Factorization of the green's operator and weak-type estimates for a random walk on a tree

Publicacions Ma e ná iques, Vol 35 (1991), 187-207 . FACTORIZATION OF THE GREEN'S OPERATOR AND WEAK-TYPE ESTIMATES FORA RANDOM WALK ON ATREE RICHARD ROCHBERG AND MITCHELL TAIBLESON 1 . In oduc ion . Le X be a ee, which is o say, a connec ed g aph wi hou loops . The o de o a e ex o he ee is he numbe o edges ha mee a he e ex . To a oid some messy, bu essen ially i ial, complica ions we will assume h oughou ha he o de o e e y e ex is a leas h ee . We say ha a e ex u is a neighbo o he e ex i u and a e connec ed by an edge . When u and a e neighbo s we w i e u - . The se o ansi ion p obabili ies {p(u, )} is said o de e mine a nea es neighbo andom walk i 0 _< p(u, ) < 1 o all u, E X and p(u, ) > 0 i and only i u - . The walk is said o be s ochas ic i j :,, p(u, ) = 1 o all u in he ee . The ansi ion p obabili ies de e mine a ansi ion ope a o , P, as ollows : Fo a unc ion de ined on he ( e ices o he) ee, P (u) = 1 : p(u, ) ( ) . VEX We o en iden i y he ansi ion ope a o wi h i s associa ed se o ansi ion p obabili ies . We assume u he ha he ansi ion ope a o P is egula in he sense ha he e is a posi i e numbe 6 such ha p(u, ) >_ b whene e u - . This implies ha he o de s o he e ices a e bounded abo e . The isi ing p obabili y, U(u, ), o u and in he ee is he p obabili y ha a walk s a ing a u will isi . The e a e wo common con en ions i u = . The s anda d con en ion .i s ha U(u, u) is he p obabili y ha he walk will isi u a some ime in he u u e, and his is ou de ini ion . The o he con en ion is ha U(u, u) = 1 . Wi h his in mind we de ine : { a(u, )  U (u, ),  i u 7É 1, i u= . P is ansien i U(u, ) < 1 o all u and in he ee . We will equi e ha P be s ongly ansien in he sense ha he e is a S > 0 so ha U(u, ) < 1 - S Suppo ed in pa by NSF G an s DMS-8701271 and DMS-8701203 . 18 8  R . ROCHBERG, M . TAIBLESON whene e u - . In he Appendix o [KPTI i is shown ha i he e is a posi i e numbe y such ha p(u, ) < 1/2 -y hen P is s ongly ansien . Le q + 1 - q(u) + 1 be he o de o he e ex u . We say ha he ansi ion ope a o P is iso opic i p(u, ) = 1 /(q + 1) whene e u - ; we say ha i is symme ic i p(u, ) = p( , u) o all u and . A ee is homogeneous i all e ices a e o he same o de . A ansi ion ope a o is homogeneous i i is de ined on a homogeneous ee and he se o ansi ion p obabili ies a each e ex a e he same . I is easy o see ha bo h an iso opic ansi ion ope a o onan o de bounded ee and a homogeneous symme ic ansi ion ope a o a e always s ongly ansien . In he sequel we assume ha P is s ongly ansien and we se Thus,0<5<1 . = sup{a(u, ) : u - } . Th oughou his pape unc ions on he ee X a e complex alued, X is supplied wi h he disc e e opology and he a omic measu e which assigns mass one o each e ex . We hink o he ee as a collec ion o e ices and o edges as a ela ionship on he ee . F om his poin o iew X is a locally compac measu e space . Fo a unc ion, , on he ee, 1/ Il 11P = (E l ( u )I P UEX li 11 . = sup i (u)1 . UEX The Laplacian ope a o , A, is de ined by A (u) = E p(u, ) ( ) - (u) . o-u Tha is, A = P - d . The G een's ope a o , G, speaking loosely, is he in e se o -A . I is de ined by a ke nel G(u, ) , he G een's ke nel, G (u) = 1 : G(u, V) ( ) . VEX 0<p<oo We de e mine G as ollows : Fix u and in X . A pa h in X connec ing u o is a ini e sequence o e ices, w = {wo,wl, .. . . w n }, whe e w o = u, w n = and o each k = 1, . . . , n, wk_ 1 is a neighbo o wk . The leng h o he pa h w is 2(w) = n . The weigh o he pa h w is n W(w) =11 p(wk-1, wk) . k=1 A i ial pa h w= {u} has leng h ze o and weigh one . FACTORIZATION ON THE GREEN'S OPERATOR  189 De ini ion . G(u, )  W(w) whe e w anges o e all pa hs ha connec u o . Obse a ions . I is immedia e ha G(u, ) = E' o p(k) (U, ) and so G=  o Pk whe e he p(k) a e ansi ion p obabili ies associa ed wi h he ansi ion ope a o P k . I is no di icul o see ha G(u, ) is he expec ed numbe o isi s o o a andom walk ha s a s a u . We no e u he ha P is ansien i and only i G(u, ) < oo o all u and . We do no use any o hese obse a ions . Fo any wo e ices, u and , in X he e is a unique pa h o sho es leng h ha connec s u o . This pa h is called he geodesic ha connec s u o . We deno e i s leng h as d(u, ), and obse e ha d is a me ic on X . Gene al e e ences o ma e s aised in his in oduc ion a e [C], [KPT], and [G] . 2 . Disk ealiza ion o he ee . The e a e wo na u al ways o gi e an o ien a ion o he edges in a ee, he disk ealiza ion o he ee and he hal -plane ealiza ion, which we desc ibe in he nex sec ion . Fo he disk pic u e, an a bi a y e ex is selec ed and is deno ed o . I is iewed as he ini ial poin o a andom walk ha is go e ned by he ansi ion p obabili ies {p(u, )} . A poin on he bounda y o X, OX, is a semi-in ini e geodesic, x = {xo, x l . ... , xk, . . . }, whe e x o = o . No e ha d(o, xk) = k o all k . Suppose u and a e in X . We say ha w is be ween u and i w is on he geodesic ha connec s u o . I x E áX hen w is be ween xand u i w is be ween u and xk o all k la ge enough . Le D = X U OX . I is be ween o and u we say ha is abo e u, o ha u is below . I x = {xo, x, .... , xk, . . . } we say ha he geodesic {xo, xi ... . . xk, . . . } connec s x o o . I ollows ha o is abo e e e y poin in D and i xE CM hen x is below xk o all k . Fo each e ex u he e a e q(u) neighbo s o u, {uj}, ha a e below u, excep o o ha has q(o) - - 1 lowe neighbo s . Fo each e ex u, u 7É o, he e is a unique e ex u - such ha u- - u and u - is abo e u . We now de ine a subbase o he opology o D . I consis s o all se s N(u), u E X whe e N(u) = { : is below u} . Wi h his opology D is compac , he es ic ion o he opology o X is disc e e, and i s es ic ion o áX is compac . I {xo, xl ... . . xk, ... } E áX hen N(xk) l 8X is e e ed o as an in e al o le el k . We say ha xk is a e ex o le e] k . We w i e Iu = N(u) l OX . We now de ine he hi ing (ha monic) measu e, p, on 8X . Deno e by F n he ( andom) e ex a s ep n o he andom walk de e mined by P wi h F o = o . p(I«) = P (3k o : Fk E N(u)  Vk > k o 1 F o = o) p ex ends o a Bo el measu e on 9X . The unc ion U(u, ) de ined in Sec ion 1, called he isi ing p obabili y, is o mally de ined by : U(u, )=P (3n>0 :F n = 1 F o =u) . 190  R . ROGHBERG, M . TAIBLESON Conside he andom walk de e mined by P s a ing a u and condi ioned o emain in N(u) om some s ep onwa d . De ine a hi ing measu e, , o his walk jus as we de ined p . Then (I u ) = 1 and o - u, 5A u- , de ine he ela i e o wa d p obabili y, 7 (u, ) = (I,) . Clea ly We say ha we a e mo ing o wa d i we mo e o a posi ion below . By P oposi ion 2 o [KPT], o u =~ o Obse e ha his o mula shows ha he p obabili y o being in N(u) om some s ep onwa d, condi ioning en he e en o being in N(u - ) om some s ep onwa d, is he same as ha o he e en o s a ing a u- , condi ioning on he e en o ne e e u ning o u- and being in N(u) om he i s s ep onwa d . Le A(u) be he p obabili y ha a andom walk s a ing a u mo es o wa d en i s i s s ep and ne e e u ns o u . We see ha A(u - ) is he denomina o in equa ion (1), so ha No a ion . neighbo s o u 7 i = l (u,uj ) 1 esul o A(o) ollows om he de ini ions . Fo u 7É o we use he ela ionship Lemma 1 . P oo . The which ollows ew i e his 7 (u, ) = (Ij = 1 . : - =u p(u - , u) (1 - U( u, u- )) UW :W - = u - P(u , w) (1 - U(w, u-» 7 (u ) = p(u, )(Á( u ) ( +u)~  i u = - . Suppose u is a e ex and u  o .  Le u j , j = 1, . . . , q be he ha a e below u . We se u - = uo, Pi = P(U) u j ), j = 0, 1, . . . , q ; aj = a(uj, u), j = 1, ... , q ; and ao = ca(u, uo ) . A(o) = 1 - U(o, o) . I u :~ o hen A(u) = p(u, u- )(1 - U(u, u - ))/U(u, u - ) . q ao = Po + E p .i al ao , j=i om he Ma ko p ope y o andom walks (see [KPT]) . We q  po q  1 ~Pia .i =1- a o =~P .i+poC1 - ~ o l . j=i  j=i F om his we ob ain FACTORIZATION ON TIIE GREEN'S OPERATOR  191 Pj( 1 - aj) = 1 - ao -Po . ao j=1 Using he no a ion in oduced be o e he s a emen o Lemma 1 we no e ha A(u) = Po(1 - «o)  pj( 1 - aj)ao ao  j-_ p o (1 - «o) when u 7É o . I is easy o see ha i {w o , wl, ... , w n} is he geodesic connec ing o o u hen n h(I .) = 117 (wk-1,wk) k=1 I w = {w o , wl, ... , w n } is he geodesic connec ing u o we de ine (7)  V(u, ) = W(w) and obse e ha n (8)  a(u, ) = Il a(wk- l , wk) . k=1 Obse a ion . I is clea ha he egula i y and s ong ansi i i y o he walk imply ha he e is a S > 0 such ha whene e u and a e e ices and u = - hen 7 (u, ) > 8 . and A(u) , . .. 1 o all e ices u . Theo em 2 . I u and a e e ices andu is abo e hen c(I  ) - V(u, ) A( ) p(I .)  V( , u) A(u) U( , u) . P oo . . Use equa ions (2)-(8), ga he e ms and simpli y . 3 . Hal -plane ealiza ion o he ee . We begin wi h he disk pic u e . Selec a poin on óX and deno e i by oo . To each o he poin , y on CM we associa e he unique doubly in ini e geodesic { . . . , y-2,y_1, yo, yl, y2, . . . } such ha {yo, yl, ... } is co inal wi h he geodesic in D ha de ines y and {yo, y_1, . . . } is co inal wi h he geodesic de ining oc . Fo his ealiza ion we call he ee Y and he ini e pa o he bounda y is 8Y, H = Y U aY . Fo each e ex u he e is a unique geodesic ha connec s u o oo ; which is o say a hal -in ini e geodesic wi h ini ial poin u ha is co inal wi h he geodesic in 192  R . ROCHBERG, M . TAIBLESON X ha de ines oo . We say ha each e ex on ha geodesic is abo e u, and ha is below w whene e w is abo e . The poin oc is abo e e e y o he poin in H, and a poin on áY is below e e y e ex ha lies on he geodesic ha connec s i o oo . A c ucial (simpli ying) di e ence wi h he disk pic u e is ha o e e y e ex, u, on he ee he e is a unique e ex, u- , ha is a neighbo o u and is abo e u . As be o e we se N(u) = { E H : is below u} . This de ines a subbase o a opology on H ha is locally compac . The es ic ion o he opology o Y is disc e e and i s es ic ion o áY is locally compac . Again we se I u = N(u) n DY . Jus as each e ex in he disk has a le el we can de ine a le e] o each e ex in he hal -plane . Selec a e e ence e ex e, and conside he geodesic {yo, y_1, y_2 . . . . } ha connec s e o oo . Thenwe se he le e] o e as 0 and each e ex, u, on he geodesic has le el -d(e, u) . Obse e ha e e y e ex on he ee is below some e ex on ha hal -in ini e geodesic . Using he ule : he le e] o u is he le el o u - plus one, a le el o each e ex is de ined . As in Sec ion 2 we can de ine he unc ions : A(u) and V(u, ) . To de ine A(u) we no longe need a special case . Fo all e ices u A(u) = p(u, u - ) (1 - a(u, u - ))/a(u, u- ) . Fu he mo e, we may de ine he ela i e o l a d p obabili ies (10)  7 (u, ) = p(u, ) (1 - a( , u))/A(u) whene e u = - . Since he bounda y is no compac we may no longe de ine he M(Iu) as p obabili ies, bu we will be able o use condi ional p obabili ies . We p oceed by choosing a e e ence e ex, o . Fo example, one may choose o = e . Se p(I o) = 1 . Ex end u o he ee by he ule : «u) = h(Iu-)7 (u,u-), ex ends o a Bo el measu e on áY . As in Sec ion 3 : Theo em 3 . I he e ex is below ¡he e ex u hen h(I,) - V(u, ) A( ) /,(Iu)  V ( , u) A(u) U ( , u) . Obse a ion . Theo em 2and Theo em 3 ha e he same o o bu he alues o he unc ion A a e de ined di e en ly in he wo ealiza ions . We no e ha p,(IJ/p(I u )is he p obabili y ha a andom walk ha is condi ioned o e en ually s ay in N(u), is e en ually in N( ) . Mo e in o mally : i is he_ p obabili y ha he andom walk hi s he bounda y in I  gi en ha i will hi he bounda y in I u . We also no e ha A(u) - 1 . FACTORIZATION ON TIIE GREENS OPERATOR  193 4 . Fac o iza ion o he Laplacian and he G een's ope a o . We begin wi h he hal -plane ; he ex ension o he disk will be gi en in Sec ion 6 . below . Recall ha e e y e ex is bo h abo e and below i sel . I we need o ule ou his possibili y we will use he locu ion s ic 1y below o s icly aboye . We de ine he ke nels o ope a o s S and T . (11)  KS(u, ) = (lu)' 0, o he wise . a(u, ) - a(u, - ),  i is aboye u 0,  o he wise . (12)  KT(u, ) Obse e ha lis(u, u) = 1 and IiT(u, u) = 1 - a(u, u - ) o all e ices u . We le , whene e he se ies con e ges absolu ely . Fo he nex de ini ions and he heo em ha ollows we use he no a ion in oduced jus be o e Lemma 1 and he o mula o A(u) in equa ion (5) . We in oduce wo di e ence ope a o s . Fo g a unc ion de ined on he ee : g(u) - 1 - a(u, u-) g(u)  1 á ( a(u, u ) ) g(u  ) Theo em 4 . Fo ¢ny unc ion g de ned on he ee A + Ag(u) = A(u) ~9(u) P oo .~ D+O 9(u) _ h P(IV)  i is below u S (u) _ 1 : Iis(u, ) ( ), T (u) _ 1 : IiT(u, ) ( ), A9(uj) - 0 9(u) 194  R . ROCIIBERG, M . TAIBLESON Sol ing o we ob ain 9 ao  E pjg(u .i) - Po(1 - %) j-i - 1 l a g(u) + 1 a a g(uo) o  o 9 Diana -  ao - po  1  u ~Po(1 - e¿,) + 1 - ceo) g() ao 9 Po(1 - ao) ~ pjg(uj) - g(u) A( )ág(u) . u 9 po(1a ó ao) z pjg(uj) + Po(1aó ao)pog(uo) j-i po(1 - ao)g(u) A he ou h s ep we used equa ion (4) . This comple es he p oo . We show nex ha A - is he in e se o T and ha -0+ is he in e se o S on app op ia e domains . Suppose is in he domain o T . I is easy o see ha T (u) = (1 - ce(u, u- )) (u) -}- a(u, u - ) T (u - ) . (u) = 1 - a ( uu- ) T (u) - 1 -(a(u ))T (u ) = A  T (u) . Since he coe icien s ha de ine T (u) a e non-nega i o and sum o one, 20° (and so 2P, 0 < p -< oo) is in he domain o T . Thus on 2P, 0 < p < oo, 0 - T=I . The e is an a such ha 0 < ca(u, u - )<_ á < 1 o all u . Consequen ly i is abo e u, n(u, ) < ád(u, ) . Conside he semi-in ini o geodesic {x o , x y , x2 ... . }, FACTORIZATION ON THE GREENS OPERATOR  195 whe e x o = u, and xk+1 = xk o k = 1,2, .... Then 0o TA - (u) =E(a(x0,xk) - a(x0,xk+1))o (xk) k=0 0o a(x0,xk)( 1 - a(xk,xk+1))x k=0 N - limo j , a(x0,xk ) ( x k) - a(x0, xk+l) (xk+l) k=0 = (x0) -  lim a(x0, XN+1) (XN + 1) X [ 1- a( k,xk+1) (xk)  1 a(a(xk,k k+l) (xk+l) N-oo p o ided a(x0,xk+1) (xk+1) = o(1) as k -> oo . This ce ainly holds i is bounded and so : Theo em 5 . 0 - is he in e se o T on ~P, 0 < p < oo . Lemma 6 . T is bounded on 2 °° wi h no m 1 . T and 0 - map ~°° on o i sel . P oo . I is easy o see ha T is bounded on ~°° wi h no m 1 and i is i ial ha 0 - is bounded on 2°° wi h no m a mos (1 + a)/(1 - á) . I ollows ha T and 0 - map P°° on o Qo° . We now conside S and 0+ . No e ha S is no de ined on 2°° . In ac i (u) - 1 hen S (u) - oo . To see his le {xkj}, J =1, . . .,j(k)',  k=0,1,2, . . ., be he j(k) e ices ha a e below u a dis ance k . Then oo 7(k)  (I =k, )  o0 S (u)=E~ i~  .1=1 : 1=00 . k=0j=1  h(I .)  k=0 Obse a ion . The e is a S, 0 < S < 1 such ha 7 (u - , u) _< 1 - S o all u . This ollows om he egula i y and s ong ansience o P . Lemma 7 . I is in £P, 0 < p < oo, hen is in ¡he domain o S . P oo .. Suppose i s ha 1 < p < oo . j(k)  j(k)  1/P (13)  (xkj) < ~~ (xkj)IP j=1  j=1 202  R . ROCHBERG, M . TAIBLESON Suppose is o le el k o . Then KT(u>V)Y'P(U) G l : gkak«ep(ku+k) u  k=0 =A E,«,qOP( ) p o ided e > 1/p(log q/ log(1/ix) - 1) . An elemen a y compu a ion shows ha a choice o such an e is possible p o ided = IXkOEP l : ú k(1+EP+10g 11lOg a) k=0 log q  _ p > 1011/ix) - pl . Le p o = max[1,p l ] . Ou a gumen shows ha T is bounded on 2P i p > po . An example is gi en in Sec ion 7 o which p o > 1 . We use a no ion in oduced by Ge l in [G] . De lni ion . A nea es neighbo andom walk is s ongly e e sible i he e is a unc ion m(u) such ha whene e u and a e neighbo s m(u)p(u, ) = m( )p( , u) and i he e a e posi i e cons an s m and M such ha m(u)p(u, )>_ m > 0 whene e u and a e neighbo s and m(u) < M < oo o all e ices u . Examples . I P is symme ic ; ha is, p(x,y)=p(y,x) o all e ices x and y, he walk is s ongly e e sible . Jus se m(u) - 1 and use he egula i y o P . A walk is iso opic i p(u, ) = 1/(q(u) + 1) when is a neighbo o u . Recall ha a egula walk is o de bounded, so i we se m(u) = 1 + q(u) we see ha iso opic walks a e s ongly e e sible . One says ha wo unc ions and g a e equi alen and w i e - g i whene e ei he is non-ze o so is he o he and he a ios o he absolu e alues a e bounded abo e and below by posi i e cons an s . Lemma 20 . I he walk P is s ongly e e sible ( egula and s ongly an- sien ) hen K T . - Ks ajad K S . -KT . P oo . Bo h esul s ollow om showing ha KS(u, ) - KT( , u) . Re e ing o (11) and (12) in Sec ion 4 we see ha bo h exp essions a e ze o i is no below u . I u = we a e asking i 1 - a(u, u- ) - 1 which ollows since 1 -« < 1 - a(u, u- ) < 1 . Thus we may assume ha is below u and ha u :~ _ Re e ing again o (11) and (12) and using Theo em 3 we wan o compa e (19) V(u, ) A( ) a( ,u) V( ,u) A(u) o (20)  a( , u) - a( , u - ) = ce( , u) (1 - a(u, u - )) . We no ed a he end o Sec ion 3 ha A(u) - 1 so we only need o show ha V(u, ) V( , u)  1 o all e ices u and . Le us ecall he de ini ions . Le w= {w0, w 1 ,. . . , w n be he geodesic ha connec s u = w o o =w n . Then (21) FACTORIZATION ON TIIE GREENS OPERATOR  203 V( ,u) 11_1p(w .i,w .i_1) . Conside wo special cases . I he walk is symme ic he exp ession in (21) is 1 and we a e done . I he walk is iso opic i is easy o check ha he exp ession in (21) is q(u) + 1 q( ) + 1 and since iso opic egula walks a e o de bounded we a e done . Mo e gene ally, i he walk is e e sible hen P( w .i-1,w .i) _- M(wi-1)p(w,i-I,wi) m(w .i) _- m(wj) p(wi,wi-1)  m(wi)p(wj,w,i-1) M(wi-1) M(wi-1) and so he p oduc in (21) is (22)  m(wi) = m(u) . ., 1 M(-j_1( ) j-1 since he walk is s ongly e e sible . This comple es he p oo . Theo em 21 . I P is s ongly e e sible ( egula and s ongly i ansien ) he he G een's ope a o is o weak ype (1,1), is bounded on ~P , 1 <p < oo, and sa is ies he conclusion o Lemma 13 . P oo .. S is bounded on P, 1 <_ p < oo, and sa is ies he conclusion o Lemma 13 . S* is o weak ype (1,1) and is bounded on QP, 1 < p < oo . Since KT - Ks ., T is o weak ype (1,1), and is bounded on 2P, 1 < p < oo . G = TSR . T, S, and R a e bounded on QP, 1 < p < oo so G is bounded on QP . Fixs>0andan E P . I {x : IG (x)I > s I < 21ISR 1I11 < Cll 111 s Now ake a ini e se o e ices, E, and le F be he subse desc ibed in Lemma 13 . Then IFI > z IEl and IIGXFII . -< IIRII .IIT(SXF)II .< IIRII .IISXFII . -<  ¡IR¡¡ . . 20 4  R . ROCIIBERG, M . TAIBLESON Theo em 22 . G= -0 -1 on U, 0 < p < oo . P oo .. Immedia e om Theo em 4, Theo em 5, Theo em 8, and Co ol- la y 12 . Co olla y . 23 . Unde he condi ions o Theo em 20, i 1 <p< oo G maps £P on o i sel . 6 . Fac o iza ion o he Laplacian and G een's ope a o -The disk ealiza ion . We p esen he e he modi ica ions needed o ca y o e he esul s o Sec ions 4and 5 . In his se ing he pa hs s a om a ini e posi ion, o ins ead o "coming om in ini y" . In his Sec ion we use he ollowing no a ion : Fo xE 9X, x = {o = xo, x1, xz, . . ., xk, . . . } . Thus, xk is always a e ex a dis ance k om o and i k >_ 1 hen (xk- 1 ) _ (xk) - . Fo k >_ 1 he q(xk) - q lowe neighbo s o xk a e deno ed xk +1 j, j  = 1, 2, . . . , q .  Fo x o we ha e q(o) -}- 1 - q -F 1 lowe neighbo s x1 j, j = 1, 2, . . . , q -I- 1 . We se andi n :~ 0 T (xn) = Fo he "backwa d di e ence" we se andi n ~ 0 k=0 0 (xn) =  1  (xn) -  a(xn>xn-1)  (xn-1) . 1 - «(xn,xn-1)  1 - a(xn,x"_1) I is easy o check ha T and A- a e in e se o each o he on he class o all unc ions on he ee . We see ha T is de ined by he ke nel KT whe e a(u, ),  i = o KT(u, ) =  a(u, ) - a(u, - ),  i u is below and 7~ o . 0,  o he wise S is de ined by he same o mula as in Sec ion 4 ; howe e , he alues u(I u ) a e de ined as in Sec ion 3 . Simila ly, T (xo) = (xo) (a(xn, xn-k) - CY(xn, X .-k-1)) ( xn - k) + a(xn, xo) ( x o) . A (xo) = (xo) A+ (xn ) -  7 (xn,xn+1,7) (xn+1,7) - (xn 7 Wi h A de ined as in Sec ion 3 we ind ha A+¿~,  (x) = R(x)O (x) . We ind again ha G= TSR and he esul s o Sec ions 4 and 5 ca y o e almos wi hou change . FACTORIZATION ON TIIE GREENS OPERATOR  20 5 7 . Examples . The simples example is an iso opic walk on a homogeneous ee o deg ee q + 1, q > 2, in he hal -plane ealiza ion . By symme y we see ha 7 (u - , u) - 7 = 1 /q o all e ices u . Simila ly, a - a(u, u - ) o all u . Using equa ion (4) we ha e 1 a =  +  q  a 2 q+1 q+1 om which i ollows ha a= 1/q . F om equa ion (5) we see ha A(u) = (q -1)/(q + 1) o all e ices u . We also see ha excep o a ac o o (q - 1)lq he ope a o s S and T a e ádjoin s . The G een's ke nel is gi en by q1 d(u, ) G(u, ) =  _l q+ 1 () which ollows by a i al applica ion o Theo em 11 . Since (GX{xo})(x) _ G(u, x o ) we see ha he G een's ope a o is no bounded on Ql . An example ha a ises om a g oup is a homogeneous ee wi h a symme ic aniso opic walk . Fo a speci ic example ake a g oup wi h iden i y e and h ee gene a o s a, b, and c ; wi h he ela ions a 2 =b2= c 2 = e . Le Y be he Cayley g aph o he g oup . I is a ee ha is homogeneous o o de h ee . Each e ex co esponds o a educed wo d and he h ee edges a each e ex co espond o he igh mul iplica ion by one o he h ee gene a o s . Assign o each edge p(g), g = a, b, o c, as he case may be, whe e p(a) -}- p(b) + p(c) = 1, and each o he p's is posi i e . Fo any such assignmen o p obabili ies he associa ed walk is egula , s ongly ansien , and s ongly e e sible . Le us now cons uc a simple example o which he G een's ope a o ails o be bounded on £P, 1 < p < oc . Take a ee ha is homogeneous o o de h ee in i s hal -plane ealiza ion . A each e ex he e is one neighbo di ec ly abo e i and wo di ec ly below . To he edge going up assign he p obabili y .4 and o he wo edges going down spli he di e ence and gi e each he p obabili y .3 . Wha makes his example wo k is ha 1/3 < 0 .4 < 1/2 . I he "up" p obabili y is g ea e han o equal o 1/2 he walk ails o be ansien and i i is less han 1/3 he G een's ope a o is bounded on h . The case whe e i is equal o 1/3 gi es he iso opic walk . Take he p obabili ies as we ga e hem . F om equa ion (4) we ha e he equa ion a= .4+ .6a 2 whe e a is he isi ing p obabili y associa ed wi h an upwa d ansi ion . F om his equa ion we see ha a is equal o 2/3 . The a gumen o Theo em 19 shows ha T, and so also G, is bounded on QP i p > log 3/ log(3/2) = p o , which is abou 2 .7 . Tes ing T agains he cha ac e is ic unc ion o a e ex we ind ha he esul is sha p and ha T ails o be bounded on QP i p < p o . A mo e ca e ul analysis shows ha T is o es ic ed weak ype (p o , p o ) . (See [SW ; p .197] o de ini ons .) 20 6  R . ROGIIBERG, M . TAIBLESON 8 . Commen s and ques ions . The p incipal boundedness esul s in his pape ollow om an isope ime ic inequali y exp essed in Lemma 13, which hen leads o he weak- ype es ima e s a ed in Theo em 14 . This ci cle o ideas is closely ela ed o he esul s ob ained by Ge l in [G], whe e he ob ains U boundedness esul s s a ing om ano he isope ime ic inequali y . While he lowe bounda y desc ibed in he p oo o Lemma 13 is no he same as he bounda y used by Ge l (and o he s in combina o ic g aph heo y) he e mus be Glose connec ions and hose connec ions should be s udied . The no ion o a lowe bounda y was in oduced in [RT] in a di e en se ing . The possibili y ha i could lead o boundedness esul s o he G een's ope a o ca ne o mind whenwe lea ned o Ge l's wo k in [G] . In [RT] Lemma 13 in he se ing o he dyadic ma ingale was used o com- pu e he K- unc ional o he spaces CMd, he space o disc e e Ca leson Mea- su es, and Po, he space o unc ions wi h ini e suppo . In his special case he dyadic ma ingale is he bounda y ma ingale o an iso opic andom walk on a ee ha is homogeneous o deg ee h ee ; he nodes o he g aph can be iewed as he index se o he dyadic in e als o R . Lemma 13 allows he ex ension o he in e pola ion esul s o he bounda y ma ingale o any s ongly an- sien andom walk on a non-homogeneous ee . In his mo e gene al se ing he nodes o he ee ep esen in e als in a nes ed sys em o in e als mo e gene al han he dyadic sys em . In Theo em 19 we ind a p o such ha he G een's ope a o is bounded on QP i p > po .  This p o depends on q + 1, he maximum o de o a e ex, and á, he "maximum isi ing p obabili y", á = sup a(u, u- ) .  Fo iso opic andom walks on a homogeneous ee o o de q + 1, qand a a e cons an s, a = 1lq, and his leads o p o = 1 . Fo iso opic andom walks on an o de bounded ee weknow ha p o = 1 bu he a gumen o Theo em 19 leads o a g óss o e es ima e o po . In [L] Lyons uses he no ion o an a e age b anching numbe o a andom walk on a ee whe e he b anching numbe a a e ex u is q(u) in ou no a ion . Lyons shows in a a ie y o p oblems ha his a e age b anching numbe beha es like q when q is a cons an . Lyons' esul s sugges ha he e migh be an "a e age is ing p obabili y" as well as an a e age b anching numbe and ha he in imum o hese alues o p o which he G een's ope a o is bounded on U could be compu ed om hese a e ages . Conside now he si ua ion when he andom walk is s ongly ansien and s ongly e e sible . In his si ua ion he G een's ope a o maps QP on o £P i 1 < p < oo and i is ne e bounded on £ 1 . Fo i i we e bounded on P hen T would be bounded on Q1, which implies ha S is bounded on £°° . Bu S is ne e bounded on Q°° as we saw in ou ema ks ollowing he p oo o Lemma 6 . This aises he p oblem o desc ibing he class o in eg able unc ions, , on he ee such ha G is also in eg able . We can de ine a kind o Ha dy space, H = { : E £ 1 , G E Q 1 }, li IIH = ji 111 - - IiG 111 . I is easy o see ha his no m is equi alen o an "a omic no m" in he sense ha he e a e unc ions called a o as and E H i and only i = 1 : Akak(x) whe e he ak a e a oms FACTORIZATION ON THE GREENS OPERATOR  20 7 and 1 : JAki < oo whe e ji IIH - in 1 : ¡Akl o e all such ep esen a ions . This is no e y sa is ac o y since he de ini ion o "a om" is es ic i e (an a om being he Laplacian o a poin mass) . One would wan a less es ic i e de ini ion o an a om as well as a maximal unc ion cha ac e iza ion be o e one would ha e a sa is ying heo y . This line o hough also sugges s ha he e should be a BMO heo y . We only ema k ha he ee as a measu e space wi h disc e e measu e and he na u al me ic induced by geodesic dis ance is no a space o homogeneous ype, in he sense o Coi man and Weiss [CW] . Fo spaces o homogeneous ype Coi man and Weiss cons uc ed a Ha dy space heo y . Re e ences [C] P . CARTIER, Fonc ions ha monique su un a b e, Symp . Ma h . 9 (1972), 203-270 . [CW] R .R . COIFMAN AND G . WEISS, Ex ensions o Ha dy spaces and hei use in Analysis, Bull . Ame . Ma h . Soc . 83 (1977), 569-645 . [G] P . GERL, Random walks on g aphs wi h a s ong isope ime ic inequali y, J . Theo e ical P ob . 1 (1988), 171-188 . [KPT] A . KORÁNYI, M . PICARDELLO, AND M . TAIBLESON, Ha dy spaces on non-homogeneous ees, Symp . Ma h . 29 (1988), 205-265 . [L] R . LYONS, Random walks and pe cola ion on ees, Ann . P ob . ( o ap- pea ) . [RT] R . ROCHBERG AND M . TAIBLESON, An a e aging ope a o on a ee, in "Ha monic Analysis and Pa ial Di e en ial Equa ions," Lec u es No es in Ma h . 1384, Sp inge Ve lag, 1989, pp . 207-213 . [RW] R . RoCHBERG AND G . WEISS, De i a i es o analy ic amilies o Ba- nach spaces, Annals Ma h . 118 (1983), 315-347 . [SW] E .M . STEIN AND G . WEISS, "In oduc ion o Fou ie Analysis on Euclidean Spaces," P ince on Uni . P ess, 1971 . Depa men o Ma hema ics Campus Box 1146 Washing on Uni e si y Sain Louis, Missou i 63130 U .S .A . @da id .wus l .ed u