Holder and Lp estimates for the solutions of the ∂-equation in non-smooth strictly pseudoconvex domains

Publicacions Ma emá iques, Vol 34 (1990), 77-91 . A bs ac HOLDER AND U' ESTIMATES FORTHE SOLUTIONS OF THE D-EQUATION IN NON-SMOOTH STRICTLY PSEUDOCONVEX DOMAINS J .M . BURGUÉS Le D a bounded s ic ly pseudocon ex non-smoo h domain in C' . In his pape we p o e ha he es ima es in Lp and Lipschi z classes o he solu ions o he ó-equa ion wi h Lp-da a in egula s ic ly pseudocon ex domains (see[2]) a e also alid o D . We also gi e es ima es o he same ype o he 86 in he egula pa o he bounda y o hese domains . 0 . In oduc ion and s a emen o esul s This pape is a con inua ion o [1] and deals wi h he es ima es o he D- equa ion on s ic ly pseudocon ex non smoo h domains . By his we mean a domain D de ined by he condi ion D = { G 0} whe e is a s ic ly p .s .h . unc ion o class C 2 de ined in a neighbo hood o bD . We ecall ha i is no assumed ha he g adien o be di e en o 0 in bD, and he bounda y ails o be a egula submani old o Cn jus in a o ally eal se . Henkin and Lei e e p o ed in [3] ha he equa ion bu = has a bounded solu ion u o any (0, q)- o m , wi h bounded coe icien s, and such ha O = 0 . In [1] i was p o ed ha he e exis s an in eg al ope a o T = K«, z)A (C) D mapping Ll(°' Q) o Llo P_1) such ha DT = i O = 0 and sa is iying he es- ima e 11T 11 LD(D) < cil 4y(D) and also he Lip 1/2 es ima e 11T IILip(1/2,D) :5 cil 11,, in he case is o class C . He e and in he ollowing he Lp spaces a e wi h espec o he Lebesgue measu e dm, and Lip (s, D) s ands o he class o con inuous unc ions on D ha ing modulus o con inui y O(P) . The Lp- no ms will be ab e ia ed by 11 Ilp . Pa ially suppo ed by he g an PB85-0374 o he CYCIT . Minis e io de Educación y Cien- cia, Spain . 78  J .M . BURGUÉS The aim o his pape is o imp o e hese es ima es ex ending o he non- smoo h case he op imal es ima es ob ained by K an z in [2] o he egula case, o o ms o a bi a y bedeg ee . The pape is o ganized as ollows . In sec ion 1 we ecall he cons uc ion o he ope a o T and he es ima es o i s ke nel K ha we e ob ained in [1] . In sec ion 2 we p o e he ollowing h ee heo ems : Theo em 1 . The ope a o T sa is ies he ollowing LP-es ima es : (a)  IIT II9 < eji IIP, 1 < p < 2n + 2, 9 = p - 2ñ+2 (b)  IIT liq c QI 111, `dq < 2n + 2 (c) Fo p = 2n + 2, IT I sa is ies ¡he es ima e I D exp{cjT (z)I sn+i }dm«) < oo o some cons an c depending on n and II II2n+2 . Theo em 2 . (a) I p > 2n -}- 2, T maps Leo 1) con inuously in o Lip (á - ~, D) (b) I ELP «Q Ql , p > 2n + 2 and l > 1, T is con inuous on D . (c) In case is o class C+7, 0 < -y <_ 1/2, hen o p > 2 , T maps Le a Ql in o Lip (min{-y, 2 - np }, D), o all a,,3 . No e in heo em 2 ha i is jus C, Hdlde es ima es can only be ob ained o (0,1)- o ms . Fo o ms o bedeg ee (a, /0), ,0 > 1, one needs ex a assump- ions on o ob ain Hdlde es ima es o a ce ain ange o p's . In case is o class C-4-1/2 , hen pa (a) holds o o ms o a bi a y bedeg ee . In o de o s a e ou hi d esul , which gi es an imp o emen o he es ima es a he bounda y, we need o ecall a de ini ion om [1] and [6] : Assuming only ha is de ined in an neighbo hood o D, we pu o (, z E D p«, z) = I (a (z), ( - z) j + I (a «), ( - z) j + II( - zjj2 This is a pseudodis ance in he sense ha iangle inequali y holds wi h some cons an C, and will be called he Ko anyi pseudodis ance . We w i e Lip,(s, D) o he subspace o Lip (s, D) such ha I (w) - (z) I = O(p(w, z) 9 ) o (, z E bD . Theo em 3 . (a) Fo 2n - - 2<p< oo, T maps LPo 1) in o Lip p ( 2 - nP l ) (b) Fo E L- 1) hen IT (z) - T (w)I <-eji II .p(z,w)' 12 Ilogp(z,w)I (c) I is o elass C2+- , 0 < y <- 1/2, hen o p > Z 1 n 27, T maps L(a,Q) in o Lip,(min(y, á - np ), D) Since CHIC - x11 2 < P(C, z) < c211 C - x11, he meaning o heo em 3 is ha he solu ions o he ó-equa ions will be, o he ange o p indica ed, wice as egula in ce ain di ec ions in bD . This, in he egula case, is a e o mula ion o he es ima es in he non-iso opic Lipschi z spaces I',,,2a in oduced by S ein . Finally, sec ion 3 con ains he es ima es o he in eg als in he p oo o he heo ems . Ou echnique di e s om ha in [2] in wo aspec s : i s o all, o cou se, he non-smoo hness makes mo e in ol ed he es ima e o he singula i y o he ke nels and, secondly, we use di ec me hods ins ead o in e pola ion esul s in ob aining he Hdlde es ima es . A main hechnical di icul y o ha is ha he domain being non smoo h we do no ha e a ou disposal he c i e ia Du(z) = O(u(z)' -1 ) o u o be in Lip (s, D) . In [1] ke nels a e ob ained o sol e á in non egula s ic ly pseudocon ex domains, o Henkin-Rami ez ype wi h weigh ac o s . Le us b ie ly ecall hei cons uc ion and main p ope ies . 1 .1 . Gene al cons uc ion . Fo U a C 1 bounded domain in C n, le s, Q : U x U -> Cn whe e s is a sec ion o Bochne -Ma inelli ype, say : o C,z E U, and ESTIMATES FORTRE Ó EQUATION  79 whene e ( E U and z E L compac in U, and Q is o class C 1 and holomo phic in z . Le also G be a holomo phic unc ion o one complex a iable de ined in a neigbo hood o U x U unde he map (C, z) - 1 + (Q(C, z), C - z) and wi h G(1) = l . Finally de ine (1)  K«, z) = en ~ (n k!1)I G(k) (1 + (Q, C - z)) s n (dQ )~ . Az(di)k -k- 1 k=0 1 . The ke nels sol ing á IIS(C, z)11= o(II( - Z11) I(S(C, z), ( - z)1 ? CLII( - x112 whe e en = ((27 i)n(n - 1)!) -1 , s" _ En j=o s j d(Cj - zj ) and Q = ~~ o Qid(Cj - zj) K is a2n - 1 o m in d(,d(,dz and dz oge he , and o 0 <_ a, a <_ n, le K,,,p he componen o bedeg ee (a, P) in z and (n - a, n- /~ - 1) in C . Then : 80  J .M . BURGUÉS Theo em 1 .1 .  Whene e Ñ > 1, i K< , Q((, z)ISEbU is 0 o z E U, he ope a o (2)  T = (_ 1)a'+# / ^ Ka,P-1 U sa is ies 8T = i EC 1 (a a)(U) . 1 .2 . The sec ion and weigh s in he s ic 1y pseudocon ex case . I D is a s ic ly pseudocon ex (non egula domain) and is a C de ining unc ion o D, le U6 = { (z) < S} and V = {-S < (z) < b} . Then Henkin and He e 's lemmas p o ides us wi h a amily o unc ions ¿j, j = 1, . . ., n, and cons an e co,eo, S o depending only on he unc ion (bu no on i s g adien , no on he a iable z), such ha (Pj EC 1 (V 6o x U6o) and a e holomo phic in z, and he unc ion 4(C, z) = E= 1 ¿j«, z)((i - zj) sa is ies : We de ine : I<p((,z)I ? co i IK-x11 >_ eo 2Ñ<P((, z) ? (C) - (z) + Cog - Z11 2 l II( - zII < Eo d«PIC-Z = dz,¿j,C = á (z) ~j«, z) _ (j «),+ . 0(11( - zII) when I1( - zII < Eo, I (z)I < 6o A(C, z) = - (z) + -¿((, z) and i x EC , (C n) , 0 < x :5 1 and x - 1 on V ! ,, de ine z and Q= (Q,, . . ., `w n ) . Qj((,z) = x (C) A«, z) W i e now, V E, 6 = {(( ; z) : 1 7 -(01 < b, I (z)1 <  z II < e} and de ine Ib ;((, z) = 4> .i(z, (),  z) _ ~¿(z, (), . á((, z)  z)+A(z, ()~¿j((, z), and ((, z) =Z :'=1 .i(C, z)(C .i - zj) . Finally, i 0 E Cm(C"), 0 < 0 :5 1, 0 - 1 on V~ s , de ine _  a,s S .i(C z) = « C, z) .i((, z) + ( 1 - 0((, z))(C .i - zi) s = (sl, .. .sn) and H = (s, ( - z) . The ollowing es ima es a e c ucial o he es ima es o he ke nel's singula - i y : 2ReA ;z~ - (() - (z) - - coll( - x112 o ((, z) E VEO,óo n (U x (1) . This implies ha , in e ms o he pseudodis ance, p : ESTIMATES FORTHE D EQUATION  81 IHI~ : c{( (() - (z))2 + ( - (() - (z))II( - zli ¡Al ^ - «) - (z) + p«, z) IHI ~~ e{(- «) - (z))II( - zjj2 + P2((, z)} 1 .3 . The esul ing ke nels and hei es ima es . Take in he o mula (1) G(w) = w n , and he sec ion and weigh in oduced be o e . De ine also : n w((,z) _ 1 : 4,i((,z)d((i - zi) j=o n w ((, z) = 1 :4> j (z, ()d((i - zi) j=1 A*((, z) = A(z, () 1 7((, z) = (()Dz w* + A*D(w A e a combina o ic compu a ion one ob ains in ha case : n-1 _ 1 ( 3 )  K((, z) = 4J A  Cn,k(  A~ ) )n-k Hn kAk (D<w)k A /n-k-1 k=0 n-1 + [ (()&A * - A*DS ) n w n w* n E C n, k(n - k - 1)( - (() )n-k A k=o (á(w)k n ,n-k-2 n-1  (() n-k - (()DSA Aw A w* A E cn,kk(  TA  ) k=1 + 11( - z11, + IMOIMO*}, Hn-kAk 1  (~ w k-1 n n-k-1 Hn-kAk+1 ) De ine now q((, z) = Ild (z)jj + li( - zil, and deno e jid (z)jj = A(z) I is clea om (3) ha K((, z) = 0 o ( E bU so Theo em 1 .1 applies and he ke nel K has he es ima e (see [1], lemmas 2 .4 and 2 .5) : ( 4 )  ¡K((, z) 1 = G(I ~~nl { - (() + q 2 ((, z)}11( - z11)} 82  J .M . BURGUÉS In e ms o he pseudodis ance, we ha e (see[1], lemma 6 .2) ( 5 )  IK((~ z)1 - ~(  1  + ( - (z)) n _  11( - zilg2((, z) 1l( - z llzn  1  [- (z)11( - x11 2 + P((, z) 2 ]n + 11(-zll ?((,z) )=de O(N1) P(('z)n+1 As showed in [1], lemma 6 .3 he ke nel K((, z) is in eg able in each a iable, uni o mely in he o he . Using a s anda d egula iza ion p ocess, one can hen show ha i E h¡oc~a,a) and DT = 0 in he weak sense hen T is a o m in Lioc(a,p-1) and DT = in he weak sense . No ice ha when z E bU, he es ima e (5) implies ( 6 )  Ih((, z) I = O(P((~ z) z-n + A(z) 2 P((, z)-n  1) =de O(N2) and also he wo se bu symme ic es ima e : z  z ¡K«, z)1 = 0( ll( - z112n-1 + 11( - x11 2 . 3P2((' z)) __de D(N3) 1 .4 . Es ima es o he di e ences . Ou me hod in p o ing H51de es ima es in ol es es ima es o he di e ences o he ke nels ¡K«, z) - K((, w)l . A sui able con ol can be ob ained in e ms o he g adien o K wi h espec o he second a iable whene e i makes sense, ha is when he coe licien s o he o m K((, z) a e C 1 in z . Fo mula (3) shows ha all e ms bu hose in ol ing D Z w * a e C 1 in z , because Dw* = E D=ob~ n d((j - zj) and D,ID~ ((, z) =D z <Pj (z, () is only con inuous, since i in ol es second de i a i es on . Obse e also ha bad e ms may appea only once (because Dz(*AD-,Y _ 0), and in he componen e K«,0 hey do no appea a all . So we can w i e he ke nel K as a sum K((, z) = K,«, z) A D=w* + K2((, z) whe e K 1 , Kz a e C 1 in z and K, ,,, o = (K 2 ) < ,,, o . The ke nels K l and IKz sa is y he same es ima es as K, ha is (5) and (6) . The g adien e in z o K l and K 2 sa is y he es ima es con ained in lemma 6 .6 o [1] and hence i ollows ha he e exis s c such ha i llz - wjl is small enough and ll( - zll > cllz - w11 1/2 , (E D, hen, o j = 1, 2 I Kj((, z) - Kj((, w)I = 0 (11z - w11M1((, z)) whe e ( 8 ') __de  I (z)in¡¡( - zl1A(z) 2  11( - z11 .1(z) 2 Nh  (  +  +  )q(, z) 11( - z112n+1 [I (z)I11( - x11 2 + P(, z) 2 ]n+l  P(, z)n+z hus, K«,o(C, z) - Ka,o(C, w) will sa is y (8) i C,z,w a e as abo e, wi hou any u he equi emen on . Fo he componen s K«,p wi h 0 > 0 w i e K(C, z) - K«, w) = {K(C, z) - K (C, w)} A azw* + K2(C, z) - K2(C, w) I ollows ha i d 2 sa is ies a Lipschi z es ima e o o de 7 ; and z, w, ( a e as abo e, hen (9)  IK«,p(C,z)-K,,p(C,w)I =o(IIz-wJIMI(C,z)+IIz-wJI`NI(C,z)) The es ima es (8), (8') and (9) will be used in he Euclidean Hólde es ima es . Fo he non-iso opic Hólde es ima es i will be con enien o es ima e he di e ence K«,p(C, z) - K«,p(C, w) jus in e ms o p . In a simila way as be o e, bu using now lemma 6 .9 o [1] ins ead o lemma 6 .6, we see ha i is jus C 2 and p(C, z) > cp(z, w) (,z,w E bU hen : (10)  IK«,o(C, z) - K«,o(C, w) I =0 (P(z, w)'/ 2 M2) whe e ( 10 ' )  M2 =de P(C, z) -n + (Z) ,P(C, z) -1-n and i EC 2 +Y hen o Q > 0 (11) whe e (11') 2 .1 . P oo o heo em 1 : ESTIMATES FORTHE a EQUATION  83 ¡K ., # «, z) - K«,p(C, w)I = O(P(z, W)1/2M2 + P(z, w) y/2 N2) N2 = P(C, z)1/2-n + ~X(z) 2 P(C, z) -n-1/2 2 . P oo o heo ems As in [2], he p oo o Theo em 1 is based on he ollowing lemmas : Lemma . Le K : R' x Rn -- i C ha e he p ope y ha K(x, .) is o weak ype s as a unc ion o y, uni o m1y in x, and K( ., y) is o weak ype s as a unc ion o x, uni o m1y in y . Then he linea ans o ma ion (x) - T (x) = J n K(x,y) (y)dy R + K,(C, z) A {0 .w* - aww*) 84  J .M . BURGUÉS de ined o : R' --> C, sa is ies 11T II Lq < Ap li II Lp whene e s > 1, 1 < p < q < oo and q =  + 9 - 1 . Mo eo e , 11T L-< A E jI II L 1 o all e > 0 . Lemma . Suppose K,s, as aboye and E L" whe e s + s, = 1, suppose also ha m = n, S2 CC Rn, and suppK C S2 x S2 . Then and c, M do no¡ depend on , bu on s and m(9) Aco ding o he lemma, we ha e o p o e ha K«, z) is o 2 2n~1-weak ype on D in each a iable uni o mély in he o he . Fo his we use he es ima e (7) and, since i is symme ic, i will be enough o p o e he ollowing esul : (12) Lemma 2 .1 . m{z : (K«, z)j > } = O( - ~ ), uni o mly in (E D This will be done in sec ion 3 . 2 .2 . P oo o Theo em 2 : Le p > 2n + 2and E Le o 1) . Fi s we p o e ha T E Lip( 1 - -+', D) Le z, w E D, b = ~~z - w~~ . We de ine i = 61/2 and es ima e T (z) -T (w), using (8), by In case mo e e m : (z)  I (C)IN,(C,z)dm(C)+J ,n(z)nD l «)¡NI«,w)dm«) B n  nD  B + 6 1 l (o1Ml«,z)dm«) B~ n(z) EC 2 + - exp((c IT (z)I )')dm(z) < M < o0 ~~ ~~La'(sl) and E LPa Q), ,Q > 1, we will ha e, aco ding o (9), one ó7  1 (C)1 Ni«, z)dm«) Using Hdlde es ima es we a e lead o he ollowing lemmms which will be p o ed in sec ion 3 : Lemma 2 .2 .1 . Fo 1 < s < 2n±2 we ha e - 2n+1 {  N i «, z) - dm«)Í 1/e =  ~ O( ~ -2n - 1 ) ~B  (z)nD Co olla y . Fo 1 < s < 2n+2 - 2n+1 Lemma 2 .2 .2 . Fo 1 < s < 22n+ i Using lemma 2 .2 .1 he i s in eg al in (12) is O(62 - ~) . The second one is also O(62 - ~), because B,,,(z) C B,,,,(w) . Finally by lemma 2 .2 .2, he las e m in (12) is O(677i2jL-2 -2n-3 ) = O(62-~) This p o es he i s pa o he heo em . Unde he condi ions in (b) we ha e one mo e e m which is O(1) acco ding o he co olla y o lemma 2 .2 .1 . Then i p > 2n+2 we ob ain a Lipschi z condi ion which exponen min  1  nn±l ) (-Y, 2 - which gi es pa (b) o he heo em . Finally, o pa (c) we simply w i e, IT (z) - T (w)I <_ II IIp{ (IK«, z)I + IK«, w)I) 9 dm«)} 1/ s B whe e ñ + 9 = 1 ESTIMATES FOR TIiE C9 EQUATION  $5 ( D~B  N I «, z)9dm«)}1/9 = O(1) n (z) { 1  Ml«,z)9dm«)}1/8 = O(17~-2n-3) B  (z) + II IIp{ J  ¡K«, z) - K(C, w)I9dm«)}lis D~B E (z) The i s e m can be made a bi a ily small by choosing e small and he second oo by choosing w close enough o z beacause he ke nels a e con inuous o he diagonal . 2 .3 . P oo o he Theo em 3 : Fi s , le A5(z) = {( : p«, z) < 6}, be he HS mande ball ela ed o p . Fo p > 2n -}- 2 and E Le o 1) we spli now he in eg als gi ing T (z) - T (w) o z,w E bD in he o m : (13)  JA .6(Z) I (C)IN2(C, z)dm«) + JAa(Z) I «)IN2(C, w)dm«) + 6112 JI (S)IM2«,z)dm«) D~A~s(z) He e 6 = p(z, w) and we ha e used (10) In case ECZ+ 7and E Le a Q) , 0 > 1, we will ha e by (11) ano he e m : ( 14 ) 6y/2  I (QN2«,z)dm«) D~A~a(z) Using HSlde 's inequali y as be o e we a e lead now o he ollowing lemmas :