A uniqueness theorem for invariantly harmonic functions in the unit ball of Cn

Publicacions Ma emá iques, Vol 36 (1992), 421-426 . A UNIQUENESS THEOREM FOR INVARIANTLY HARMONIC FUNCTIONS IN THE UNIT BALL OF en Abs ac JOAQUIM BRUNA Dedica a la memó ia den Pe e Menal We p o e a bounda y uniqueness heo em o ha monic unc ions wi h espec o Be gman me ic in he uni ball o Cn and gi e an applica ion o a Runge ype app oxima ion heo em o such unc ions . Le B be he uni ball in en and S i s bounda y . The in a ian laplacian 0 in B is he Laplacian associa ed o he Be gman me ic and i is gi en in coo dina es by Pa ially suppo ed by DGICYT g an PB89-0311 . (6i7 - zizj)DiDj . The e m in a ian comes om he ac ha i commu es wi h all au- omo phisms kP o B : 0(u oT) = Du o T . Co espondingly, hose unc ions u E C 2 (B) annihila ed by O a e called in a ian ly ha monic o M-ha monic (see [4, chap e 4] o he mo e ele an p ope ies o hese unc ions) . The aim o his no e is o gi e a bounda y uniqueness heo em o M- ha monic unc ions and an applica ion o a Runge ype app oxima ion p oblem . 1 . The uniqueness heo em essen ially s a es ha S, hough A com- ple ely degene a es he e, is non-cha ac e is ic o a ce ain Cauchy p ob- lem : 42 2  J . BRUNA Theo em . Le U be a ball cen e ed a ~ E S, and le uEC°° (U 1 B) sa is y Du= 0 in U . Then om n u= a n =0 onuns i ollows ha all de i a i es o u a e ze o on U l S (hence u - 0 i i is eal-analy ic acc oss S) . The p oo will show in ac ha u and onñ de e mine all he o he s de i a i es o u on S . The s a emen sugges s ha he ollowing Cauchy- Kowale ski ype heo em is p obably ue : i , g a e eal-analy ic unc- ions de ined on U n S, he e is ano he ball V C U con aining S and a eal-analy ic unc ion u in V such ha Du= 0 in V n B and u = , án ñ = g on V 1 S . P oo . Le Ao = Ei j-1(Si7 -ziz j )DiD j and le R= ~ iz j D jbe he adial (holomo phic) de i a i e ; w i e R= N + iT, hen N = -2- and T is a eal angen ield o S . We will show ha u alone de e mines N3 u, j <n- 1 and ha u, N' 2 u de e mine all de i a i es a poin s o S . A compu a ion shows ha AoN - NAo= Do +N 2 +T 2 D O T - TDo = 0 . TN-NT=0 . F om his i easily ollows by induc ion on k ha DoN k = Pk(N)Do + kNk+1 + Rk(N,T) whe e Pk(x) is a monic polynomial o deg ee k and Rk(x, y) is o deg ee < k + 1 in x, y, bu o deg ee < k in x . Nex , he ollowing no mal- angen ial decomposi ion in [21 is needed Do = I i 2 {(1 - Iz1 2 )RR + A + (n - 1)N} z He e A is he box-laplacian on S ; i s pa icula exp ession will no be needed, only he ac ha i is a angen ial ope a o . I ollows ha a poin s w E S (4)  (Do )(w) = (A )(w) + (n - 1)N (w) . UNIQUENESS THEOREM FOR HARMONIC FUNCTIONS  423 Applying his o u we see ha Nu = 0 on S . Assume by induc ion we ha e p o ed N( k )u = 0 on S whene e u E C°°(U n B) is M-ha monic and ze o on U n S, k < n - 1 . Then by (3) and (4) on U n S . By (2), ( n- 1)N (k+l) u = D o N(k) U = kNk+1U +R k (N , T) u R k (N,T)u = i<k i+j<k+1 N Z Tju and by (1), Tju is also M-ha monic and ob iously ze o on S . By he induc ion hypo hesis, Rk(N,T)u = 0 on U n S . Then we conclude ha N(k+1) u = 0 i k < n-1 . I k = n-1 we canno conclude N(n)u = 0 bu i is clea ha i his is known o hold, hen he induc ion can con inue and so NW u = 0 o all j on U n S, which p o es he heo em . The e is some connec ion o his esul wi h a esul om Folland [1] acco ding o which an M-ha monic unc ion u in he whole ball o class Cn up o he bounda y mus be in ac plu iha monic . 2 . As an applica ion o he heo em we p o e : Theo em . Le K C B be a compac se such ha B K is connec ed . Then, e e y sa is ying O = 0 in a neighbou hood o K is he uni o m limi on K o a sequence o M-ha monic unc ions u n in B, con inuous on B . I mus be poin ed ou ha his esul can be p o ed as well by com- bining a gene al esul o [3] on analy ic-hypoellip ic ope a o s and [4, 5 .5 .4] . Ou p oo p oceeds by duali y and elies on some well-known ac s ha we p oceed o ecall . The e is a decomposi ion o mula, alid a leas o u E C 2 (B), U(Z) = I P(~, z)u(~) do,«) + I Au(S)G(~, z) d>,«) s  s ha co esponds o he Poisson-G een o mula in Euclidean space . He e do, is he no malized Lebesgue measu e on S, d>,«) = (1-1 ~ j2)-n-1dV«) is he in a ian measu e, P«, z) is he in a ian Poisson (o Poisson- Szegó) ke nel _  2 n P«, z) = Í 1- (Z I 2 , (ES, zEB, 42 4  J . BRUNA and G«, z) is he G een unc ion wi h pole a z, (1 _ )n-1 G«, z) = G(wz«), 0 ) = cn  ,~,  d , ~~Pz(S)j2 c, z being a cons an and ~o z he au omo phism o B, unique up o uni a y ans o ma ions, ha sends z o 0 and 0 o z . Mo eo e , 1 - I(Pz«)I 2 = (1 - IZI 2 )(1 - I(12) 11 - zl2 so ha G is in ac symme ic . Oneway o ob aining (5) is o w i e p ecisely he Poisson-c een o mula o u o co z a 0 and change a iables in he esul ing in eg als . A second (and be e ) way o looking a (5) is h ough he c een iden i y in he Be gman me ic o AC B (u0 - 0u) dA _ a (U á - 8 ) a  a  dS . He e is he ou wa d uni a y no mal (by he Be gmann) me ic o áA and dS is he induced measu e on 8A . Fo mally, one ob ains (5) by specializing o A = B, (~) = G,«)=G«, z) and checking ha OCG«, z) dA(~) = Sz,  z E B G«, z) = 0  E S (6)  wC G«, z) dS = P(~, z) da  E S, z E B (in a igo ous way one should choose as A he ball o adious < 1 and hen make --> 1) . Fo mula (5) implies he ollowing ac s : (a) The gene al o m o an M-ha monic unc ion u in B, con inuous on B is u(z) = P[ ](z) = P«, z) «) do , s wi h E C(S) ; equi alen ly, P[ ] is he unique solu ion o he Di ichle p oblem Du= 0 in B, u = on S . (b) Fo u E CZ(B) o compac suppo u (z) = B Du(S)G«, z) dN(~) Le . u coincides wi h he G een po en ial o i s Laplacian . The simme y o G hen implies ha o a measu e wi h compac suppo in B, he G een po en ial Gp a n (e)  P(~, z) = cn 9 n G( (' z) = 1 '  E S, z E B . This is because wha (6) eally means is, as said be o e, UNIQUENESS THEOREM FOR HARMONIC FUNCTIONS  425 Gj¿(z) = I G(~, z) dp(~) s sa is ies (AGí ) dA = dM in he weak sense, and in pa icula OGil = 0 in he usual sense o he suppo o p, . Finally, we will need a e o mula ion o (6) in e ms o he Euclidean no mal, which is P«, z) = lim aG( (, z) do, . Since = c,, (1 - ) n á and dS = c, (1 - ) 1-2 ' do , (we deno e by c ', all cons an s depending on n), (c) ollows by L'Hopi al's ule . Al e na i ely, (c) can be p o ed o cou se by di ec compu a ion . No e ha ~G«, z)j = 0(1- I(j2)n o a ixed z . Hence á ; G( (, z) = 0 o ( E S, j =0, . . . . n- 1 . P oo o he heo em : Le p be a measu e on K which is o hogonal o all M-ha monic unc ions in B, con inuous on B . By (a) abo e his is equi alen o I P«, z) dp(z) = 0 o ( E S . x We conside he G een po en ial o p G(p)(w) = I G(z, w) dM(z) x so ha OG(p,) = 0 o K . Mo eo e G(p) is s ill de ined and is eal analy ic in a neighbou hood o B, o K, because so is each cp, z o z E K . No e ha all such po en ials sa is y 7 á ~ G(p)«) = 0, j = 0, . . .,n - 1,  E S . 426  J . BRUNA By (c), (7) says ha also (z) = B O «)G«, z) d, «) = L x O «)G(~, z) dA«) . Hence by Fubini's heo em n Ó n G(P)(~) = 0,  ES . The e o e, by he heo em in Sec ion 1, all de i a i es o G(p) anish a S and, since B K is assumed o be connec ed, we conclude ha G(M) is iden ically ze o in B K . Le now as in he s a emen ; mul iplying by a es unc ion, we can assume ha is compac ly suppo ed in B . By (b), L (z) dp(z) =  B~ x O (~) {Ix G«, z) dp(z) } dA«) = x -  A «)Gli(~) dA(~) = 0 ~B~x which inishes he p oo , by Hahn-Banach's heo em . Re e en es 1 .  G . FOLLAND, The sphe ical ha monic expansion o he Poisson- Szegd ke nel o he ball, PAMS 47 (1975), 401-408 . 2 .  D . GELLER, Some esul s on HP heo y o he Heisenbe g g oup, Duke Ma h . J . 47, no . 2 (1980), 365-390- 3 .  B . MALGRANGE, Exis en e e app oxima ion des solu ions des equa- ions aux de i ees pa ielles e des equa ions de con olu ion, Ann . Ins . Fou ie 6 (1956), 271-355 . 4 .  W . RUDIN, "Func ion heo y in he uni ball o en," G undleh en 241, Sp inge -Ve lag . Depa amen de Ma emá iques Uni e si a Au ónoma de Ba celona 08193 Bella e a (Ba celona) SPAIN Rebu el 24 de No emb e de 1992