An Integral formula on submanifolds of domains of Cn

Publicacions Ma emá iques, Vol 35 (1991), 559-569 . AN INTEGRAL FORMULA ON SUBMANIFOLDS OF DOMAINS OF Cn Abs ac A Bochne -Ma inelli-Koppelman ype in eg al o mula on submani olds o pseudocon ex domains in C' is de i ed ; he esul gi es, in pa icula , in eg al o mulas on S ein mani olds . The me hod o in eg al ep esen a ions in se e al complex a iables has been p o ed o be qui e e icien in cons uc ing holomo ph_ic unc ions and mo e gene al analy ic objec s (di e en ial o ms sol ing he á-equa ion, sec ions o holomo phic ec o bundles e c) ; see, o example, Henkin and Lei e e [4], [5], Henkin and Polyako [6] and Range [10] . This me hod e ol es abou Bochne - Ma inelli-Koppelman's in eg al o mula : i D C C'' ís a bounded domain in C' wi h smoo h bounda y óD hen (1 .1)  .= uAK,- á .AK q +8(J UAK y - 1) D D  D o e e y (0, q)- o m u wi h C 1 -coe icien s on D, whe e K 9 a e app op ia ely cons uc ed ke nels (see O elid [9]) . The in eg al o mula (1 .1) canbe modi ied o p oduce a a ie y o o he o mulas wi h which a ious p oblems (such as he á-equa ion and in e pola ion p oblems) can be sol ed ; hus (1 .1) is he i s main s ep in se e al cons uc ions . The pu pose o his pape is o cons uc an analogue o (1 .1) i D is eplaced by a _submani old M o D . Mo e p ecisely le D _be a bounded domain in C' and D a pseudocon ex one wi h D C _D ; le M be a (closed and comp_lex) submani old o D and le M = : D l M . Assume ha áM = (0D) n M _ is smoo h . Then we will cons uc ke nels K,«, z) de ined o z) E M x M wi h ( ~ z so ha (1 .2)  u (z) =  U«) A Kq«, z)- SEaM TELEMACHOS HATZIAFRATIS 1 . In oduc ion CEM  S -bu«) A Kq (S, z) + az ( EM U(S) A K9-1 ( S, z)) 56 0  T . HATZIAFRATIS o u E Cho a) (M) and z E M . In [3] we de i ed an in eg al o mula like (1 .2) in he case M is a comple e in e sec ion (i .e ., i he e exis unc ions hl, .. ., h p , holomo phic on D _ , so ha M = {hl = . . . = h p = 0} and dhl n . . . n dh p :~ 0 on M) ; he cons uc ion in he gene al case (Le ., when M is no necessa ily a comple e in e sec ion) will be based on he esul o [3] . A simila cons uc ion was ca ied ou by Be nd sson [1] . As a ma e o ac ou ke nels a e equi alen o hose o Be nd sson bu w i en in a di e en way . We hink howe e ha ou e sion o he cons uc ion as well as he di e en p oo o he in eg al o mula a e o u he in e es . He e is an ou line o he cons uc ion . Fi _s we co e M by su icien ly small open se s {U,} (open in C - ) so ha each M n U o is a comple e in e sec ion . Then he esul in [3] gi es ke nels K9 o which (1 .2) holds in M n U, . Such ke nels a e no unique in he sense ha he e a e ce ain choices ha can be made, in pa icula , K9 depends on He e decomposi ions o he holomo phic unc ions which de ine M n U o (as he se o he common ze os) . Bu using a esul o Be nd sson [1] we show ha such He e decomposi ions can be chosen app op ia ely so ha K9 =K9 onmnu,nu, ; hus we can de ine a global ke nel K 9 (by se ing K 9 = : K9 on M n U,) . Then we ha e o show ha (1 .2) holds ; and his is done along he same lines as in [3]] . He e is an ou line o his p oo in he case u is a holomo phic unc ion on M . In his case we ha e o show ha (1 .3)  u(z) = ~  u«)Ko«, z),  z E M CEa~u Fix a z E M and pick a U o wi h z E U, . Bu , as a compu a ion shows, dCK0 = 0 ( o each T and ( 7~ z) and hence dC[u«)Kó «, z)] = 0 ; hus (1 .3) is equi alen ( ia S okes' heo em) o u(?) =  u«)Ko«, z) CEaU, which holds by he esul o [3] (o [2] o ha case) since Ko = Kó in U n M . Ou esul gi es, in pa icula , in eg al o mulas o domains D CC X, i X is a S ein mani old, ia he heo em ha S ein mani olds admi embedding in some ON . In eg al o mulas on S ein mani olds ha e been cons uc ed p e iously, using di e en echniques, by Henkin and Lei e e [4] . Rela ed is also he wo k o Be nd sson [1], Ho mann [7], Palm [9] and S ou [11] . AN INTEGRAL FORMULA ON SUBMANIFOLDS  561 2 . No a ion Le h = (h1, . . . . h p ) whe e he his a e holomo phic unc ions de ined in some open se o Cn and suppose ha {hij (~, z), j = 1, . . . , n} is a He e decomposi- ion o h i , Le ., h?j «, z) is holomo phic in bo h ( and z and n hi(S) - hi(z) = E hij(S,z)(Sj - zj),i = 1, . . .,p . j-1 We associa e o hese da a he ollowing di e en ial o ms . Fi s n-P-1 5D(2 .1)  1 ah«, z ) = cde [hlj, . . . , h p j, yj, (C +  z)yj] whe e y = (yl, . . . , yn ) is some smoo h unc ion (we will be mo e speci ic abou y la e ) and c is no malizing cons an : c = (-1) -(-2- 1)  1  1  whe e m = : (2- i ^~ ' T c! n- p . In he de e minan o (2 .1), j uns om j = 1 up o j = n o ming he (gis + az)yl n ows o i ; also he column o di e en ial o ms in he de e minan (n -p - 1)- imes (as i is de e minan s see [4] . Also de ine (2 .2) whe e (a~ + " C7z)yn n -P 1 8h 1 h ah ¡oh(C)12 de  á~P , d(j Ioh(S)1 2 =  y :  1 a(h1 . . . . . , hP) (~) 2 . 1<j l < . . .<jp<n  a(Sjl, . . . , (j ) indica ed) ; o p ope ies o such o cou se Q'«) is de ined o ( wi h Joh(« :yÉ 0 . Wi h his no a ion se K h (~, z) = ah«~ z) n ~3h(~) . Le ce, be he pa o a h which is o ype (0, q) in z and (0, n -p- q - 1) in ~, i . e ., Also le Kq (~, z) = a9 «, z) nQ h (~) . - )  _~- _ c q = c C n  q - 1  de [hlj, . . . . h p j, yj, azyj,  aCyj is epea ed O cou se Kq depends on he choice o he He e decomposi ions {hij} o hi, al hough we do no indíca e his in he no a ion ; bu i will be clea om he con ex which choice we will be using in each case . Also we ha e been agueabou he domains in which he abo e unc ions and di e en ial o ms a e de ined, because he e, we simply in oduced no a ion and we will be e y speci ic abou his la e . 56 2  T . HATZIAFRATIS 3 . Cons uc ion o he ke nels _ Fi s we desc ibe he se ing . Le_ D, D be domains in C" wi h D bound_ed, D pseudocon ex and D C D . Le M be a closed complex su_bmani old o D o (complex) dimension m ._ Se M = DnM and aM = (OD)nM and assume ha aD is smoo h and ha M mee s aD ans e sally so ha aM is also smoo h . Le ,y = (-y,, . . ., -yn) : D x D - {~ = z} - C' be a smoo h map sa is ying (3 .1) ((j  zj) = 1 and  (- z' o 0 < j( - zi < 6 o some small 6 > 0 j=1  zl2 n  (j 7j  ^y ., I- (an examPle o  ls g . l en bY  -z' all (, z wi h (  z) . 7  hj =  Z In his se ing we will now con _ s uc he ke nels . Le {U a } be a se o small con ex open se s o en so ha M C U a U aand mo eo e le ha = (he,', ... . ha) : D ~ CP, p= : n- m, be holomo phic maps so ha MnU, = {z E U a : ha (z) = 0} and _  17h a l ~ 0on M _ nU a ; ha such unc ions há exis ollows om Ca an's Theo em A since D is assumed o be pseudocon ex . Fu he mo e he e exis p x p ma ices A,T = [(A ,)ik]1<i,k<p o unc ions hi ~  í hi (A,T)ik, holomo phic in U a n UT, so ha A, T Le ., (3 .2) The exis ence o such ma ices A .T ollows om Ca an's Theo em B since U a n U T is con ex . Lemma 1 . ah ; ah, ; _ A uT  ~eu a nuTnM . ahp ahp ac ; aS ; P oo :: Di e en ia ing (3 .2) we ob ain ahá a5j ahk (A,T)ik - + k=1 a(j k=1 p há = E(A,T)ikhk, i = 1, . . . , p . k=1 hT a(AoT)ik . k a(j on u, nu T , Since hk = 0 o C E U Q n U T n M, he o mula o he lemma ollows immedi- a ely . The ollowing lemma is p o ed by Be nd sson [1, p .414] ; i_ s P oo is based on Ca an's Theo em B, using again he pseudocon exi y o D . Lemma 2 .  The e exis unc ions h ~ «, z), i = 1, . . . . p, j = 1, . . . , n, holo- mo phic in (C, z) E (U, n M) x M so ha hij i  ihlj (3 .3) 1 h- j j  h- pi Fu he mo e h ~ «, z) Na e holomo phic ex ensions in C in a neighbou hood (in en) o U a n M, sa is ying Then T P oo . De ining AN INTEGRAL FORMULA ON SUBMANIFOLDS  563 n o j=1, _, n,CEu,nu,nM,zEM (3 .4)  1 : h íj (C, z) (SJ - zj) = h° «) o zE M . j=1 Lemma 3 . Le A E CPXP and B, C E CnxP so ha A - BT = C T (T deno es anspose) . and, hence, a hh ' = de (A,) de [C, * * *] = de (A) de [B, * * *] whe e * * * deno e app op ia e di e en ial o ms ( he same on bo h sides o he equa ion) so ha he ma ices [B, * * *] and [C, * * *] a e n x n . P oo : This is a s aigh o wa d compu a ion based on he mul ilinea i y o he de e minan s (as unc ions o hei columns) and he de ini ion o he de- e minan s . (This gene alizes he ac ha he de e minan o he p oduc o squa e ma ices is equal o he p oduc o he de e minan s o hese ma i- ces) . Lemma 4 . I aho is he o m (2 .1) associa ed o he He e decomposi ions {h9'j ande¿ h is de ned simila ly, hen (3 .5)  ah' = de (A, T ) . a h ' o ( E U o n U T n M, z E M and simila ly H T , we ob ain om Lemma 2 ha A oT - (HT) T = (Ha ) T o (E U, n U, n ú, z E M ; hus Lemma 3 applies and gi es (3 .5) . 564  T . HATZIAFRATIS Lemma 5 . We ha e (3 .6)  ph = P oo .. By Lemma 1 we ob ain The e o e 1 de (  ph °  E U Q n U Tn M . ah°  ahi =A aT  ] ~N9k 1<i,k<p  N .9k 1<i,k<p . a(hi, . . .,hP) = de (AQT)  a(hi ... .,hP) , ~ ; p )  a«il , .. . , ~ ;p) and (3 .7)  j7h o l 2 = 1 de (A,)1 2 jVhT1 2 ( his also shows ha de (A, T ) ,-~ 0 on U Qn U T n M) . Also se ing and simila ly o BT we see ha Lemma 1 gi es Hence, by Lemma 3, 8h1 aC, . . . acá &h,' ah, aC  . . . aCn 1T . ( B T ) T = (Bu )T (3 .8)  de  ah1 , . . . , ah P C ;  a( ; , d  = de (A,) - de Now (3 .6) ollows om (2 .2), (3 .7) and (3 .8) . We a e now eady o de ine he ke nels : n-p ahi  ahT ~ , a(, . . . ,  , dj a(j K«, z) = : K h ' (~, z) = ah «, z) A Qho (~) o ( E U Qn M, z E M, ~ ,~ z . By Lemmms 4and 5, i ollows ha K(~, z) is well-de ined o  E M, z E M, z . Simila ly we de ine K 9 = aq  ,Q h o q > 0 and K_1 = 0 . AN INTEGRAL FORMULA ON SUBMANIFOLDS  565 4 . The in eg al o mula Wi h he ke nels jus cons uc ed we will p o e he ollowing heo em . Theo em 1 . I u E Cho e) (M) (0 < q < m) and z E M hen u(z) = ~  u«) n Kq(C, z)- (EaM Fi s a lemma : Lemma 6 . We ha e ( ,9 c + c 9z)K(S, z) = 0 . P oo . : I su ices o show (4 .1)  (8S + á z ) K ho (C, z) = 0 o cE U o l M, zE M, c =~ z . Now obse e ha (4 .2)  «I - zj)a ho (S, z) = de P oo o Theo em 1 : I su ices o show ha o W E (Có (M))(m, m - q) . CEM - bu«) A K9 (S, z) + áz IEM u«) A K 9 -1 (C, z)] . (4 .3)  J  U(Z) A W(z) = J  EBM u n K q A W- zEM  zEM ~ n,- -1 hay  . . .  ha y 7j  (ac + az)yi J 2<j<n (in he abo e de e minan j uns om j = 2 o j = n o ming he 2nd up o he n h ow o i ) . We ob ained (4 .2) in he ollowing way : (SI - z1) mul iplied he i s ow o he de e minan which de ines aho «, z) and hen we added o ha i s ow he j h- ow mul iplied by (Si - zj) (j = 2, . . . , n) . Then (4 .2) ollows in iew o he i s o he assump ions in (3 .1) and (3 .4) . Now (4 .2) easily implies ha (á C + Ó z ) a ho = 0 ; since, mo eo e , 5CQ«) = 0 (by [2, Co olla y 1, p . 76]), we ob ain (4 .1) which comple es he p oo o he lemma . áuAK y Acp+ J  (az(]  uAK 9 -1))A9 zemCEM  zEM  (EM 56 6  T . HATZIAFRATIS Le us poin ou he way in which he a ious o ms in he igh -hand side o (4 .3) dependon he a iables ~ and z : u_  «5u«)  u«), K 9 = K,«, z), K 9 -1 = Kq-1 «, z), cp = ;o(z) and 8u =  . By deg ee easons, (4 .3) is equi alen . o (4 .4)  L  u(z) n w(z) =  u n K A cp- EM ó(MxM) (in ob aining (4 .4) we used also he ac ha J lS~,z)  uAKAW=  uAKAW E(óM)xM  8 (M x M) which holds since u has compac suppo in M) . By S okes' o mula (4 .5) a(mxm)uAKAW=Ld[UAKAW]+L uAKA<p whe e C E = {((, z) E M x M : j( - zi = E} . By Lemma 6 and deg ee easons L buAKA(p+~ (áz(J uAK))AW MxM  M M 1 ~~  d[uAKAW] =  auAKAW-(-1)9J  uAKn77cp ; MxM-{~~-z~<E} Mxm  MxM hence (4 .4) will ollow om (4 .5) as soon as we es ablish he ollowing (4 .6)  lim  u A K A ;o =  u(z) n W(z) . E-0 £,z)EQ  . EM Bu we may assume wi hou loss o gene ali y ha supp(u) C Ua o some u . Then o E small enough uAKAcw= uAK h 'n ;o C E  C E and (4 .6) ollows om (4 .7)  lim  uA Kho A cp =  u (z) 1 w(z) E-0 c E  IzEM Bu (4 .7) is exac ly wha is p o ed in [3, p . 339 341] (in ha pa o he p oo , (3 .4) plays an impo an ole) . This comple es he p oo o he heo em . his is an analogue o Cauchy's in eg al o mula on M . Con inuing o assume D o be s ic ly pseudocon ex and he gj's as abo e, le us se and AN INTEGRAL FORMULA ON SUBMANIFOLDS  567 As we men ioned in he in oduc ion he in eg al o mula ha we p o ed can be used o de i e a ious o he in eg al o mulas . He e we discuss wo examples . Wi h no a ion as be o e assume u he mo e ha D is a s ic ly pseudocon ex domain . Then, by a classical cons uc ion o Henkin and Rami ez (see [51), he e exis unc ions g j (~, z), j = 1, ... , n de ined o «, z) E (aD) x D, which a e smoo h in ( and holomo phic in z, so ha G(~, z) _ : ~~ i (~ j -z j )gj «, z) 0 . Thus i we se n-p-1 C«, z) _ [C(~, z)1 -p de [hi , . . . ; h'i, g~, DZgj 1 A aho (C) o ( E U naM and z E M hen C«, z) is w ell-de ined o  E aM and z E M, holomo phic in z and (as i ollows' om heo em 1) i ep oduces holomo phic unc ions on M, Le, o E O(M) l C(M)we Na e ?7,(S,z,~)=(1- .1) i'- zI2+aG(S'z) o (EaD,zeD, .1E[0,11 L9(~,z,, )=CCn-p-1/ 4 5 . Applica ions (z) = CEaM (0C«, z), 9 n-P-9-1 de [hi , . . . , hpi, l  a  ;, (aS~+ da  n ah~  E U o n aM, z E M ; hen L .«, z, . ) is well-de ined o -C E 8M, z E M . Also se B,«, z) = : L«, z, )1>,-o ; zeM ; hen i is clea ha B,(~, z) is de ined o ~, z E M, ( :,A z .  Using he abo e ke nels we de ine he ollowing ope a o s : T 9 u(z) =  u(S) A Le - 1( S, z, ) +  u (S) n BQ-1(S,z),z E M (C,a)E(aM)X[0,1]  CEM