BOUNDED Ho-CALCULUS FOR ELLIPTIC OPERATORS HERbERt ...

Download PDF
3 downloads 0 Views 363KB Size Report
Comment
HERbERt AmAnn AnD MAtthiAs HiEbER. Math. Institut, UniversitХat ZХurich ... Ave., Los Angeles, CA90024-1555, USA. In memoriam Peter Hess. Abstract.
    !#"$%&#'(" )*,+.-/*,+1032547698:8;6;[email protected]:CD69EF)5CG*,-/*,+ H#IKJML1NPOGQSRTJTU JTVPJXWZY:ùja?]'ýg͛÷fñ1ù ÷ (ð@S ðáÍ ÷ÆõFðáñFð`> 8×ø 0 ý ZÚ #ù ’$œK–SË™X½†–n™ 3 •¾œDTWVj[RTlknmoXnmqpl\X"Z [R\5•Ê‡•«™7ØÒ%È«“,ɍœ™K“ ] æcç%]'4ý /5ùæcç%]'4ý /íKõ1ù 0 !@Á,¿,¿/“œX’¯™K½†–v™rS  F–nÃÇÀõ5ð`> 83ø­Û:–Z†Ç™K½†–v> ™ 3éë]'ý SÀ÷F> J

J

ø-cs> #



–n”{ÒhœX•¾ # S

ý ZÚ ù

g .nï dIc#d ò ïjðéï d ò ï t@ý6#SÞù65 Ýâ_^ 0 å š›’h†Òh’#•Ê™t¼“È«È¾“vÙ7œ¼”X“‘ý ZÚ *jùϙX½†–n™ ]'ý SÔ÷">Àë;ýgõ†÷fø9ùÚ ½†’‡Â¼“ZȾȫ“vÙt•¾,ÉtȾ’h‘F‘–tœf½,“vÙ7œ/™K½†–v™;]á•Dœ:œf™K–n؆ȫ’­Á,ÃÇ@’h”aœfÁ,•«™K–ZØ,Ⱦ’Ï–ZÇ,Ç@•«™X•¾ÄZ’­¿/’h”X™XÁ†”X؆–n™X•¾“Z†œ%Ú š›’h”K’#–n†Ç•¾™K½,’(¼“ZȾȾ“vÙt•«,É@Û t #ù  œX•«9ý¼ø- c7> 83ø3÷  ýO>cÿø " AAý c 3 ù

 

Ùt½†’h”K’  ýLfùæcç ¼“Z” áõ,ۆ–n†Ç  ýLfùæcç õ¼“Z” ð3õ,Ú jJ!$K G/³ G/³#"%$'&(&  =*)l 3 ë4]'ýSÔ÷F>ÀëÞýgõ,÷Xø9ù0 ýg•G2 ù 13+54  76   .- & -".- /  *+98  ;:ÔÇ,“Z‘ý34Fù áÇ@“‘ý43ù  4ýd J 3ù e_f ð?E 8 ÷ dÔëâ_^ ÷ ý ZÚ=ÿð3ø "t #ԙK½,’héï d J ?ï  ­Û–n†Ç ï d J ?ï D #œX•«:ýgø(cs> 8áø­Ú š›’h†Òh’ZÛ,ɍ•«Ä’h(E ë;ýTõ,-÷ vùKÛ,Ù’#œf’%’ ™K½†–v™ 4'ý d J J 3ù e_f ð`E 8:nÛ,¿,”K“vÄj•¾Ç@’PÇ ý  J SÞù 4 f   f e   c  ý >’   æzç 'õ5¼“” ëÕ(AÚ ½jÁ†œ :ù 0 Žd™½†–ZœØÒ%’h'œX½,“vÙtØj Ë Ò!Ž1™X“1œf½ÿö ,™K½†–v™tú‡4ý 3ù­•¾œ–#Ù’hȾÈÊАÇ@’hþ†,’%ÇFȾ•¾,’%–Z”“Z¿/’h”{–v™K“Z”Ï•¾ / –Z†Ç3™X½†–n™™X½†•¾œÇ,’!þ†,•«™X•¾“ZÝ•DœÒh“Z†œX•¾œf™X’%1™Ùt•Ê™K½?™X½,’’P–n”KÈ«•¾’h”“,’¼“Z”úÔë ýO> ë ýTõ,÷fø9ùÛhÙ’ÏœX–SË7™X½†–n™ 3Þ½†–Zœ:ù 0 ½jÁ†œ Ù’¯•¾@¼’h”¼”K“Z‘ ýL#,Ú jù™X½Ã–v™ %.ý43ù  M ð ý6 J â  ù % ¼“”%'ë ý >:ù!ÛA–n†Ç $'&(&   3 )W ? 3 ë]'ýSÔ÷">:ù -) ý 6 ÷F>:ù  Ú @A ·Ã¶v·   K@¶P :³ :³ ">$'&(&  4* )l 3 ë4]'ýSÔ÷F>:ù ÷ Ùt½†’h”K’$6 ' æzç :ù  ZÛ,™X½,•Dœt¿,”K“vÄZ’PœÏ™K½,’$–ZœKœf’%”f™K•«“'•«Â ×çÝõ,Ú !@Á,¿,¿/“œX’,“vÙ ™X½Ã–v™ æzç  ?õ7–nÃÇ#¿,Á@™ EÞæcç *W(t t–n†Ç Þæzç   ý DcsEaù eRf S  ý >  Ú ›È¾œX“FÈ«’h™ 37ò$æzç J 3é–n†Ç ò1ý d/ùæzçéý d J 37òvù eRf Ú ½,’%;ý ZÚ *jù•¾‘F¿,È«•¾’%œ ò ýdAù ð  #S ï d J  ï ð  S  ýO >3 ë(]'ý4/ù›–n†Ç À•¾œ(–”K’%–ZÈ9•¾1™X’%”X¿/“ZÈD–v™K•«“Ô¼Á,†Ò!™X“”5ý«÷ cù Π¼“”(œX“Z‘F’ C×ë ýTõ,÷ Sù–nÃÇë;ö Z÷  TÚ ½,’hÞ•«™¯½†–Zœ(Ø/’h’% œf½†“vÙtÿØjË @ “Z”K’Ôö ?+ϙX½Ã–v™ 3 ¿/“œKœf’PœXœX’%œ#– Ø/“ZÁ††Ç@’%Ç АÒh–nÈDÒ!Á†È«Á†œ“ /  Ú < ù  ) ý  Zí6 ÷">:ù¼“”t’%–Ò{½ ÃÛ   ¼“”œX“Z‘F’ ;:ZÚ “Z†œX’ F1Á,’%1™XȾËZÛ9¼”X“‘ ý  Ú  1ùÛý  Ú '” (ð # ð ‡Ú ›“n™X’(™K½†–v™  –n†Ç    '–n”K’¿/“œX•«™X•¾ÄZ’hȾ˽,“‘F“nРɍ’h,’%“ZÁ†œ“ZÂaÇ@’%ÉZ”K’h’ 9c (•«$  @Ú 5 •¾ÄZ’%UC  ë aÖFÙt•«™X½'ï Cï@! ð J #@ÛhØjË •«†Ç,Á†Ò™K•«“#•Ê™$'&&  * )l ÿ÷S ë Õ G  Ýõ5ð3ñSò>8áñ983ø 0 )W 3)W-"7 O !      ) $ ) * )l #>:   %ÏæAÕ  ü  ýT ÿù    * +98  ; :Àý ,Ú jù #   :    )W 6o. /I. - & )  + $    %  * +98  ;

:×ý @Ú %õjù) ;ý ,Ú Sù +  -U . /I  3õ #O% + . 6 6q   =3 )W   %.ý %hù!ý ¾÷ †ùë  Ö '  Õ Ö ÷" ýg ù  ý ,Ú   ù   ï †ï 1  1 ï    %.ý %hù!ý ùhï@ ð .œfÁ†¿  ï %Aý d/ùhï:í)dë :ûâ ä  .nï êAï  ï †ï  t@6ý #S ù 5  ý ,Ú  *jù +   - 5ë Õ G  ×ï Cïj ð J #0 (¶v·:·M ³ ؆œf’%”X蒛™K½†–v™ B  c %vý ù  ç3c ï Ãï B  %vý  ù  ÷ ë Õ G  0 ý ,Ú ' ù ›È¾œX“F,“n™K’(™X½†–n™%Û@™K½†–nA 'jœ™K“Ôý ,Ú jù{Û B  c %vý  ù  a ö%ïf–n”KÉêAï 3ø(cÿñ ò  ö  t S&ð ï êAïjð`$ S  0 ý ,Ú 1ù  h Àö t,Lý #SÞùð ï ê/ï1ð Ï÷ `S J ÷ –Z†Ç5¿,Á@™ æz ç h  f "Ú ½,’%ý @Ú  jù9•«‘F¿,Ⱦ•«’Pœ‡™X½†’t ’ ,@•¾œf™X’%†Ò!’t“ZÂA'æcç A/ýSÔ÷Xñ1ù 3õ#œXÁ†Ò{½5™X½Ã–v™ B  d J %vý  %ù  a ö%ï ê/ï `"A ÷ d ë é÷ Fë Õ G  0 

%







Œl ›š ’h†Òh’Ù’Ç@’PÇ@Á†Òh’¼”X“‘ :’%‘5‘– *ÃÚ –nÃÇ Â¼”K“Z‘Ìý @Ú 1ù¯™K½,’’ ,@•Dœ^™K’h†Òh’“ZÂ(–ÞÒh“Z†œf™K–Z™

œXÁ†Ò{½™X½†–n™  d J %ný  ù  eRf ð ÷ dë ÷ ë Õ G  ÷ ý ,Ú .8 8    &>$ tÍÏίæcç Í­ÎjýgÕ­Ö­÷K ÿù0 * 6   $'&&   3 )W $  ,  &j ý4†Ú vù0 )W

ë-   F÷  ýTÍ Î ù  4   ý ù÷F ýTÍ Î ù  0 





(¶v·:·M ³ ½,’–ZœKœX’h”X™X•¾“Z¼“Z”.™K½,’œXËj‘¯Ø/“ZÈZÒhȾ–œXœ ¼“ZȾȫ“vÙ7œ.¼”X“‘ ™K½,’Ï”K’%œXÁ,Èʙ{œ” #ð!ð ð  Ú $'&&  r3 )W  2  + . -& 9c tð ï C?ï 8#   æcç ?3 )W -$   #  4* )l  æcç   % 1  ÷ 1 1  e_f  O3 ) %  ëÍ    ՇÖÏ÷" ýg ù  0 )l  ë( ý  Î & ÷XÍ­ÎZù +  -U . /I  ë ý c Z÷ ùU  ð ný  ù  ‘– , %     0       1 1  eRf  





QŒ

(¶v·:·M ³ ½,“j“œX’ –¯œX‘F“j“n™X½¼Á,†Ò™K•«“ÀÙt•Ê™K½œXÁ,¿,¿/“Z”X™•¾é–Z†ÇØ/’h•¾,ÉÒh“Z†œf™K–Z™KÈ«ËF’FÁÖnÈ ™K“ (“Z™X½,’$œXÁ,¿,¿/“Z”X™7“n‡ø­ÚA½,’%Ç@’!þ††’ý SùØ1Ë'”X’%¿,Ⱦ–Ò!•¾,ÉF•«™K½,’$Ò!“†œ^™K”XÁÃÒ™X•¾“Z“nÂtý¼ø vù ™K½,’(¼Á,†Ò!™X•¾“Zø Ø1ËÏÚ ò 8áø +t # 0 )W ÔÜ  3 )W  :      &Y

On n /I. 6 : ( -"  - +I n -"; :(6=8    $  $ U  6=8    /= :(-" $'&  Í­Î@ýgՇ֭÷X ù 0 ›ÈʙK½,“ZÁ†ÉZ½×’%ȫȾ•¾¿@™X•DÒ'“¿Ã’%”K–n™X“”Kœ¯“Z×ՇֽÖSÄZ’'Ø/’h’%3œf™XÁ†Ç,•«’PÇ ØjË;‘–njË;–nÁ@™K½,“Z”{œh Û ½,’%“nÐ ”K’h‘ ,Ú *–n†Ç “Z”K“ZȾÈD–n”KË , Ú œX’h’%‘ ™K“Øҏ†’hÙ •« ™X½†•¾œ¯ÉZ’h†’h”{–nȾ•Ê™^ˍÚA—”X’%Äj•«“Á†œ(ÉZ’%,’h”{–v™K•«“ ™K½,’h“”X’%‘œ.”X’ F1Á,•«”K’­‘¯ÁÃÒ{½$œ^™K”X“,ÉZ’%” C Òh“Z†Ç,•Ê™K•«“†œ:–v™9•«@þÏ,•Ê™^Ë E%¼“Z”.™K½,’Òh“j’!ßÒ!•¾’h1™Kœý¼’Ú ÉÃÚ«Û1ö gù{Ú Žd™7œX½,“ZÁ†È¾Ç–nÈDœf“5ØÒ(“Ø†œf’%”Xč’%ÇF™K½†–v™™K½,’(”X’Pœf“È«Ä’h1™’%œf™X•¾‘–v™K’%œ“n ½,’h“”X’%‘ ,Ú *5–Z”X’ Á,,•«Â¼“Z”K‘ Ùt•«™X½”K’%œX¿Ã’PÒ™t™K“™K½,’$Ò!ÈD–ZœKœ“nÂaÁ,,•«Â¼“Z”K‘FÈ«Ë ý6 ÷fñSòvùMÐd’hȾȫ•¾¿@™X•DÒ(“Z¿/’h”{–v™K“Z”{œœX–n™X•Dœ^¼Ëj•¾,Éý ,Ú 1ùÚ ŽÀ™X½,’5–nØ/“vÄZ’¯™X½,’%“Z”K’h‘  Òh–ZÔØÒF–n –n”KØ,•«™X”{–n”KË'‘F“@Ç@Á,ȾÁ†œ“nÂÒh“Z1™X•¾jÁ,•Ê™^ËÚ ’5”X’Pœ^™K”X•DÒ™ †“vÙ ™X½,’ Ò!ÈD–ZœKœ“nÂ5–ZÇ@‘F•DœXœX•«Ø†È«’Ô‘F“@Ç@Á,Ⱦ•™X“3ØÒޖnØ,Ⱦ’Ô™K“׿,”K“vÄZ’™K½†–v™ J Üà¿Ã“1œXœX’%œKœf’Pœ'– Ø/“ZÁ††Ç@’%Ç АÒh–nÈDÒ!Á†È«Á†œ%Ú " !;Jj·Ã¶J :³: " ³  :Y3 )W )8 & 3 )W n = + )W . - / †Ú * r  &  4  $'&(&  =* )l f  f  4ý fù   8  ý †Ú ?1ù 0 ò  )l s* )l -" .  -  

   : > áõ- $ )(3 )W  J Ü  ë ) ýgÍ Î í Ô÷fø-cÿñjù +   -=   ) $  + . - / 6=8 ý 6 ÷fñSònù (F 6 6  &    &  -. -7Ü  Õ Ö  * +98  ; :×ý ,=Ú l ;*Y6NZ+7aL]„*]M*\Ï:\zI{QSI|*L]‰SN TÃ!\^+dQS+W QSRSUDNvW ‰a VP0RTJT:\]dEI{5yDNv1-H#$b*IK+-JT7L1Nvu%-+7„%+5b!N *{Y6€z:\EggBifk+7)SkolLahMW@*{iYGi #|QX(pn*ifSŒ 1QP$ N*+ 7c* OS- N* +MW1H#IKJTL1NZ`@_N eP‚=ƒ gTifkq!Is hMW,ifqkXp%ŒlPiN  i i „xÏ!N …­IKJM„S@W F #)S H7"#*†W&\E7* XY* 8?A:9  OP 3* +dWZu%‚P]MUcQS‹\^‡] T/\^]MyzI{‹SEW ˆ9\Šw ‰Ã„9] MnWjifk N  if>Œ ÀoAN …‡VPƒ7I{QS„{§G‹„(I{QP|FH5&N ˆaI{‹!I{RM\AW `ÃRM\^VS|S„{§D|SU ‹Z\f]T\^QhJTUzI{yA„‚n\^]dIKJT„]MR­w ÀoPN …‡„ƒ7IKJMRTV1=W j= 9I1-""*$EOP*|(- +8*{Y6*-OP 3* +B3/t&uA/:ŽK&W /goW `†I{bfU ®Sb RSNPH#IKJTL1PN ‚0gTifk hMWvŒqrXp Q{Š S //  e0ggifk Is hMWjqkXpniii  /{//g"W RSNZH#IKJML1Nuh„%b"N R!I{‚vI{Q  e0ggifk ok hdW1ŒlrXp%ŒŒl  /Iu  e0ggifk ok hMW1ŒŒPidp%ŒŒq  u@W Rv)N TnI{b!Nnu%b^UDNvY9QPU [SN x@„ Mh¨!„SWZuh\fbJfNnOG‰ H#IKJTL1@N eE‘DgTifk!s{ol hM"W Q!s QXp Qk S T‡ O eP‚zggifk!s{oŒ hMW@idp QPN  i^Š ÿuZN ; N …‡]M\ ’“cQ1_W ?A7 G  !"#$_%'-)!"#*!+BD”I17\cQPOSI17^WZ‰aƒt\^]XNSH#IKJML1Nuh„%bNc!W `Ã]M„X[%Uc|S\^QSb^\W UÏN OdNcWifk!s{ŒPN  if>r ÀH5N ‰N …‡]dI{RTQS„RM\^"y • R M%UcUD!W `,N `,N`I{¬S]T\fbU M„S=W Œr e5I{QPœ‹ U