Remarks on Kato's square-root problem

Publicacions Ma emá iques, Vol 35 (1991), 299-321 . REMARKS ON KATO'S SQUARE-ROOT PROBLEM JEAN-LIN JOURNÉ 0 . In oduc ion Le T be a maximal acc e i e ope a o on a Hilbe space I - l, wi h domain V . The e is a well-de ined squa e- oo o T, T'I' which sa is ies he equa ion T 1 / 2 U = 1 A -1/2 (T - i - A) -1 Tu dñ, o o all u in V[1] . When T is sec o ial, Ka o conside s he p ehilbe ian s uc u e on V de ined by Suppo ed by he N .S .F . <u, >y = <Tu, >+<u,T >+<u, >, whe e < . , . > deno es he scala p oduc on 'H . Ka o conjec u ed ha he domain o£ T1/2 was he comple ion o V in 'H o he p ehilbe ian s uc u e de ined by < . , . > y . This conjec u e was disp o ed by A . McIn osh [2] . He obse ed ha he ailu e o his conjec u e was connec ed wi h he ailu e o he inequali y 11 I AI B - BI AI 11  <  C  11 AB - BA li, o gene al sel -adjoin ope a o s, ¡Al being (A')'/' . Since he boundedness o he i s commu a o o Calde ón [3] is a ( ue) special case o he abo e ( alse) gene al inequali y, McIn osh sugges ed ha Ka o's conjec u e migh be ue when T is a di e en ial ope a o on R' o he o m di AV, whe e A is a ma ix- alued unc ion such ha , o some S > 0 and o all x in R" and 1 in Cn, Re < 1, A(x)j >  >  $ 1112 . I is his special case which is now known as Ka o's conjec u e o , mo e p ecisely, Ka o's squa e- oo p oblem . The connec ion suspec ed by McIn osh u ned ou o be qui e signi ican and led simul aneously o he solu ion o Ka o's conjec u e in dimension 1 and o he p oo o he L 2 -boundedness o£ he Cauchy-ke nel on Lipschi z g aphs [4], conjec u ed by Calde ón and p o ed by himsel in he case o£ small Lipschi z cons an s [5] . 300  J .-L . JOURNÉ A simple escaling on A shows ha i is enough o conside he case whe e ~~ A-I Iloo < 1 . In dimension la ge han 1, Ka o's conjec u e has been sol ed when A is a small pe u ba ion o he iden i y, ha is i 11 A-I jj '> " < e n , whe e e n depends only on he dimension n and decays exponen ially wi h n [6],[7] . A na u al way o a ack Ka o's conjec u e is he e o e o ind bes possible lowe bounds o e n . He e we ob ain explici lowe bounds o e n , decaying like n -1 / 2 . In pa icula , i n < 5, one can ake e n o be 1/3 . Also we answe a ques ion o A . McIn osh conce ning he solu ion o Ka o's conjec u e in dimension 1 . Rescaling A so ha , o all x in Rn, Re A(x) > I, one may assume ha ¡¡A -1 -Ijj,, < 1 . In dimension 1, he solu ion is ob ained by expanding an ope a o depending on A in a se ies o ope a o s depending mul ilinea ly on he new a iable B = I - A -1 , hen by showing ha hese ope a o s a e bounded on L 2 and ha he no ms can be summed when 11 B ~j '> " < 1 . In dimension la ge han 1, a simila p ocedu e does no seem o b ing he same kind o simpli ica ion and hence, i was sugges ed by A . McIn osh, as a i s s ep o unde s and he highe -dimensional case, o ep o e Ka o's conjec u e in dimension 1 wi hou using he a iable B . This is wha we do a he end o his pape . The con en o he sec ions o his pape a e as ollows . In he i s we in oduce some no a ions and ecall some o mulas . In he second we ecall some basic ac s abou Ca leson-measu es . In he hi d, we s a e a heo em abou mul i-linea ope a o s, which eadily implies he imp o emen on he o de o magni ude o e n , and whose p oo we ou line . In he ou h and i h we p o e some echnical es ima es on ke nels o ope a o s and in he six h we conclude he p oo o he a o emen ioned heo em . In he se en h we show how o use a ious p ope ies o he ope a o s a ising in Ka o's p oblem o imp o e he lowe bound o e, gi en by ou heo em . In he eigh h we gi e a di ec p oo o Ka o's conjec u e in dimension 1 . 1 . No a ions andFo mulas The ollowing is pa ly bo owed om [7] . I we make he change o a iable A = -2 , we see ha he o mula gi ing T'I' can be ew i en as T'/ 2 =2  T  d . 0 2 T + I Le Dj _- i jai , D = V and D* = (D I , . . ., D n ) . We deno e by {A} he ope a o o poin wise mul iplica ion by he ma ix A . Then T can be w i en as D*{A}D . Le U = A- I . Then T = D*{U}D +,n, . Le P = ( 2 A+ I)-1 . Then ( 2 T + I) -1 can be w i en as j :(-1)'P, [ 2D*{U}DP ]~, j ;1 0 as long as 11 U jj,,,< 1 . And in his case, Le Q = D * P and R be he ma ix o ope a o s whose en ies a e Dó' . Obse e ha 2 DD*P = R(I - P ) . The e o e he Sobole space W1,2 will be in he domain o T 1 / 2 i he se ies o ope a o s Q * [{U}R(I - P )]j d j>- o is con e gen in L 2 -ope a o no m . Tha each summand is bounded wi h a no m domina ed by C,j 11 U j1i is known [6],[7] . Obse e ha i A is bounded and s ic ly acc e i e, 11 AA- I 11,,~< 1, o small enough . Hence one can always educe o he case whe e 11 U 11 < ,~< 1 . The e o e Ka o's conjec u e will be sol ed i one can show ha , o all e > 0, each summand is bounded wi h a no m a mos C,,(1 + e)j when 11 U jj" .< 1 . One can also do a powe se ies expansion in he a iable V= I - A-1 . S a ing om we ob ain KATO'S SQUARE-ROOT PROBLEM  30 1 - =L . .(-1)jp [ 2 D{U}D*P ] j D*{A}D = j :(-1)jD*P [ 2 {U}DD*P ] {A}D . j_>0 ( 2 D*{A}D + I) D* = D*{A}(I + 2 DD*) - D*{A} (I - {A}-1) (2)  T D * R [{V }R ]' d j30 0  = D*{A} (I - {V}(I + 2 DD*) -1 )(I + 2DD*), ( 2 D*{A}D + I) -1 D*{A} = D*(I + 2 DD*) -1 (I - {V}(I + 2 DD *) - 1 ) . Le R = (I + 2 DD*) -1 . Then i 11 V li~~< 1, one ob ains ha W1,2 will be he domain o T 1/2 i he se ies o ope a o s is con e gen . I is easy o check ha D*R = Qi and ha R = I -R+RP, . Hence he e is a e y close esemblance be ween he wo expansions (1) and (2) . In dimension 1 howe e , R = I and R = P , so ha he expansion in 302  J .-L . JOURNÉ he a iable V is much easie o handle han in he expansion in he a iable U . By mul iplying A by a la ge numbe so ha Re A > I, one can educe o he case whe e II V II<, .< 1 . The e o e one can y o p o e Ka o's conjec u e by showing ha o all e > 0, he summands in he se ies (2) a e domina ed by C E (1 + e)j II V IIi in ope a o no m . This is how i was done o iginally in dimension 1 [4] . In highe dimension i is no clea ha one o he se ies (1) o (2) has an ad an age o e he o he . One can ask in pa icula i heie is a simple ela ion be ween hei adii o con e gen e . In dimension 1, i is easy o see a p io i ha hey a e he same . Indeed, a unc ion a is such ha II a -1 II .< , Ao < 1, i and onl y i II a-1 -1/(1- ió) II<, ,< A0/(1- Aó) . The e o e i Ka o's conjec u e is ue when II a - 1 II~< o < 1, i is also ue when II (1 - A2 )a` - 1 II,,,, A o , and also when II a-1 - 1 II,< A o a e escaling . I is a li le su p ising, bu easy o see, ha his emains ue in highe dimension . I is a consequence o he ollowing ac . I one pu s on GL n (C) he dis an e induced by he ope a o -no m on Cn, d(A, B)  e,C uéu=1 II Uj - V I II, hen in e sion does no in gene al map halls o o he balls . Howe e i does map a ball cen e ed a ound a mul iple o he iden i y o a ball o he same na u e, jus as in C* . This is wha he nex lemma exp esses . Lemma 1 . Le 0 < Ao < 1 and A in GL,(C) . Then IIA-III<ao ~ II(1-Aó)A - '-III<ao . P oo . . Le w be a uni ec o in C' . We wan o show II (1- aó)A - 'w -w II ,< A o . Making he change o a iables = A-1w, i is enough o show ha II ( 1 - Aó) - A jj,< ao 11 A ll Bu and A a e jus wo ec o s x and y such ha II y - x II< - o II x II . The e o e we can check his inequali y in C le ing x = 1 and y = 1 + z wi h IzJ < A o . A e di ision by Iyi(1 - ió), his is equi alen o I(1 + z) -' - (1 - ~`ó)-'I  >,0(j - ió)-' This, in u n, ollows om he ac ha he in e sion in C* maps B(1, A o ) o B((1 - Aó)-1, A 0 (1 - Aó)-1) . This p o es Lemma 1 . Since he in e sion is an in olu ion on GL  (C), i ollows ha a ball cen e ed a ound a mul iple o he iden i y is mapped exac ly on o a simila ball . F om he p e ious lemma we see ha , i A is bounded and ReA> S > 0, min II AA- I II,, .= min II AA - 1- I II  . a>o  a>o KATO's SQUARE-ROOT PROBLEM  30 3 I ollows ha i he se ies (1) con e ges when 11 U II, > , > < A o < 1, o some Ao > 0, hen W l,z is he domain o T 1 1 2 when mine,>o II AA- I II,,~< Ao, and he e o e when mina > o II AA -1 - I II,, .< Ao . So he se ies (2) mus con e ge when 11 V II < , .< A o . And con e sely . Ha ing obse ed ha he adii o con e gence o (1) and (2) a e he same, we would like o eco e his ac di ec ly om he s udy o he mul i-linea e ms o (1) and (2), and o show ha his adius is 1 . As we shall see, he me hod we use gi es wo dis inc alues o he adii o con e gence o (1) and (2), and his shows ha i is no op imal . 2 . P elimina ies on Ca leson Measu es Le p deno e he ope a o o con olu ion wi h he Poisson-ke nel . Then a Ca leson-measu e M on he hal -space R++ 1 i s a measu e o which one has he es ima e (3)  IR-+1 Ipj(X)1 2 d1,(X, ) '< C(,1) II II2 A necessa y and su ñcien condi ion o a measu e p o ha e his p ope y is he exis ence o a cons an C > 0 such ha o all cubes Q in R n, (4)  11(Q x [0, si) S Cm, whe e IQI and 6 deno e he Lebesgue measu e and he side-leng h o Q . Le us deno e by c P he bes cons an in (4) . Then, o some absolu e cons an C, C(p) <, Cc, . Also, i we eplace he Poisson-app oxima ion by some o he app oxima ion o he iden i y (p ) >o sa is ying app op ia e es ima es, hen he bes cons an in (3) will p esumably change . Howe e , i one looks a he size o he di e ence % JR++1 (pj - pj)(,)jzdp(,, ) we see ha i depends only on a cons an h,, which we de ine o be he bes cons an in he inequali y 1,(Q x [6/2,61) <, CIQ1 . O cou se h,, < c, . The poin is ha in gene al, and in wo king on Ka o's p oblem in pa icula , one ies o es ima e c P o measu es o which one al eady has a good con ol o h,, . Lemma 2 . Le p and p be as aboye . Then, o some C > 0, J Rn+ ,I(p -p )(x)1 2 dp(x, ) < Ch,, 11 II2 30 4  J .-L . JOURNÉ P oo : Obse e ha he ope a o s Tk de ined o k E Z by < 9, Tk >=  [(P - P )(x)] [(p,9 - P 9)(x)] dp(x, ), R .^ X 12k ,2k+11 sa is y he assump ions o he Co la -S ein Lemma wi h a cons an depending only on h,, . This p o es Lemma 2 . F om Lemma 2 we see ha 2 1/2 [Jx .^+1 P (x)1 d/ (x, )1  [Cocwl2+C1h1,l2] 11 112, whe e Co is independen o he app oxima ion o he iden i y bu C1, o cou se, is no . Lemma 3 . The cons an C o can be chosen equal lo 2 in all dimensions . P oo . . By Lemma 2 we know ha we can choose P as we like o es ima e he bes Co . By he same a gumen , we can eplace P" (x) by S (x) = mQ( x) (x), whe e Q( , x) is he dyadic cube con aining x o size 2 k , wi h 2k-1 < < 2 k . I is easy o see ha IS (x)l'dp(x, )~ 1/2 < cN' IIm>ó IS m0,12 and i is a classical ma ingale-inequali y ha l maxiS (x)j  <, 2ll ll2 . >o  z Tol e he wi h Le n na 2 applied wi h S ins ead o p , hese wo inequali ies eadily imply Le n na 3 . A ob ious bu usc ul ema k is ha , in he inequali y /2 [JR^+i ll íe (x)ll ' dl (x, )~  1< ( 2 cl~ /2 +C1hl , / 2 )ll ll2 . we can eplace dIÁ(x, ) by dp(x, u ), whe e 0 < u < 1, and eplace P by p,,, o so ne w> 1 . KATOS SQUARE-ROOT PROBLEM  30 5 3 . A Mul ilinea Es ima e Le (Ai)iEN be a amily o ma ix- alued unc ions sa is ying IIAjii , ,~ S 1 . Le (Kj)iEN be a sequence o con olu ion ope a o s mapping C d - alued unc- ions o C 1 - alued unc ions . We assume ha he symbols u(Ki), which a e ma ix- alued, a e o he o m ((Sk, ( )/IIIZ)) whe e he Skj's a e homogeneous polynomials o deg ee 2 . Le (MiEN be a sequence o numbe s . Fo > 0and i E N we de ine he ope a o Ki = AiI - - Ki(I - P ) . We assume ha he symbols o he K¡, 's and he K¡'s, which a e ma ix- alued, a e con ac ions on Cd, o all 1 E R d . Theo em 1 . Fo all e > 0 he e exisis C E > 0 such ¡ha¡, o all F E LC, (R d ), [Ja~112 d / 1IQj{Aj}Ki, . . . K ._l, {An}xn,iF 2-J1 2 < CE(1+2~+ =)nJIFI12 . The p oo which we shall ou line ollows as usual om Ca leson-measu es es ima es . The imp o emen o e [6] and [7J comes om Lemma 3 and om he ac ha o he Ca leson-measu es ha en e in o he p oo , i is e y easy o ob ain agood con ol o h,, . I (Fí) >o is a amily o ec o - alued o scala - alued L 2 - unc ions we shall de ine I I IF 111 by IIIF III = lJo[100 IIF ll z-l 1/ 2 . Ske ch o P oo . Fi s we educe (9) o a simila es ima e whe e K,, is eplaced by P . To do his we domina e by IIIQ {Aj}K,, IIIQ {A1}Ki, . . . K ._i, {A .}P Á' .FIII . . . K  _l, {An}K ., Flll +  IIIQ {A~ }h1, ... Kn_1, {An}(~nI + hn)FIII Fo all n > 0, X n , Y n and i n deno e he sup o hese 3 quad a ic exp essions when IIFII2 < 1 . Since {An}(AnI -}- K n ) is a con ac ion, Y,, < X,,_1 . Hence (10)  X n 1< Y  + X ._1 So i Y,, g ows a mos like (1 + 2 -}- e)n o all s > 0, hen so does X n . 30 6  J .-L . JOURNÉ To es ima e IIIQ {A1}K1, . . . K,l, {A n }P FIII he classical hing o do is o decompose he exp ession in he no m as a sum o he ype {Q {A1}K1, . . . {An_1}Kn_1, An}P F+ e o e m, whe e he e o is o he o m L F wi h L l = 0 . The main e m is es ima ed using a Ca leson-measu e es ima e which i sel ollows om an L 2 -es ima e a he o de n - 1 . As we shall now see, his p ocedu e can be imp o ed o yield be e cons an s . Le (w ) >o be some adial smoo h app oxima ion o he iden i y, such ha w 1 is non-nega i e and suppo ed in he uni ball . Now le = w *w . Obse e ha con olu ion wi h w o is a con ac ion on L 2 o L°° . Now we domina e and IIIQ { A l}Ki, . . . Kn_1, {An}P FIII by he sum o he h ee ollowing e ms ( 11 )  IIIQ {Al}Ki, . . . Kn_1, {An}(P1 - 2n )FIII, (12) 111 Q {A1}Kl, . . . K n _ 1 , {A n } 2n F- {Q {A1}K1, . . . K n_ 1 , ¡ A n } 2n F1  , (13)  111 {Q {A1}K,,, ... Kn_1, An} 2n Fl The, i s e m is less han III(P - 2  )FIII, which is domina ed by n II1(Pi - !)FIII + , III( 2k - zk-l ) lll k=1 and llena : by C(1 + n)IIFII2 . The second e co !s also o he YI>c IIIL FIII wi h L l = 0 o all > 0, and can be es i na ed di ec ly wi hou using induc ion on n . The co esponding es ima e g ows slowe han exponen ially . To es i na e (13) one has o es ima e e ! , and h ! , o he Ca leson-measu e The cons an h , can be es a na ed wi hou induc ion and g ows slowe han exponen ially . To es i na e c ! , we choose a cabe Q o side-leng h 6 and we wan o es ima e jQ /2^ {A}K1, /2 n . . . K n _ 1 , /2n An(x) 2 d dx I .EQ, í KATO'S SQUARE-ROOT PROBLEM  30 7 We shall see ha because o he ac o 2n, his is essen ially domina ed by an exp ession o he o m up o e o e ms, wi h e n 's such ha 11° ° 1( 1 +E  ) < oo . O cou se, he e he no m o A n has o be aken in he Hilbe -Schmid sense and his is wha in oduces a ac o -,íd - . In conclusion one ob ains an inequali y o he o m o lix - y¡¡ > . X n-1II An 11i 2 ((1+e,)Q) , Y n <2X n _ 1 ((1 + En) -líd) + e o e ms . Combining his wi h (10), one ob ains (14)  X n <, (1 + 2V d)Xn_1 + e o e ms, which implies he heo em, modulo app op ia e con ol o he e o e ms . 4 . Technical P elimina ies An ing edien in mos subsequen es ima es is he ollowing . Lemma 4 .  The ke nels o he ope a o s Ki, sa is y ¡he inequali y llhi, (x - y)¡¡ This lemma ollows easily om he asymp o ic p ope ies o he Fou ie ans- o m o 1/(1 + 2), which decays exponen ially as well as i s de i a i es . We omi he de ails .. La e in he p oo s, we shall no need he ull o ce o he exponen ial ac o . A polynomial ac o o su icien ly high deg ee, depending on he dimension, would be enough . Such decay, howe e , can only come om su icien smoo h- ness o he symbol o K¡, which is i + ( 2111 112 /1 + 2ll1ll2 ) x Requi ing ha he symbol o K ; be egula enough, and in a scale-in a ian way, o ces 110 2 u(Ki) which is homogeneous o deg ee 2, o be a polynomial . This jus i ies ou assump ion on a(Kj) a leas in la ge dimension . A consequence o Lemma 4 is he ollowing . Lemma 5 . Le¡ and g be wo L 2 - unc ions such ha d (supp( ), supp(g)) = b > 0 . Then i < b/n, (15)  <g,K1 {A1} ... {An_1}Kn >  Cn11 J12119112e 2n 31 4 Theo em 2 . The adius o con e gence o ¡he Ka o unc ional is a leas a -1 , whe e a is ¡he la ges posi i e oo o ¡he equa ion (31) Be o e ske ching he p oo o his heo em, le us indica e ha app oxima e nume ical alues o a-1 in dimensions 2, 3, 4, and 5 a e espec i ely .474, .416, .376, and .347 . Ske ch o p oo . . Fo simplici y we shall p oceed as i he ke nels o (I - P )R and Q we e suppo ed in {Ilx - y¡¡ < } . This o cou se is no ue, bu , as he p oo o Lemma 7 shows, his is ue o all p ac ical pu poses . The ac ha R de ines p o emen o (10) : (32) To see his we jus need o obse e ha o a unc ion F in L z, JIP112 - - 11(I - R)F112 = JIF112 . Hence X n < maxaz+ , z= 1 AX n _1 -}- MY,,, which is exac ly (32) . By he posi i i y o R we can w i e i as ~, whe e S is a con ac ion . Since K = I - (I - P )R = I - (I - P ) + S we can ew i e i as I ollows ha 1  (33)  Z n < 2 (1 (1+C2)(Zn-1 + YQXn_1), + negligible e o e ms . No e ha , by (32), (34) Combining (30) and (34) we ob ain J .-L . JOURNÉ CX-21 Xz-1=XId . an o hogonal p ojec ion yie1ds he ollowing ¡ni- X n <  Yaz +Xn_ 1 . ZP +2(I-(I-P,)S) . n Xn < C E Y2 k=1 1  n-1 (35)  Z n < 2 Zn-1 +  E Z~  + negligible e o e ms . j=1 1) Le us igno e he e o e ms in (35) . Le C > 0 and /a > 1 be such ha o 1. < n, Zj < CQj . Then Z n < CQn i 9 is such ha a n i 1 p n-1 + ~ld-  pn-1 , 2  1_ KATOS SQUARE-ROOT PROBLEM  31 5 ha is, i i > a . This implies Theo em 2 modulo he handling o he e o e ms, which is i ial . Le us men ion ha in wo king wi h he se ies (1) ins ead o (2) one does no seem o be able o imp o e (10) in o (34) . Hence one ob ains signi ican ly wo se es ima es o X n o (1) han o (2), while, as he nex sec ion will sugges , one should expec a disc epancy g owing slowe han exponen ially . 8 . A di ec p oo o Ka o's conjec u e in dimension 1 We wish o p o e he ollowing : Theo em 3 . Fo all E > 0 he e exis s a con,s an C E such ha o all n  1, and al ELe (R), 1 < i < n, and E LZ(R), (39) Fo all u E]0,1], we shall p o e (36)  QL  ({aá}(I-PL))  < CE(1+e)n  ~laj1j .JI II2 . The p oo we shall gi e clea ly yields mul ilinea es ima es . Howe e we shall wo k wi h one single bounded unc ion a o no m 1, and we shall deno e {a} by a . This symbol will s and o {a}) o any powe j, o ins an e in (43) and (44) . P oo ..  The p oo o Theo em 3 elies on a i ial ex ension o one iden i y o [4], namely : (37)  Q aP = P {P a}Q + {Q a}P - Q {Q a}Q . Le S i be such ha P S L = P a o some a > 0 . Then Q S = DP S, = DP « =ó Qa This, oge he wi h (37) and a escaling immedia ely gi es : Lemma 9 . Fo all u > 0 and a > 0, (38)  Qu aPa = u Pu {Pu a}Qa + {Quia}Pa - IXQu {Qu a}Qa " 11 IQu (a(I - P »n aP I < C(1 + E)n ~I IJ2 . This clea ly implies (36) . The eason o in oducing he pa ame e u will become appa en du ing he p oo . Le p be an in ege possibly equal o l . We w i e n = qp+ = q+(p - 1)q+ = q+s, wi h < p . Le a > 1 and j = a9 . Finally, o -y > 1, we se Qé= P -Py . 31 6  J .-L . JOURNÉ We wish o educe he s udy o 111 Q (a(I - P »n aP 111 o he s udy o 8+1 (40) I I Qu a(I - P« )a(I - Pa, )a . . . a(I -P . q - ) (a(I - Pp ))  aPp  , whe e we shall be able o ake ad an age o he ac o  in Lemma 9 . To do his we eplace each P by he co esponding P- y , whe e -y is some app op ia e powe o a . S a ing his p ocess om he igh and using Minkowsky's inequali y we domina e he le hand side o (39) by he sum o (40) and he ollowing exp essions . (41) (42) Qu (a(I - P )) n aQQ Qu (a(I - p,» n - 1 a&aPp ¡Qu (a(I - P )) n-2 aOAa(I - Pp )aPp Qu (a(I - P )) -1 aoA (a(I - Pp )) 9 aPp Qu (a(I - P )) 9-2 aQ 9-1 (a(I - pR »s+1aPp II s+1 Qu aQ a(I - Pa2 )a . . a(I - P,,,,-l ) (a(I - PP ) / aPp To s udy (40) we expand each (I - Py ) and eg oup he esul ing 2n e ms aeco ding o he loca ion o he i s P y which appea s when going om he le . A new applica ion o Minkowski's inequali y hen shows ha (40) is less han he sum o he ollowing exp essions : (43) (44) IQu aP« a(I - P,,2,)a . . a(I - P a ,~,,) (a(I s+i Qu aP .,, , (a(I - Pp ) / aPp ~ Qu aPp (a(I - Pp )l 9 aPp 111 111 Q« aPp aPp Q- aPp ] 11 KATO'S SQUARE-ROOT PROBLEM  31 7 An applica ion o Lemma 9 pe mi s o decompose each o hese exp essions in he sum o h ee e ms, he wo las o which can be handled ia he usual Ca leson-measu e a gumen , hanks o he ollowing lemma . Lemma 10 . Leí 0 < wo . . . < w, Then he ke nel  I' (x, y) o P,,,oa(I-P .,)a(I-P,) ... (I-P,,,  _1)aP .  is domina ed by C (w  /wo)i12 (l+n)3 (_ . )1,+,,,,1,2 . ~ , zThe same is ue i P,, o is eplaced by Q,,, o . We de e he p oo o his lemma un il he end o he sec ion . We in oduce some no a ions o bes cons an s in quad a ic es ima es, o which we shall ob ain es ima es by induc ion . These cons an s depend on n which, o he ime being, is ixed . Recall ha u E]0,1] . The i s cons an DQ s is he bes cons an in he inequali y ( 45 )  ( 40 ) - C11 11 2 . (46)  1I Qu (a(I - P ) I k aP 111 <, Cli 112 . Pp » s+laPp The cons an E', independen o a, is he bes cons an in he inequali y We wan o de i e an inequali y o DQ s . As we al eady obse ed, (40) is less han he sum o (q+s+1) quad a ic gauan i ies, (q-1) o {43) and (s+2) o 31 8  J .-L . JOURNÉ (44) . Each o hese quan i ies can be domina ed by h ee o he s using Lemma 9 . I will ollow ha D9 , is domina ed by he sum o 3(q+s--1) numbe s . Co esponding o he i s e m in Lemma 9 we ha e he ollowing (q+s+1) numbe s : ~Dq-1,s, - D q -2,s, ... , aq 1 D i ,, -E s , QE s -1, . . . ,  Eo, ~CO . a 2  - He e c o deno es he bes cons an in he inequali y IIIQ , 111 < CII . ll2- Those numbe s a e ob ained by escaling and using he ac ha he P 's a e con ac ions on L 2 and L°° . The wo las e ms in Lemma 9 can be eg ouped in a single one . Lemma 10 and he usual Ca leson-measu e a gumen gi e a global es ima e in C 1 / 2 (1 -}- n) 4 . In doing his, one uses ha he measu e IQ u a1 2 dxd / is a Ca leson-measu e uni o mly in u . The p e ious ema ks yield he ollowing q-1  s ( 47 )  D ,s <  .Z Dq-j,s + u co + zs  E m + Caq/ 2 (1 + ,n)4 . j-1 a  p  l "'=o The s a egy is o use (47) o show ha i (48)  ER, < C1(1 + E)' o some e > 0 and uni o mly o m E N and u E]0,1], hen o some e' < E, (49)  Du ,s < C2 (q, s)( 1 + E')q(1 + e)9 whe e C2(q, s) has he o m C(q + s + 1)4 and is independen o u E]0,1], This will equi e an app op ia e choice o a and p . Then one shall show ha o some e" < e, C > 0, and uni o mly in u E]0,1], (50)  E uu m < C(1 + e")"` . By i e a ing his p ocedu e one can malee e as small as we wan , hus p o ing he heo em . Le us be mo e p ecise . Le e > 0 and choose p such ha (51)  21/2 +1  <  1 + e < 21/2p-2 . Then we cla,im ha i (48) holds, hen (49) holds o any (52) 1/3 e' > ((1 +E)2 -2/  -1 . No ice ha , by (51), (1+E2  1/3 p _)  - 1 <e . Also we claim ha in (50) we can choose any e" such ha (53) (56) KATO'S SQUARE-ROOT PROBLEIII  319 e" > (1 + e) 1-1 /p(1 -I- e ')1/p - 1 . Since 2 1 / 2 p+ 1 i s he only ixed poin o he ans o ma ion x ___> (  2  )1/3 x2p-2 (55)  a(1 +e') > 2 . Obse e ha , by (55), j :i~, l (a(1 + E')) inc easing one jus needs i ollows by i e a ion ha we can make e as close as 21/2p+1 - 1 as we wan . Then, by inc easing he alue o p we can ge e as close o 0 as we wan . Hence, o inish he p oo o he heo em, we jus need o p o e wo claims and Lemma 10 . P oo o he i s claim : We assume (48) and (51), and we wan o conclude (49) o any e' such ha (54)  1+e' > ( (1+e) 2 p -2 ) Le a = (1 + e) 2 p -2 (1 + E') 2 . No ice ha , by (54), In o de o deduce (49) om (48), we shall use an induc ion on q based on (47) . Then i will be su icien ha C 2 be so ha , o all 0 < k' < q, k'-1 C2 s k ~ -j) i C2 (s,k')(1+E)9(1+S')k , E  ( k .  (1+E)'(1+e')k- j-1 +Ca lo / 2 (s + k' + 1)4 + Cl s k, 1 (1 + e)' . < 1 . Hence, since C2(s, .) will be (57)  C2 (s, k ' )( 1 + e)" (1 + e')k' > C 3 ce k '/ 2 (s -1- k' + 1)4 -F- s á k , 1 (1 + e)9, whe e C3 is some cons an which emains bounded i eand e' s ay away om hei minimum alues and Cl emains bounded . Le us se C2(s, k') = 2C3(S+ 32 0  J .-L . JOURNÉ k' + 1)a and check ha wi h his choice (57) is sa is ied . Equi alen ly we need ha o all 0 < k' < q, (5g)  2(s + k' + 1) 4 (1 + e),( 1 + e l ) k ' > a k '/ 2 (s + k' + 1)4 + s a , 1 (1 + e)9 . In (58), he second e m o he igh hand side is ob iously less han hal o he le hand side . To see ha his is he same o he i s e m, we i s obse e han when k' = q i ollows om a 9/2 = (1+e)(p-1)Q(l+e')a ' (1+e),(1+e')9 . To deduce i o smalle alues o k', jus obse e ha a l/2 > (1 + e') . Hence, (58) is p o ed, om which ollow (57) and (56) . Hence (49) ollows om (48) by induc ion, using (56) and (47) . P oo o he second claim : Wi h ou choice o C 2 , (49) eads as ollows : (59)  D9 , , < C(q + s + 1) 4 (1 + e) , ( 1 + e')9 . We wan o deduce (50) om (48) and (59) , assuming ha e" sa is ies (53) . The decomposi ion o he le hand side o (39) in (41)+(42)+(40) implies ha Eñ is domina ed by he sum o (s + 2) + (q - 1) + 1 e ms . To handle he (s + 2) i s ones, we decompose Qá  P as - 2Pu Qu L . By (48), his gi es a con ibu ion o c o logp o he i s one and Cl log /l(1 + e»,  0 j <, s, o he (s + 1) ollowing . By (59) one has con ibu ions o he o m C(q - j + s + 1)4 j log a (1 + e)9 (1 + e') 9- j, 1 < j < q - 1 o he e ms cons i u ing (42) and inally C2 (q, s) (1 + e)' (1 + e')Q o (40) . Summing hese es ima es, and hen using (53) and he ac ha (q + 1)(p - 1) >, s we ob ain Eñ '< C(q + s + 1 ) 6 ( 1 + e)'(1 + e')9 < C E ,,(1 + e1L)n . This p o es he second claim . P oo o Lemma 10 : An ob ious es ima e is l(x, Y),<¡¡ PWO (x - -)¡1211PW  L( . - Y)¡¡ 2 = C1-1(WOWn)-1/2 . This is su icien when Ix - y¡ < C(1 + n) 3/2 Wn since in his case, )_1/2  Wn L/2 )3 Wn _1 (WOW n  < C ~-)  (1 +12 WO  (Wn ) 2 + (x -y)2 I IX - M > C(1 + 7L) , / 1 W n , we unca e P,,, ol (x - .) and P,,, al ( . - y) on balls espec i ely cen e ed a x a ld y and o diame e Ix - y J/4 . The a -away pa s ha e L2- no ins o he a de o I x - yj -s /2 wo and IX - yI -s /2 w n espec i ely . Hence, hey will gi e con ibu ions o a nos CW n Ix - y¡-s/2(wo )-1/2 . I e nains o con ol he con ibu ion o he local ha s . He e we use ha hei suppo s Na e a dis ance Ix-y¡/4 . As we inen ioned in he ema le ollowing he p cxl o Lemma 5, we can apply i in ou si ua ion and his gi es a con ibu ion C~nC~z_bI/~W^L(W~Wn)'~/2 -~, This concludes he p oo o Lemma 10 and o Theo e n 3 . KATO'S SQUARE-ROOT PROBLEM  32 1 Re e ences 1 .  T . KATO, "Pe u ba ion Theo y o Linea Ope a o s," Sp inge -Ve lag, 1966 . 2 .  A . MCINTOSH, On he compa abili y o A 1 / 2 and A* 1 / 2 , P oc .A .M .S . 32 (1972),430-434 . 3 .  A . P . CALDERÓN, Commu a o s o Singula In eg al Ope a o s, P oc . Na . Acad . Se¡ . U .S .A . 53 (1965), 1092-1099 . 4 .  R . C0IFMAN, A . MCINTOSH AND Y . MEYER, L'in ég ale de Cauchy dé ini un opé a eu bo né su L Z pou les cou bes lipschi ziennes, Ann . o Ma h . 116 (1982), 361-388 . 5 .  A .P . CALDERÓN, Cauchy in eg als on Lipschi z-cu es and ela ed ope - a o s, P oc . Na . Acad . Se¡ . U .S .A . 74 (1977), 1324-1327 . 6 .  E . FABES, D . JERISON AND C .E . KENIG, Mul ilinea Li lewood-Paley es ima es wi h applica ions o pa ial di e en ial equa ions, P oc . Na . Acad . Se¡ ., U .S .A . 79 (1982), 5746-5750 . 7 .  R . C0IFMAN, D .G . DENG AND Y . MEYER, Domaine de la acine ca ée de ce ains opé a eu s di é en iels acc é i s, Ann . Ins . Fou ie 33 (1983), 123-134 . 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