ÙÒÑÒØÐ ÃÐÖ ÓÖÑ ω ÚÒ Ò ÐÓÐ ÓÓÖÒØ Ý ω = i n g 2 kj dz k dz j. k,j=1 Ï ÙÑ ØØ Ø ÑØÖ ÒÓÖÑÐÞ Ó ØØ ÊÐÐ ØØ ω n = 1. PSH(, ω) := { L 1 (, ω) : dd c ω, C ()} ÛÖ Ù ÙÐ d = d c = i 2π ( ) Ò C () ÒÓØ Ø Ô Ó ÙÔÔÖ ÑÓÒØÒÙÓÙ ÙÒØÓÒ º Ï ÐÐ Ø ÙÒØÓÒ ØØ ÐÓÒ ØÓ PSH(, ω) ω¹ôðùö ÙÖÑÓÒ ω¹ô ÓÖ ÓÖصº ÌÖÓÙÓÙØ Ø ÔÔÖ Û ÐÐ ÙÑ ØØ ÐÐ Ø ÙÒØÓÒ Û ÓÒ¹ Ö Ø Ý Ø ÜØÖ ÓÒØÓÒ ¾º½µ sup 1. Ì ÓÒØÓÒ ÒÓØ Ö ØÖØÚ Ò Û ÓÒ ØÒØ ØÓ ÙÒØÓÒ Ò ÖØÒ ÖÐÐ Ð Ø ÒÛ ÙÒØÓÒ Ð Ó ÐÓÒ ØÓ ØØ Ð º ÆÚÖ¹ ØÐ ¾º½µ ÛÐÐ ÓØÒ ÚÖÝ ÐÔÙÐ ÓÖ ÓÙÖ ÔÙÖÔÓ º Ï ÛÓÙÐ Ð ØÓ Ò Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ (ω u ) n := (ω dd c u) n
3 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½½ ØÒ ÓÒ ω¹ô ÙÒØÓÒ º ÁØ ÛÐÐ ÒÓÛÒ ØØ Ø ÒÒÓØ ÓÒ ÓÖ ÐÐ ω¹ô ÙÒØÓÒ ÓÖ ÓÙÒØÖÜÑÔÐ Ò ÑÒ ÓÒ ¾ ¾ µº ÌÖÓÖ ÓÒ ÓÙÐ Ö ØÖØ ÓÒ Ð ØÓ ÑÐÐÖ Ð Ó ω¹ô ÙÒØÓÒ º Ï ÒÓØ Ø ÑÜÑÐ Ð Ó ω¹ô ÙÒØÓÒ ÓÖ Û Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ ÛÐÐ Ò Ý E(, ω) ÓÖ Ù Ø Ý E ÓÖ ÑÔÐØݺ ÁÒ Ø Ø ØÓÖÝ Û Ú ÓÑÔÐØ ÖÔØÓÒ Ó Ø Ð Ù ØÓ Ó Ð µº Í Ò ÓÒ Ò Ð Ó Ö Ø Ð E ÓÒ ÃÐÖ ÑÒÓÐ º ÄØ Ù Ö Ø ÖÐÐ ÓÑ ÓÒ ØÖÙØÓÒ ÓÖ ÓÙÒ ω¹ô ÙÒØÓÒ ÈÖÓÔÓ ØÓÒ ¾º½º ÄØ u ÓÙÒ ω¹ô ÙÒØÓÒº ÌÒ ÓÒ Ò Ò Ø ÔÓ ØÚµ ÙÖÖÒØ ÅÓÖÓÚÖ ω n u ωu k := (ω dd c u) (ω dd c u), k = 1,...,n. }{{} k times ÔÖÓÐØÝ Ñ ÙÖ ÓÖ ÚÖÝ ÓÙÒ ω¹ô ÙÒØÓÒ uº ÈÖÓÓº ÁØ ÒÓÙ ØÓ Ò Ø ÙÖÖÒØ ÐÓÐÐÝ ºº Ò ÓÓÖÒØ ÖØ ÛÖ Û Ú ÓÒØÒÙÓÙ ÔÓØÒØÐ ÓÖ ω ÙÒØÓÒ v Ù ØØ dd c v = ωµº ÙØ ØÒ uv ÑÔÐÝ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒº ÀÒ ÓÒ Ò Ù Ø Ð Ð Ö ÙÐØ ÖÓÑ Ì¾ ØÓ Ò ÓÙÖ ÙÖÖÒØ º ÆÓØ ØØ Ø ÒØÓÒ ÓÖÒØ Ò Ò ÒØÖ ØÓÒ Ó ØÛÓ ÖØ º Ì Ð Ø ÖØÓÒ Ó Ø ÔÖÓÔÓ ØÓÒ ÓÐÐÓÛ ÖÓÑ Ø Ø ØØ Û Ò ÓÑÔÓ n ωu n = ω n n! k!(n k)! (ddc u) k ω n k k=1 ÛÖ Ø ÐØØÖ ØÖÑ ÔÔÒ ØÓ ÐÓ ÙÖÖÒغ ÓÖ ÑÓÖ ØÐ Û ÖÖ ØÓ ½ Ò ¾ Ø ÐØØÖ Ò Ø n = 2µº ÒØÙÖÐ ÕÙ ØÓÒ ÙÒÖ ÛØ Ò Ó ÓÒÚÖÒ Ø ÓÔÖØÓÖ ÓÒØÒÙÓÙ º ÌÓ ØÙÝ ÓÒØÒÙØÝ Ö ÙÐØ ÓÒ Ò Ò Ø ÔØÝ cap ω Ý ØØÒ { } cap ω (A) := sup ωu n : u PSH(, ω), 0 u 1 A ÛÖ A Ò ÖØÖÖÝ ÓÖÐ Ù Ø Ó ÓÖ ÑÓÖ ØРý µº ÊÐÐ ØØ ÕÙÒ u j ÓÒÚÖ ØÓ u ÛØ Ö ÔØ ØÓ ÔØÝ t > 0 lim cap ω ({ u j u > t}) = 0. ÈÖÓÔÓ ØÓÒ ¾º¾º Ì ÅÓÒÑÔÖ ÓÔÖØÓÖ Ò ÓÚ ÓÒØÒ¹ ÙÓÙ ÓÒ Ö Ò ÕÙÒ Ò PSH(, ω) L ()º ÁØ Ð Ó ÓÒØÒÙÓÙ ÛØ Ö ÔØ ØÓ ÓÒÚÖÒ Ò ÔØÝ cap ω º ÈÖÓÓº Ë Ã¾ º
4 ½¾ ˺ ÒÛ ÀÚÒ Ò ÑÒ Ø Ö ÙÐØ ÓÒ Ò ÛØÖ Ø ÔÓ Ð ØÓ ¹ Ò Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ Ð Ó ÓÖ ÙÒÓÙÒ ÙÒØÓÒ º Ï ÛÓÙÐ Ó ÓÙÖ Ð ØÓ Ô Ø ÔÖÓÔÖØ ºº ÓÒØÒÙØÝ ÓÒ Ö Ò ¹ ÕÙÒ º ËÓ Û Ñ Ø ÓÐÐÓÛÒ ÒØÓÒ ÒØÓÒ ¾º º ÄØ u PSH(, ω)º Á ÓÖ ÚÖÝ ÕÙÒ Ó ω¹ô ÙÒØÓÒ u j PSH(U, ω) L (U) Ö Ò ØÓ u ÓÒ ÓÑ ÓÔÒ Ù Ø U Ó Ø ÓØ ÕÙÒ ω n u j ÛÐÝ ÓÒÚÖÒØ ÓÒ Uµ Ò Ø ÐÑØ Ñ ÙÖ M(u) ÒÔÒÒØ Ó Ø ÕÙÒ Û Ò ω n u := M(u) ÓÒ Ø Ø U ÛÖ Ø ÓÒÚÖÒ ÓÐ º ÀÖ PSH(U, ω) ÒÓØ Ø Ø Ó ÖÑ Ó ω¹ô ÙÒØÓÒ Ò ÓÒ Uº Ì ÒØÓÒ ÓÖÒØ ÓÒ ÒØÖ ØÓÒ º Ì Ð Ó ÙÒØÓÒ u Ò ÒØÓÒ ¾º Ø ÑÜÑÐ Ð Ó ω¹ô ÙÒØÓÒ ÓÖ Û ÓÒ Ò Ò Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ Û Û Ú ÒÓØ Ý Eº ÊÑÖº Ç ÓÙÖ ÓÒ Ò Ù ÕÙÒ ØØ ÓÒÚÖ ØÓ u ÚÖݹ ÛÖ ÓÒ º Ï ÓÓ Ø ÐÓÐ ÒØÓÒ ÒÓØ ÓÒÐÝ Ò ÓÖÖ ØÓ Ù ÓÒ¹ ÒØÓÒ ÛØ Ø ØÓÖÝ ÙØ Ð Ó Ù ÐÓÐ ÔÔÖÓÜÑØÓÒ ºº ÛØ ÑÓÓØ ω¹ô ÙÒØÓÒ ÚÖÝ ÐØ ÑØØÖ Ò ÓØÒ ÖÕÙÖ Ö ØÖ¹ ØÓÒ ÓÒ Ø ÓÖÑ ω Ò Ø ÙÒÖÐÝÒ ÑÒÓк ÁÒ Ø ÐÓÐ ÓÒØÜØ Û Ò ÔÔÖÓÜÑØ ÐÝ Ù Ò ÓÖ ÜÑÔÐ ÓÒÚÓÐÙØÓÒ ÛØ ÑÓÓØ Ö¹ Òк Ç ÓÙÖ ÒØÓÒ ¾º Ó ÚÖÝ ÑÐÐ ÔÖØÐ Ù Û Ú ØÓ ÐÐ ÓÒÚÖÒØ ÕÙÒ µº Ì ÓÐÐÓÛÒ Ö ÙÐØ Ñ Ø ÒØÓÒ ÑÓÖ ÑÒÐ ÌÓÖÑ ¾ºº ÄØ u PSH(, ω)º Ì ÓÐÐÓÛÒ ÓÒØÓÒ Ö ÕÙÚ¹ ÐÒØ ½º u Eº ¾º ÓÖ ÚÖÝ x ØÖ Ü Ø ÒÓÙÖÓÓ U x Ù ØØ ÓÖ ÚÖÝ ÕÙÒ u j PSH(U x, ω) L ÛØ u j ց u Ø ÕÙÒ ( u j ) n p 2 du j d c u j (ω uj ) p ω n p 1, p {0,...,n 2}, Ö ÛÐÝ ÓÙÒº º ÓÖ ÚÖÝ x ØÖ Ü Ø ÒÓÙÖÓÓ U x Ù ØØ ØÖ Ü Ø ÕÙÒ u j PSH(U x, ω) L ÛØ u j ց u Ù ØØ Ø ÕÙÒ ( u j ) n p 2 du j d c u j (ω uj ) p ω n p 1, p {0,...,n 2}, Ö ÛÐÝ ÓÙÒº ÈÖÓÓº ËÒ Ø Ö ÙÐØ ÐÓÐ ÓÒ Ò Ù Ø ÖÙÑÒØ ÖÓÑ ÈÖÓÔÓ ¹ ØÓÒ ½º½ ÓÒ ÑÓÖº Á v ÐÓÐ ÔÓØÒØÐ ØÒ u j v ց u vº ÆÓÛ Ø Ö ÙÐØ ÓÐÐÓÛ ÖÓÑ Ó³ ØÓÖÑ Ò Ø Ø Ð µº
5 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½ Í Ò ÒÐÓÓÙ ÖÙÑÒØ ÓÒ Ò ÔÖÓÚ ØØ ÑÓ Ø ÐÓÐ Ö ÙÐØ ÖÓÑ Ø Ø ØÓÖÝ ÖÑÒ ØÖÙ Ò Ø ÃÐÖ ÑÒÓÐ ØØÒº ÁÒ ÔÖØÙÐÖ Û Ú Ø ÓÐÐÓÛÒ ÓÖÓÐÐÖÝ ÓÖÓÐÐÖÝ ¾ºº Á u E Ò w PSH(, ω) ÛØ u w, ØÒ w Eº ÆÓÛ ÓÐÐÓÛÒ ½ Ò ½ Û Ö ÖÝ ØÓ ÒØÖÓÙ Ø Ó ÐÐ ÖÐÐ Ð ÒØÓÒ ¾ºº ÄØ E p ÒÓØ Ø Ð Ó u PSH(, ω) Ù ØØ ØÖ Ü Ø u j PSH(, ω) L (, ω) ÛØ u j ց u Ù ØØ sup j ( u j ) p ω n u j < Ò Ø ÔÔÖ ÓÖ ÑÔÐØÝ Û ÙÑ p 1 ÐØÓÙ Ø ÒØÓÒ Ñ Ò ÓÖ ÚÖÝ p > 0µº ÊÑÖº ÁÒ Ø Ø ØØÒ ÓÒ Ð Ó ÓÒ Ö Ð ÛØ Ø ØÓÒÐ ÔÖÓÔÖØÝ sup (dd c u) n <. j Ì ÓÚÓÙ ÐÝ Ø Ò ÓÙÖ ØØÒ Ò ÐÐ ÒØÖÐ Ì (ω u j ) n Ö ÕÙÐ ØÓ 1º Ç ÓÙÖ ÓÙÒ ÙÒØÓÒ ÐÓÒ ØÓ E p ÙØ E p ÓÒØÒ ÑÒÝ ÙÒ¹ ÓÙÒ ÙÒØÓÒ º Ì ÓÛÚÖ ÒÒÓØ ÚÖÝ ÒÙÐÖ Û ÛÐÐ Óй ÐÓÛ ÖÓÑ Ø Ö ÙÐØ ÐÓÛ ÑÓ Ø Ó ØÑ Ö ÒÖÐÞØÓÒ Ó Ø ÒÐÓÓÙ Ö ÙÐØ Ò ¾ µº ÈÖÓÔÓ ØÓÒ ¾ºº ÄØ u E p º ÌÒ u c E p ÓÖ ÒÝ ÓÒ ØÒØ cº ÊÑÖº ÏÒ Û Ó ÒÓØ ÔÖÓÖ ÙÑ ØØ ÐÐ ÙÒØÓÒ ÓÒ Ö Ö ÒØÚ Ø ÓÒØÓÒ Ò Ø ÒØÓÒ Ó ÖÐÐ Ð ØÓ ÑÓ ÐØÐÝ Ò ¾ µ ÒÑÐÝ Ò Ø Ó ( u j ) p Û ÒØÖØ u j p º ÈÖÓÓ Ó ÈÖÓÔÓ ØÓÒ ¾ºº Ì ÖØÖ ÑÔÐ Ó ÖÚØÓÒ Ù Ø ÓÙÖ Ò¹ ØÐ ÙÑÔØÓÒ ¾º½µº ÁÒ u j ÕÙÒ ÖÓÑ Ø ÒØÓÒ Ó E p ÓÖ ÙÒØÓÒ uµ ØÒ u j c Ù ÕÙÒ ÓÖ u cº ÇÒ Ù Ø ØÓ Ù Ø ÅÒÓÛ ÒÕÙÐØÝ Û Ù Ø Ò ω uj c Ö ÐÐ ÔÓ ØÚ Ñ ÙÖ º ÈÖÓÔÓ ØÓÒ ¾ºº Á u 1,...,u n E ØÒ ω u1 ω un ÔÖÓÐØÝ Ñ ÙÖº ÛÐÐ Ò ÈÖÓÓº ÐÐÝ Û ÖÔØ Ø ÖÙÑÒØ ÖÓÑ ÈÖÓÔÓ ØÓÒ ½º Ó ¾ º ÁØ Ù ØÓ Ò Ø ÔÓ ØÚµ ÙÖÖÒØ u k ω u1 ω uk 1.
6 ½ ˺ ÒÛ ËÒ Û Ò ÛÖØ ω u1 ω uk := ω u1 ω uk 1 ω dd c (u k ω u1 ω uk 1 ) Ø ÒÓÙ ØÓ ØØ u k ω u1 ω uk 1 ω n (k 1) <. ÏÒ k = 1 Ø Ó ÓÙÖ ÓÐ ÓÖ ÐÐ u PSH(, ω). ÌÓ ØØ u 2 ω u1 ÛÐÐ Ò Ø ÒÓÙ ØÓ Ù Ø Ñ ÐÙ¹ ÐØÓÒ Ò ¾ º ÇÒ ØÓ ØØ du i d c u i ω n 1 <, i = 1, 2, Ì Ì Ç ÖÚ ØØ du i d c u i ω n 1 ( u i) n 2 du i d c u i ω n 1 <, ÛÖ Û Ú Ù Ø ÒØÓÒ ÓÖ Ø ÓÒ ÒÕÙÐØÝ Ò Ø ÓÒØÓÒ ¾º½µ ÓÖ Ø Ö Øº ÆÓÛ Û Ò ÔÖÓ Ý ÒÙØÓÒº ÁÒ Ù Ò Ò Ò ÖÓÑ ¾ Û Ú u k ω u1 ω uk 1 ω n (k 1) = u k ω u1 ω uk 2 ω n (k 2) du k d c u k 1 ω u1 ω uk 2 ω n (k 1) u k ω u1 ω uk 2 ω n (k 2) ( du k d c u k ω u1 ω uk 2 ω n k1) 1/2 ( du k 1 d c u k 1 ω u1 ω uk 2 ω n k1) 1/2. Ì Ö Ø ÒØÖÐ ÓÒ Ø ÖØ Ò ÓÙÒ Ý ÒÙØÓÒ Û Ú k 1 ÙÒØÓÒ µº ÓÖ Ó Ø ÒØÖÐ Ò Ø ÔÖÓÙØ Û ÔÖÓ Ò Ø ÓÐÐÓÛÒ ÛÝ du k d c u k ω u1 ω uk 2 ω n (k 1) = = u k dd c u k ω u1 ω uk 2 ω n (k 1) u k ω uk ω u1 ω uk 2 ω n (k 1) u k ω u1 ω uk 2 ω n (k 2)
7 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½ Ò Û Ò ÙØÖØ ÓÙÒ ÒØÖк Ï ÒÓØ ØØ Û Ú ÓØ Ö Ó u k 1 Ò Ø Ö Ø ØÖÑ Ò Ó u k Ò Ø ÓÒº Ì ÑÔÐ ØØ ÒØÖÐ ÒÚÓÐÚÒ k ØÒØ ÙÒØÓÒ Ö ÓÒØÖÓÐÐ Ý ÒØÖÐ ÛØ k 1 ÙÒØÓÒ Ø ÒØÖØ ÙÒØÓÒ Ð Ó ÔÔÖ Ò Ø Û ÔÖÓÙصº ÇÙÖ ÓÐ ÛÐÐ ØÓ ØÑØ Ø ÒØÖÐ Ý ÒØÖÐ ÛÖ Ø ÒØÖØ ÙÒØÓÒ ÔÔÖ Ò Ø Û ÔÖÓÙØ Ø Ð Ø ØÛ Ò Ó ÓÒº ÁÒ Ø Ò Û Ø ÒØÖÐ Ó Ø ØÝÔ u j ωu k j ω n k, Û Ö ÒØ Ý ÒØÓÒº ÓÖ Û ÔÖÓ ÙÖØÖ Û Ñ ÐØ Ù ØÑÒØ º ÁÒ Ø Ó Ø ÙÒØÓÒ u 1,..., u k Û Ø ØÖ ÓÙÒ ÔÔÖÓÜÑÒØ ÒÑÐÝ u s,js := max{u s, j s }º Ï Ó Ó Ò ÓÖÖ ØÓ Ò ÙÖ ØØ ÐÐ ÒØÖÐ Ö ÒØ Ò ØÒ Û ÓÛ ØØ Ø ØÑØ Ö ÙÒÓÖѺ Ï ÖÓÔ Ø Ò j s Ò ÛØ ÓÐÐÓÛ º Í Ò Ø Ñ ÓÖ Û Ø u k ω uk ω u1 ω uk 2 ω n (k 1) [ u k ωu 2 k ω u1 ω uk 3 ω n (k 1) u k ω uk ω u1 ω uk 3 ω n (k 2)] 1/2 [ u k 2 ω uk 2 ω uk ω u1 ω uk 3 ω n (k 1) u k 2 ω uk ω u1 ω uk 3 ω n (k 2)] 1/2 u k ω uk ω u1 ω uk 3 ω n (k 2). ÆÓÛ Û ÒÓØ Ý M l = max{ Ì u i 1 ω l u i1 ω ui2 ω uik l ω n (k 1) : i j {1,...,k}, i 1 i j, j 1} Ø ÑÜÑÙÑ ÓÚÖ ÐÐ ÒØÖÐ Ù ØØ Ø ÙÒØÓÒ Û ÒØÖØ ÔÔÖ Ò Ø Û ÔÖÓÙØ l ØÑ Ø Ð Ø ÒÕÙÐØÝ Ò Ö M 1 C (M 1 C) 1/2 (M 2 C) 1/2 ÓÖ ÓÑ ÓÒ ØÒØ C ÒÔÒÒØ Ó M 1 Ò M 2 º Á M 1 Ò M 2 Ö ÒØ Ø Ø ÔÓÒØ ÛÖ Û Ò Ø ÔÔÖÓÜÑÒØ µ Û Ø ÙÒÓÖÑ ÓÒ¹ ØÖÓÐ Ó M 1 Ò ØÖÑ Ó M 2 º ÈÖÓÒ ÒÐÓÓÙ ÐÝ Û Ø M l ÓÒØÖÓÐÐ Ý M l1 º ÈÙØØÒ Ø Ö ÙÐØ ØÓØÖ Û ÓÒÐÙ ØØ Ø ÒØÐ ÒØÖÐ ÓÙÒ ØÙ ÔÖÓÚÒ ÓÙÖ ÐѺ
8 ½ ˺ ÒÛ Ï ÒÓØ ØØ Ù Ò Ø ÓÚ ÖÙÑÒØ ÓÒ Ò Ø Ø ÓÐÐÓÛÒ ÔÖÓÔÓ¹ ØÓÒ ÈÖÓÔÓ ØÓÒ ¾ºº Á u 1,...,u n E 1 ØÒ E 1 L 1 (, ω u1 ω un )º ÈÖÓÓ º ¾ ÈÖÓÔÓ ØÓÒ º¾ µº ÁÒ ÖÓÑ ÈÖÓÔÓ ØÓÒ ¾º Ø ÓÐÐÓÛ ØØ E Ò E 1 Ö ÓÒÚÜ Ø ÓÒ Ò ÓÑÔÓ ω (uv)/2 ÒØÓ 1 2 (ω uω v ) Ò ØÖÑ Ò Ù Ø Ö ÙÐØ ÓÖ ÑÜ ØÖÑ µº ÙØ ØÒ ÓÖ ÚÖÝ u n1 E 1 u n1 ω u1 ω un (n 1) n1 Û Òغ u 1 u n1 n 1 ω n u 1 u n1, n1 ÈÖÓÔÓ ØÓÒ ¾º½¼º ÄØ u PSH(, ω)º ÌÒ ( u) ε E p ÓÖ ÑÐÐ ε > 0º ÈÖÓÓº Ò ÐÑÒØÖÝ ÓÑÔÙØØÓÒ ÓÛ ØØ ( u) ε PSH(, ω) ÓÖ ε < 1 Û Ù Ø ÒØÐ ÓÒØÓÒ ¾º½µµº ÆÓÛ Ø Ö ÙÐØ ÓÐÐÓÛ ÖÓÑ Ø Ø ÒÐÓÙ µº ÁÒ ( u v) ε PSH v Ù ÙÐ ÐÓÐ ÔÓØÒØÐ ÓÖ ωµº ÆÓÛ ( u v) ε ( u) ε v Û Ø ÒØÚ ÔÓØÒØÐ Û ÔÓ Ð Ò Û Ò ÓÒ ØÒØ ØÓ Ø ÔÓØÒØе Ò Ø Ö ÙÐØ ÓÐÐÓÛ ÖÓÑ Ø ØÐØÝ Ó E p ÙÒÖ ØÒ ÑÜÑÙÑ º ÓÖÓÐÐÖÝ ¾º½½º Ì Ñ ÙÖ ω u1 ω un ÔÙØ ÒÓ Ñ ÓÒ ÔÐÙÖÔÓÐÖ Ø ÓÖ u 1,..., u n E 1 º ÈÖÓÓº ÁÒ ÈÖÓÔÓ ØÓÒ ¾º Ò ¾º½¼ ØÐÐ Ù ØØ ( u) ε ÒØÖÐ ÛØ Ö ÔØ ØÓ Ù Ñ ÙÖ ÓÖ ÑÐÐ ÔÓ ØÚ εº ÙØ ÒÝ ÔÐÙÖÔÓÐÖ Ø ÓÒØÒ Ò {u = } ÓÖ ÓÑ u PSH(, ω) ½ µ Ò Ø Ñ ÙÖ ÒÒÓØ ÔÙØ ÒÝ Ñ ÓÒ ØØ Øº ÊÑÖº ÑÐÖ Ö ÙÐØ Ð Ó ÓÐ ÓÖ Ø Ñ ÙÖ du 1 d c u 1 ω u2 ω un. º ÅÒ Ö ÙÐغ ÁÒ ¾ Ø ÙØÓÖ ÔÓ Ø ÓÐÐÓÛÒ Ì ÈÖÓÐѺ ÄØ PSH(, ω) ÙÒØÓÒ Ù ØØ ( )p ω n ÒØ ÓÖ ÓÑ p 1µº Ó ØÖ Ü Ø ÕÙÒ j PSH(, ω) L () Ö Ò ØÓ Ù ØØ sup j Ì ( j) p ω n j < Ï ÔÖÓÚ ØØ Ø Ø º Ï ÓÛ ÑÓÖÓÚÖ ØØ Û Ò ÓÓ j ³ Ò Ù ÛÝ ØØ lim Ì ( j) p ω n j = Ì ( )p ω n. ÊÑÖº Ò Ø ÙÖ Ø ÔÖÓÐÑ ÛÒ p > 1º ÓÖ p = 1 ÐÙÐØÓÒ ÑÐÖ ØÓ ØÓ Ò ¾ º Ð Ó ÌÓÖÑ º¾ ÐÓÛµ Ú Ù Ø Ö ÙÐغ Ï ÑÒØÓÒ Ø Ù Ù Ò ¾º½µ Ò L p ÒØÖÐØÝ Û Ø
9 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½ L 1 ÒØÖÐØݺ Ì ÝÐ E 1 Ò Û Ò Ù ÐÐ Ø ÑÒÖÝ Ò ÖÓÑ ËØÓÒ ¾º Ï ØÖØ ÛØ ØÒÐ ÐÑÑ ØØ Ú Ù Ø ÑÒ ØÓÓÐ ÓÖ ÓÙÖ ÐØÖ ØÙÝ ÒØÖØÓÒ Ý ÔÖØ º ÄÑÑ º½º ÄØ ÓÖ Ò j := max{, j}º ÌÒ ÓÖ ÐÐ ÙÖÖÒØ T Ó Ø ÓÖÑ ω m j ω s ω n 1 m s m µ Ø ÒÙÑÖ Ì ( )p ω T, Ì ( )p ω T Ò Ì ( )p 1 d d c T Ö ÓÙÒ Ý ÓÒ ØÒØ ÒÔÒÒØ Ó j m µ Ì ( )p dd c j T = Ì jdd c ( ) p T º ÈÖÓÓº Ï ÐÐ Ù ÒÙØÓÒ ÓÒ mº Ï ÐÐ ÔÖÓÚ a 0 µ Ò (a m ) (b m ) (a m )&(b m ) (a m1 )º ÈÖÓÓ Ó a 0 µº ÖÓÑ Ø ËØÓ ØÓÖÑ Û Ú ( ) p (ω ω) T = p ( ) p 1 d d c T 0. Ì ËÓ Ø ÒÓÙ ØÓ Ø ÓÙÒÒ Ó Ø ÒÙÑÖ ( )p ω T º ÁÒ Ø ÔÐ Ø ÒÙÑÖ Ö ÒÔÒÒØ Ó jº Í Ò Ø ÓÚ ÒÕÙÐØÝ Û Ú ( ) p ω k ω n k ÈÖÓÓ Ó (a m ) (b m )º Ï Ú ( ) p dd c j T = lim k ( ) p (ω ) n <. ( k ) p dd c j T dd c j T Ò ÒÓØ ÔÓ ØÚ Ñ ÙÖ ÙØ Ò ÛÖØØÒ ¹ ÖÒ Ó ØÛÓ ÔÓ ØÚ ÔÖÓÐØÝ Ñ ÙÖ Ò Û Ò Ù ÑÓÒÓØÓÒ ÓÒÚÖÒµº ËÓ Û Ú ( k ) p dd c j T = j dd c ( k ) p T ÛÒ ÓØ ÙÒØÓÒ Ö ÓÙÒ ÒØÖØÓÒ Ý ÔÖØ ÐØÑصº ËÓ j dd c ( ) p T ( ) p dd c j T = lim k k k j dd c (( ) p ( k ) p ) T ( jdd c ( ) p T k =: (A k B k ). k k ) jdd c ( k ) p T
10 ½ ˺ ÒÛ ÆÓÛ Û ÐÐ ØÑØ A k Ø Ó B k ÑÐÖµ Ù Ò Ø ÀÐÖ Ò¹ ÕÙÐØÝ A k jp ( ) p 1 dd c T j p(p 1)( ) p 2 d d c T k k ( jp ( ) p 1 ω T ) ( ) p 1 ω T j k k k p(p 1)( ) p 2 d d c T ( ) (p 1)/p ( jp ( ) p ω T k k ( ) (p 1)/p ( jp ( ) p ω T k ( jp(p 1) k k ) 1/p ω T ) 1/p ω T ) p 2 ( ) p 1 d d c T ( p 1 1/(p 1). d d c T) k ÁÒ ØÖÑ Ø ÓÒ ØÓÖ ØÒ ØÓ ÞÖÓ Ò ÐÐ Ø Ñ ÙÖ ÚÒ ÓÒ ÔÐÙÖÔÓÐÖ Ø Ò Û ÒØÖØ ÓÚÖ Ø ØØ Ö ØÓ { = }µº Ì Ö Ø ØÓÖ Ö ÓÙÒ Ý (a m ) Ó Û ÓØÒ k A k = 0º ÒÐÓÓÙ ÐÝ B k 0 Ò Û Ö ÓÒº ÈÖÓÓ Ó (a m )²(b m ) (a m1 )º ÏÖØ T Ì T = ω j Sº Ò Ø ÔÖÓÓ Ó a 0 µ Ø ÒÓÙ ØÓ ØÑØ Ø ÒÙÑÖ ( )p ω T º Ï Ú ( ) p ω T = = ( ) p ω j ω S ( ) p ω ω S ( ) p dd c j ω S. ÆÓÛ Ì Ý (a m ) Ø Ö Ø ØÖÑ ÓÙÒº ÁØ Ý ØÓ ØØ Ø ÓÙÒ Ý ( )p (ω ) n º Ð Ó Ý (b m ) Û Ò ÒØÖØ Ý ÔÖØ Ò Ø ÓÒ ØÖÑ ØÓ ÓØÒ ( ) p dd c j ω S = = p j dd c ( ) p ω S ( j )( ) p 1 dd c ω S p(p 1) j ( ) p 2 d d c ω S. ÙØ j Ò j < 0 Ó Ø ÓÒ ØÖÑ ÒÓÒÔÓ ØÚ Ò Ø Ö Ø Ò ØÑØ Ý
11 p ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ( j )( ) p 1 dd c ω S p ( ) p ω 2 S, Û ÓÖÒ ØÓ (a m ) Ò ÓÙÒ Ý ÓÒ ØÒØ ÒÔÒÒØ Ó jº ÆÓÛ Ø ÓÐÐÓÛ ÖÓÑ ÓÙÖ ØÑØ ØØ Û Ò ÓÙÒ ÐÐ Ø ÒØÐ ÒÙÑÖ Ý (p 1) nì ( )p (ω ) n º ÊÑÖº ÆÓØ ØØ Ò Ø ÔÖÓÓ Û Ù ÓÑ ÔÐ ØÙÖ Ó j º Ï ÓÒØÙÖ ØØ ÒØÖØÓÒ Ý ÔÖØ ÓÐ Ò ÑÙ ÑÓÖ ÒÖÐ ØÙØÓÒ Ò Ø Ø ¾ µº ÁÒ ÔÖØÙÐÖ Ø Ñ ØØ ÒØÖØÓÒ Ý ÔÖØ ÐØÑØ ÛÒÚÖ ÓÒ Ó Ø ÒØÖÐ Òغ ÆÓÛ Û Ö ÖÝ ØÓ ÔÖÓÚ ØØ j ÕÙÒ ÓÐÚÒ ÓÙÖ ÔÖÓÐÑ ÌÓÖÑ º¾º sup j ÈÖÓÓº Ü jº Ï Ú ( j ) p ω n j ( j ) p ω n j <. ( ) p ω n j ÙØ Ý ÄÑÑ º½(a n ) Ø Ð Ø ØÖÑ ØÑØ Ý (p 1) nì ( )p (ω ) n Û ÒØ Ý ÙÑÔØÓÒº Ì ÖÙÑÒØ Ò Ø Ø Ñ Ò ÄÑÑ º¾ Ó ¾ º ÌÓ ÔÖÓÚ ÓÙÖ ÒÜØ Ö ÙÐØ Û Ò ÑÓÖ ÐØ ØÑØ º Ï Ö Ø ÔÖÓÚ ÄÑÑ º º ÄØ Ù Ò ω¹ô ÙÒØÓÒ Û ÐÓÒ ØÓ Eº ÌÒ Ò Ø Ò Ó ÙÖÖÒØ º ω dd c max(u, j) χ {u> j} (ω dd c u) ÈÖÓÓº ÒÐÓÓÙ Ö ÙÐØ ÓÖ ÓÙÒ ÙÒØÓÒ Ö ÛÐÐ ÒÓÛÒ Ò Ò ÓÙÒ Ò Ì¾ þ º ÀÖ Û Ù ÑÐÖ ÖÙÑÒØ º ÄØ S ÔÓ ØÚ ÓÖÑ Ó Ö (n 1, n 1)º Ï Ú ØÓ ÔÖÓÚ ØØ (ω dd c u j ) S χ {u> j} (ω dd c u) S. ÁØ ÒÓÙ ØÓ Ú Ø ØÑØ ÓÒ ÓÑÔØ Ù Ø K Ó {u > j}º Ü ε > 0º Í Ò ÕÙ ÓÒØÒÙØÝ Ó PSH(, ω) ÙÒØÓÒ ½ µ ÓÒ Ò Ò Ò ÓÔÒ Ø U ÛØ cap ω (U) < ε Ò u = u 0 ÓÒ \U ÓÖ ÓÑ ÓÒØÒÙÓÙ u 0 º ÄØ u s ց u s u s PSH(, ω) C() Ò V t := { j < u 0 t} (t > 0)º Ï Ú { j < u s t} ÓÒ V t \Uº Ì ÒÝ ÓÔÒ V Ù ØØ K V V t Uº ÌÒ K (ω dd c u) S lim inf s V U (ω dd c u s ) S lim inf s V \U ½ (ω dd c u s ) S 2jCε
12 ½¼ ˺ ÒÛ ÓÖ ÓÑ ÓÒ ØÒØ C ÔÒÒ ÓÒ S ÙØ ÒÓØ ÓÒ jµ Û ÓÐÐÓÛ ÖÓÑ Ø ÄÆ ÒÕÙÐØ ÄÆ µº ÁÒ Ø ÒÓÙ ØÓ ÔÖÓÚ Ø ÒÕÙÐØÝ ÓÖ ÑÐÐ ÓÑÔØ ÓÒØÒ Ò ÓÓÖÒØ ÖØ Ò ØÒ ÔÔÐÝ Ð Ð ÄÆ ÒÕÙÐØ º ÆÓÛ lim inf s V \U ÄØ V ց K ØÓ Ø K (ω dd c u s ) S 2jCε lim inf s (ω dd c u) S K ØÒ ÐØ t ց 0 ØÓ Ò ÙÔ ÛØ K (ω dd c u) S V \U (ω dd c max{u s t, j}) S 2jCε. (ω dd c max{u t, j}) S 2jCε, K (ω dd c max{u, j}) S 2jCε. ÆÓÛ Ò ε ÖØÖÖÝ Ò C ÔÒ ÓÒÐÝ ÓÒ S ÙØ ÒÓØ ÓÒ ε Û Ø Ø Ö ÓÒÐÙ ÓÒº ÌÓÖÑ ºº ÄØ E L p (, ω n ) Ò j := max(, j)º ÌÒ lim ( j ) p ω n j = ( ) p ω n. ÈÖÓÓº ÁØ ÓÐÐÓÛ ÖÓÑ ØÒÖ Ñ ÙÖ¹ØÓÖØ ÖÙÑÒØ ØØ lim inf Ó Û Ú ØÓ ÔÖÓÚ ØØ ( j ) p ω n j ( j ) p ω n j Ì ÔÖÓÓ ÛÐÐ ÒÙØÚº Ï ÐÐ ÔÖÓÚ ØØ ( j ) p ω k j ( ) p ω n, ( ) p ω n. ( ) p ω n l ω l. ÓÖ k = n Ò l = 0 Û Ø Ø Ö Ö ÙÐغ ÄØ Ù ØÖØ ÛØ k = 1 Ò l ÖØÖÖÝ ( j ) p ω j ω l ω n k 1 ( ) p ω l ω n k. ÈÖÔ ÑÙ ÑÔÐÖ ÖÙÑÒØ ÛÓÙÐ Ó Ò Ø ÙÒØÓÒ ÓÒ ØÒØ Ò Û Ú ÓÒØÒÙØÝ Ö ÙÐØ ÓÖ Ö Ò ÕÙÒ Ò E 1 º Ï ÒÚÖع Ð ÔÖÓÖÑ Ö ÓÑ ÐÙÐØÓÒ Ò Ø ÑÒ ÔÖÓÓ Ð Ó Ù ÑÐÖ
13 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½½ ØÑØ ( ) p ω j ω l ω n l 1 = = ( ) p ω ω l ω n l 1 j dd c ( ) p ω l ω n l 1 j ( ) p ω ω l ω n l 1 ( p(p 1) j ( ) p 2 d d c ω l ω n l 1 p( j )( ) p 1 dd c ω l ω n l 1 ÆÓÛ Ý ÑÓÒÓØÓÒ ÓÒÚÖÒ Û Ù Ø Ò ÄÑÑ º½µ Ø ÕÙÐ ØÓ p(p 1) ( ) p 1 d d c ω l ω n l 1 = (p 1) ( ) p ω l1 ω n l 1 ( ) p dd c ω l ω n l 1 ( ) p ω l1 ω n l 1 = p p ). ( ) p dd c ω l ω n l 1 ( ) p ω l ω n l ( ) p dd c ω l ω n l 1, Û Û ØÓ ÔÖÓÚº ÙÑ Ø Ö ÙÐØ ÓÐ ÓÖ k 1 Ò ÖØÖÖÝ lº Ï ÐÐ ÔÖÓÚ Ø ÓÖ k ( j ) p ω k j p ( ) p ω k 1 j j dd c ( ) p ω k 1 j [ p(p 1) ω l1 ω n k j ( ) p 1 dd c ω k 1 j ( ) p ω k 1 j ) j ( ) p 2 d d c ω k 1 j ] ω l1 ω n k l
14 ½¾ [ p(p 1) p p j ( ) p 1 ω ω k 1 j j ( ) p 1 ω k 1 j ˺ ÒÛ j ( ) p 2 d d c ω k 1 j 1 ( ) p ω k 1 j ] ω l1 ω n k l. ÆÓÛ Û Ò Ù ÄÑÑ º ØÓ ÓÙÒ ÖÓÑ ÓÚ Ø Ö Ø ØÛÓ ØÖÑ ÓÒ Ø ÖØ ÛØ χ {u> j} ØÑ Ñ ÙÖ ÒÔÒÒØ Ó jº ÁÒ p(p 1) j ( ) p 2 d d c ω k 1 j p p(p 1) p j ( ) p 1 ω ω k 1 j j ( ) p 2 χ {u> j} d d c ω k 1 j ( ) p 1 χ {u> j} ω ω k 1. ÌÒ Û Ò Ù ÑÓÒÓØÓÒ ÓÒÚÖÒ ÓÖ ØÓ ØÖÑ µ Ò ÒÙØÓÒ ÝÔÓØ ÓÖ Ø ÒÜØ ØÛÓº ÏØ Û Ø Ö p(p 1) = p ( ) p 1 d d c ω l ω n l 1 (p 1) ( ) p ω l1 ω n l 1 ( ) p ω l ω n l (p 1) Û Ò Ø ÔÖÓÓº p ( ) p ω l ω n l ( ) p ω l ω n l = ( ) p ω l ω n l Ï Ò Ø ØÓÒ ÛØ Ò ÒÐÓÓÙ Ö ÙÐØ Ò Ø Ø ØÓÖݺ ÄØ Ù ÖÐÐ ÓÑ ØÖÑÒÓÐÓݺ ÓÑÒ Ò C n ÐÐ ÝÔÖÓÒÚÜ Ø ÑØ ÒØÚ ÜÙ ØÓÒ ÙÒØÓÒ º PSH ÙÒØÓÒ f Ù ØØ {z f(z) < c} ÓÖ ÐÐ c > 0º ÄØ E 0 Ø Ø Ó ÓÙÒ ÜÙ ØÓÒ ÙÒØÓÒ Ò E Ø Ø Ó PSH ÙÒØÓÒ ÓÖ Û ÓÒ Ò Ò ØÖ ÅÓÒÑÔÖ Ñ Ò Ù ÛÝ ØØ Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ ØÐÐ ÓÒØÒÙÓÙ ÓÒ Ö Ò ÕÙÒ ¾ µº ÄØ F p Ø ÙÐ Ó E ÓÒ ØÒ Ó ØÓ ÙÒØÓÒ g ÓÖ Û ØÖ Ü Ø ÕÙÒ g j E 0 Ö Ò ØÓ g Ù ØØ º½µ sup j ( g j ) p (dd c g j ) n <, sup j (dd c g j ) n <.,
15 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½ ÒÐÐÝ ÐØ F Ø ÙÐ Ó E ÓÒ ØÒ Ó ØÓ ÙÒØÓÒ g ÓÖ Û ØÖ Ü Ø ÕÙÒ g j E 0 Ö Ò ØÓ g Ù ØØ sup j (dd c g j ) n < ÓÖ ÑÓÖ ØÐ ÓÒÖÒÒ Ø ØÓÔ Û ÖÖ ØÓ ½ ¾ µº ÇÒ Ò ÔÖÓÚ Ø ÓÐÐÓÛÒ Ö ÙÐØ ÌÓÖÑ ºº ÄØ h F ÙÒØÓÒ Ù ØØ ( h) p (dd c h) n <. ÌÒ h E p ºº ØÖ Ü Ø ÕÙÒ Ó ÙÒØÓÒ Ö Ò ØÓ h Ò Ø ÝÒ º½µ Ò ØÖ p¹òö ØÒ ØÓ Ì ( h)p (dd c h) n º ÈÖÓÓº ÄØ h j := max{h, j}º Ì ÙÒØÓÒ Ò ÒÓØ ÐÓÒ ØÓ E 0 ØÝ Ò ÒÓØ ØÒ ØÓ ¼ ÓÒ Ø ÓÙÒÖÝ ÙØ Ó ÓÙÖ ØÝ ÐÓÒ ØÓ Fµ ÙØ Û Ø ÒÝ w E 0 ØÒ w j := max{h j, m j w} E 0 ÓÖ ÒÝ ÔÓ ØÚ m j º ÆÓÛ w j ց h j m Ì j ÙØ Û Ô j ܵ Û Ò Ü m j Ó ÐÖ ØØ w j1 w j Ò (( w j) p (dd c w j ) n ( h j ) p (dd c h j ) n ) < 1/j Ö Û Ù Ø ÓÒØÒÙØÝ Ó Ø ÅÓÒÑÔÖ ÓÔÖØÓÖ ÓÒ Ö Ò ÕÙÒ µº ÌÖÓÖ Û Ò Ö ØÖØ ÓÙÖ ÐÚ ØÓ Ø ÕÙÒ h j ( h j ) p (dd c h j ) n = p p ( h) p (dd c h j ) n = ( h j )( h) p 1 (dd c h j ) n p(p 1) h j dd c ( h) p (dd c h j ) n 1 h j ( h) p 2 dh d c h (dd c h j ) n 1 ( h) p dd c h (dd c h j ) n 1 p(p 1) h j ( h) p 2 χ (h> j) dh d c h (dd c h) n 1 ÛÖ Û Ú Ù Ø Ø ÚÖÒØ Ó ÄÑÑ º Ò ÒØÖØÓÒ Ý ÔÖØ Û ÐØÑØ Ò F ½ Ò ¾ µº ÀÒ ( h j ) p (dd c h j ) n p ( h) p dd c h (dd c h j ) n 1 p(p 1) ( h) p 1 dh d c h (dd c h) n 1 n p k (p 1) ( h) p 1 dh d c h(dd c h) n 1 p n ( h) p (dd c h) n k=1
16 ½ ˺ ÒÛ = p(p n 1) ( h) p 1 dh d c h (dd c h) n 1 p n ( h) p (dd c h) n = (p n 1) ( h) p (dd c h) n p n ( h) p (dd c h) n = ( h) p (dd c h) n. º ÄÓÐ ÓÑÔÖ ÓÒ ØÓÖѺ ÁÒ Ã½ Ø ÙØÓÖ ÔÖÓÚ Ø ÓÐÐÓÛ¹ Ò Ö ÙÐØ ÌÓÖÑ º½º ÄØ u, v PSH(, ω) C() ÙÒØÓÒ ÓÒ ÓÑÔØ n¹ñò ÓÒÐ ÃÐÖ ÑÒÓк ÌÒ {u<v} ω n v {u<v} ÁÒ ØØ ÔÔÖ Ø ÙØÓÖ ÒÐÝÞ ÓÒÐÝ ÓÒØÒÙÓÙ PSH(, ω) ÙÒØÓÒ ÒÚÖØÐ Ø Û ÐÑ Ø ÖÑÖ ØÖ ÌÓÖÑ ¾º½ Ò Ã½ µ ØØ Ø ÓÒØÒÙØÝ ÙÑÔØÓÒ ÖÙÒÒغ ÁØ Û Ð Ó Ù Ø ØØ Ø Ò¹ ÖÐ Ó ÓÙÒ PSH(, ω) ÙÒØÓÒ ÓÙÐ ÔÖÓÚ Ý Ù Ò ÕÙ¹ ÓÒØÒÙØÝ ÖÙÑÒغ Ì Ò Ð Ó ÓÒ Ý Ù Ò ÖÒØ Ö ÙÐØ ÖÓÑ Ã ÒÑÐÝ ÓÒ Ò ÔÔÖÓÜÑØ ÒÝ ÓÙÒ PSH(, ω) ÙÒØÓÒ Ý Ö Ò ÕÙÒ Ó ÓÒØÒÙÓÙ ω¹ôðùö ÙÖÑÓÒ ÙÒØÓÒ º ÀÖ Û ÔÖÓÚ ØØ Ø Ö ÙÐØ ØÐÐ ÓÐ ÛÒ u, v E p ÓÖ ÐÐ p 1º Ì ÔÖÓÓ ÖÔØ ÖÙÑÒØ Ó ½ ÖÓÑ Ø Ø ÓÒØÜغ Ï ÐÐ ÓÒÒØÖØ ÓÒÐÝ ÓÒ ØÓ ÔÓÒØ ÛÖ ÐØ Ù ØÑÒØ Ö Ñº ω n u. ÄÑÑ º¾º ÄØ E p Ò j = max{, j}º ÌÒ {u<v} ω n lim inf {u<v} ω n j. ÈÖÓÓº ÆÓØ ØØ j ÕÙÒ Ò Ø ÒØÓÒ Ó E p ÓÖ Ù ØÓ Ö ÙÐØ Ò ËØÓÒ µº ÆÓÛ ÓÒ ØÓ ÖÔØ Ø ÔÖÓÓ Ó ÄÑÑ º Ö Ø ÔÖص Ò ½ º ÌÓÖÑ º º ÄØ Ù Ú E p ÙÒØÓÒ ÓÒ ÓÑÔØ n¹ñò ÓÒÐ ÃÐÖ ÑÒÓк ÌÒ {u<v} ω n v {u<v} ω n u. ÈÖÓÓº ÄØ v j, u j Ò ÓÚº ÌÒ {u<v} ω n v lim inf lim k {u k <v} ω n v j lim inf k {u k <v j } ω n v j.
17 ÖÐÐ Ð ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ½ ÆÓÛ ÌÓÖÑ º½ ÝÐ lim inf k {u k <v j } ω n u k k {u<v j } {u<v j } ω n u = ω n u k {u<v} ÛÖ Û Ú Ù ÑÓÒÓØÓÒ ÓÒÚÖÒ ÓÖ Ø Ð Ø ÕÙÐØݺ ÒÓÛÐÑÒØ º Á ÛÓÙÐ Ð ØÓ ØÒ ÈÖÓ ÓÖ ËÛÓÑÖ ÃÓÓ¹ Þ ÓÖ ÑÒÝ ÐÔÙÐ Ù ÓÒ Ò ÓÑÑÒØ º ω n u, ÊÖÒ Ì½ ̾ Ð Ã ½ ¾ ÄÆ ½ ¾ ý þ ÃÐ º ÓÖ Ò º º ÌÝÐÓÖ Ì ÖÐØ ÔÖÓÐÑ ÓÖ ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÁÒÚÒغ Åغ ½µ ½º ÒÛ ÔØÝ ÓÖ ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ Ø Åغ ½ ½¾µ ½¼º º Ó Ì ÓÑÒ Ó ÒØÓÒ Ó Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÑÖº º Åغ ½¾ ¾¼¼µ ½ ¼º º Ó Ò Ëº ÃÓÓÞ ÇÒ ÖÙÐÖÞØÓÒ Ó ÔÐÙÖ ÙÖÑÓÒ ÙÒØÓÒ ÓÒ ÑÒÓÐ ÈÖÓº ÑÖº Åغ ËÓº ØÓ ÔÔÖº ͺ ÖÐÐ ÈÐÙÖÓÑÔÐÜ ÒÖÝ Ø Åغ ½¼ ½µ ½¾½º Ì ÒÖÐ ÒØÓÒ Ó Ø ÓÑÔÐÜ ÅÓÒÑÔÖ ÓÔÖØÓÖ ÒÒº ÁÒ Øº ÓÙÖÖ ÖÒÓе ¾¼¼µ ½½º ÜÔÐØ ÐÙÐØÓÒ Ó ÅÓÒÑÔÖ Ñ ÙÖ Ò Ø ÊÓÒØÖ ³ÒÐÝ ÓÑÔÐÜ ÈÓØÖ ½µ ØÐÒØÕÙ ÈÓØÖ ¾¼¼¾ ¾º ˺ ˺ ÖÒ Àº Áº ÄÚÒ Ò Äº ÆÖÒÖ ÁÒØÖÒ ÒÓÖÑ ÓÒ ÓÑÔÐÜ ÑÒÓÐ Ò ÐÓÐ ÒÐÝ ÈÔÖ Ò ÀÓÒÓÖ Ó Ãº ÃÓÖµ ÍÒÚº ÌÓÝÓ ÈÖ ÌÓÝÓ ½ ½½½ º κ Ù Ò º Ö ÁÒØÖÒ ÔØ ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓРº ÓѺ Òк ½ ¾¼¼µ ¼ º ÅÓÒÑÔÖ ÓÔÖØÓÖ ÓÒ ÓÑÔØ ÃÐÖ ÙÖ ÖÚ»Ñغλ ¼¼¾ º ˺ ÃÓÓÞ Ì ÅÓÒÑÔÖ ÕÙØÓÒ ÓÒ ÓÑÔØ ÃÐÖ ÑÒÓÐ ÁÒÒ ÍÒÚº Åغ º ¾ ¾¼¼ µ º Ì ÓÑÔÐÜ ÅÓÒÑÔÖ ÕÙØÓÒ Ò ÔÐÙÖÔÓØÒØÐ ØÓÖÝ ÅѺ ÑÖº Åغ ËÓº ½ ¾¼¼µ ÒÓº ¼º ź ÃÐÑ ÈÐÙÖÔÓØÒØÐ ÌÓÖÝ ÐÖÒÓÒ ÈÖ ½½º ÁÒ ØØÙØ Ó ÅØÑØ ÂÐÐÓÒÒ ÍÒÚÖ ØÝ ÊÝÑÓÒØ ¼¹¼ ÃÖÛ ÈÓÐÒ ¹ÑÐ ÐÛÓÑÖºÒÛѺٺٺÔÐ ÊÚ º½¼º¾¼¼ Ò Ò ÒÐ ÓÖÑ ¼º½º¾¼¼ ½¼ µ
j j ( ϕ j ) p (dd c ϕ j ) n < (dd c ϕ j ) n <.
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µ(, y) Computing the Möbius fun tion µ(x, x) = 1 The Möbius fun tion is de ned b y and X µ(x, t) = 0 x < y if x6t6y 3
ÈÖÑÙØØÓÒ ÔØØÖÒ Ò Ø ÅÙ ÙÒØÓÒ ÙÖ ØÒ ÎØ ÂÐÒ Ú ÂÐÒÓÚ Ò ÐÜ ËØÒÖÑ ÓÒ ÒÖ Ì ØÛÓµ 2314 ½¾ ½ ¾ ¾½ ¾ ½ ½¾ ¾½ ½¾ ¾½ ½ Ì ÔÓ Ø Ó ÔÖÑÙØØÓÒ ÛºÖºØº ÔØØÖÒ ÓÒØÒÑÒØ ½ 2314 ½¾ ½ ¾ ¾½ ¾ ½ ½¾ ¾½ ½¾ ¾½ Ì ÒØÖÚÐ [12,2314] ½ ¾ ÓÑÔÙØÒ
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