Sharp bounds for general commutators on weighted Lebesgue spaces

a Xi :1002.2396 2 [ma h.CA] 8 Ma 2011 SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. Abs ac . We show ha i a linea ope a o Tis bounded on weigh ed Lebesgue space L2(w) and obeys a linea bound wi h espec o he A2cons an o he weigh , hen i s commu a o [b, T ] wi h a unc ion bin BMO will obey a quad a ic bound wi h espec o he A2cons an o he weigh . We also p o e ha he k h-o de commu a o Tk b= [b, T k−1 b] will obey a bound ha is a powe (k+ 1) o he A2 cons an o he weigh . Sha p ex apola ion p o ides co esponding Lp(w) es ima es. In pa icula hese es ima es hold o Tany Calde ´on-Zygmund singula in eg al ope a o . The esul s a e sha p in e ms o he g ow h o he ope a o no m wi h espec o he Apcons an o he weigh o all 1 < p < ∞, all k, and all dimensions, as examples in ol ing he Riesz ans o ms, powe unc ions and powe weigh s show. 1. In oduc ion Singula in eg al ope a o s a e known o be bounded in weigh ed Lebesgue spaces Lp(w) i he weigh belongs o he Apclass o Muckenhoup . Recen ly he e has been enewed in e es in unde s anding he dependence o he ope a o no m in e ms o he Apcons an o he weigh , mo e p ecisely one seeks es ima es o he ype, kTkLp(w)≤ϕp([w]Ap) 1 < p < ∞, whe e he unc ion ϕp: [1,∞)→[0,∞) is op imal in e ms o g ow h. The i s esul o his ype was ob ained by S. Buckley [4] who showed ha he maximal unc ion obeyed such es ima es wi h ϕp( ) = cp 1 p−1 o 1 < p < ∞, and his is op imal (see [21] o ano he ecen p oo ). This p oblem has a ac ed enewed a en ion because o he wo k o As ala, Iwaniec and Saksman [2]. They p o ed sha p egula i y esul s o solu ions o he Bel ami equa ion, assuming ha he ope a o no m o he Beu ling- Ahl o s ans o m g ows linea ly in e ms o he Apcons an o p≥2. This linea g ow h was p o ed by S. Pe e michl and A. Volbe g [32] and by Pe e michl [30,31] o he Hilbe ans o m and he Riesz T ans o ms. In hese pape s i has been shown ha i Tis any o hese ope a o s, hen (1.1) kTkLp(w)≤cp,n [w]max{1,1 p−1} Ap1< p < ∞, 2010 Ma hema ics Subjec Classi ica ion. P ima y 42B20, 42B25. Seconda y 46B70, 47B38. Key wo ds and ph ases. commu a o s, singula in eg als, BMO, A2,Ap. The hi d au ho would like o acknowledge he suppo o Spanish Minis y o Science and Inno- a ion ia g an MTM2009-08934. 1 2 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. and he exponen max n1,1 p−1ois bes possible. I has been conjec u ed, and e y ecen ly p o ed [16], ha he same es ima e holds o any Calde ´on-Zygmund ope a o T. By he sha p e sion o he Rubio de F ancia ex apola ion heo em [12], i su ices o p o e his inequali y o p= 2, namely (1.2) kTkL2(w)≤cn[w]A2. We emi he eade o [10] o a new p oo and o gene aliza ions and applica ions. The linea g ow h in L2(w) has been shown o hold o dyadic ope a o s (ma ingale ans o m [34], dyadic squa e unc ion [15], dyadic pa ap oduc [3]), o o ope a o s who ha e lo s o symme ies and can be w i en as a e ages o dyadic shi ope a o s (such us he Hilbe ans o m [30], Riesz ans o ms [31], Beu ling ans o m [32], [13]). All hese es ima es we e ob ained using Bellman unc ions. Recen ly all he abo e esul s ha e been eco e ed using di e en se s o echniques, and ob aining linea bounds o ala ge class o Haa shi ope a o s [20], [8]. In pa icula , in he la e pape [8], no Bellman unc ion echniques no any wo weigh esul s a e used, and he me hods can be ex ended o o he impo an ope a o s in Ha monic Analysis such as dyadic squa e unc ions and pa ap oduc s, maximal singula in eg als and he ec o - alued maximal unc ion as can be ound in [9]. The sha p bound (1.2) o any Calde ´on-Zygmund ope a o Thas been p o ed in [16] by T. Hy ¨onen. Hy ¨onen’s p oo is based on app oxima ing Tby gene alized dyadic Haa shi ope a o s wi h good bounds combined wi h he key ac ha o p o e (1.2) i is enough o p o e he co esponding weak ype (2,2) es ima e wi h he same linea bound as p o ed in [29]. A di ec p oo a oiding his weak (2,2) educ ion can be ound in [17]. A bi ea lie , in [22], he sha p Lp(w) bound o Twas ob ained o alues o pou side he in e al (3/2,3) and he p oo is based on he co esponding es ima es o he in insic squa e unc ions. I should be men ioned ha un il he A2-conjec u e was p o ed, only he ollowing special case (1.3) kTkLp(w)≤Cp[w]A11< p < ∞, had been shown o be ue o any Calde ´on-Zygmund ope a o . Obse e ha he condi ion imposed on he weigh is he A1weigh condi ion which is s onge han Ap bu he e is a gain in he exponen since i is linea o any 1 < p < ∞(compa e wi h (1.1)). This has been shown in [23,24] and we emi he eade o [28] o a su ey on his opic. The main pu pose o his pape is o p o e es ima es simila o (1.1) o commu a o s o app op ia e linea ope a o s Twi h BMO unc ions b. These ope a o s a e de ined o mally by he exp ession [b, T] =bT( )−T(b ). When Tis a singula in eg al ope a o , hese ope a o s we e conside ed by Coi man, Rochbe g and Weiss in [7]. Al hough he o iginal in e es in he s udy o such ope a o s SHARP BOUNDS FOR GENERAL COMMUTATORS 3 was ela ed o gene aliza ions o he classical ac o iza ion heo em o Ha dy spaces many o he applica ions ha e been ound. The main esul om [7] s a es ha [b, T] is a bounded ope a o on Lp(Rn), 1 < p < ∞, when bis a BMO unc ion and Tis a singula in eg al ope a o . In ac , he BMO condi ion o bis also a necessa y condi ion o he Lp-boundedness o he com- mu a o when Tis he Hilbe ans o m. La e on a di e en p oo was gi en by J. O. S ¨ombe g (c . [33] p.417) wi h he ad an age ha i allows o show ha hese com- mu a o s a e also bounded on weigh ed Lp(w), when w∈Ap. This app oach is based on he use o he classical C. Fe e man-S ein maximal unc ion and i is no p ecise enough o u he de elopmen s. Indeed, we may hink ha hese ope a o s beha e as Calde ´on-Zygmund ope a o s, howe e he e a e some di e ences. Fo ins ance, an in e es ing ac is ha , unlike wha i is done wi h singula in eg al ope a o s, he p oo o he Lp-boundedness o he commu a o does no ely on a weak ype (1,1) inequali y. In ac , simple examples show ha in gene al [b, T] ails o be o weak ype (1,1) when b∈BMO. This was obse ed by he hi d au ho in [26] whe e i is also shown ha he e is an app op ia e weak-L(log L) ype es ima e eplacemen . This shows ha he ope a o canno be a Calde ´on-Zygmund singula in eg al ope a o . To s ess his poin o iew i is also shown by he hi d au ho [27] ha he igh ope a o con olling [b, T] is M2=M◦M, ins ead o he Ha dy-Li lewood maximal unc ion M. In he p esen pape we pu sue his poin o iew by showing ha commu a o s ha e an ex a “bad” beha io om he poin o iew o he Ap heo y o weigh s ha i is no e lec ed in he classical si ua ion. Ou a gumen will be based on he second p oo o he Lp-boundedness o he commu a o p esen ed in [7]. This p oo is in e es ing because he e is no need o assume ha Tis a singula in eg al ope a o , o show he boundedness o he commu a o i is enough o assume ha he ope a o Tis linea and bounded on Lp(w) o any w∈Ap. These ideas we e exploi ed in [1]. The i s au ho showed in [5] ha he commu a o wi h he Hilbe ans o m obeys an es ima e o he ollowing ype, (1.4) k[b, H]kL2(w)≤C[w]2 A2kbkBMO, and he also showed ha he quad a ic g ow h wi h espec o he A2cons an o he weigh is sha p. The echniques used in ha pape ely e y much in dyadic conside a ions and he use o Bellman unc ion a gumen s. Using ecen esul s on Haa shi s ope a o s, he also deduced he quad a ic g ow h o commu a o s o Haa shi ope a o s and ope a o s in hei con ex hull, including he Riesz ans o ms and he Beu ling-Ahl o s ope a o , see [5] o he Also, i should be men ioned ha he e is a co esponding e sion o (1.3) o com- mu a o s o any Calde ´on-Zygmund ope a o wi h quad a ic g ow h as in (1.4). This is p o ed in [25] whe e an endpoin es ima e can also be ound. By comple ely di e en me hods, we show in his pape ha i an ope a o obeys an ini ial linea bound in L2(w), hen i s commu a o will obey a quad a ic bound in L2(w). In ligh o he posi i e esolu ion o he A2-conjec u e, we conclude ha he 4 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. commu a o o any Calde ´on-Zygmund singula in eg al ope a o and a BMO unc ion obeys a quad a ic bound in L2(w). In ac we show ha i an ope a o Tobeys a bound in L2(w) o he o m ϕ([w]A2), hen i s k- h o de commu a o wi h b∈BMO, Tk b:= [b, Tk−1 b], will obey a bound o he o m ck nk!ϕ(γn[w]A2)[w]k A2kbkk BMO. Obse e ha i we conside he special case o Calde ´on-Zygmund ope a o s wi h ke nel K hen Tk b( )(x)ZRn (b(x)−b(y))kK(x, y) (y)dy, and he la ge kis, he mo e singula he ope a o will be, because he exponen in [w]k A2becomes la ge . Co esponding es ima es in Lp(w) a e deduced by he sha p e sion o he Rubio de F ancia ex apola ion heo em ound in [12], and a e shown o be sha p in he case o he Hilbe and Riesz ans o ms (in any dimension) o all 1 < p < ∞, and o all k≥1. I will be in e es ing o eco e he esul o he highe o de commu a o s wi h he Haa shi ope a o s (and hence o he Hilbe , Riesz and Beu ling ans o ms) using he dyadic me hods, bu so a we do no know how o do his. Recen ly ex ensions o ou esul o wo weigh se ings, ac ional in eg als and mo e ha e been ob ained by D. C uz-U ibe and Kabe Moen, see [11]. The emainde o his pape is o ganized as ollows. In Sec ion 2we ga he some basic esul s. In Sec ion 3we gi e he p oo o he main esul . In Sec ion 4we show wi h examples ha he main heo em in he pape , and i s co olla ies a e sha p. Finally, he las sec ion is an appendix whe e we show a esul ha i is claimed bu ne e p o ed in he li e a u e, a sha p e e se H¨olde ’s inequali y o A2weigh s. 2. P elimina y esul s 2.1. A Sha p John-Ni enbe g. Fo a locally in eg able b:Rn→Rwe de ine kbkBMO = sup Q 1 |Q|ZQ|b(y)−bQ|dy < ∞, whe e he sup emum is aken o e all cubes Q∈Rnwi h sides pa allel o he axes, and bQ=1 |Q|ZQ b(y)dy. The main ele ance o BMO is because o i s exponen ial sel -imp o ing p ope y, eco ded in he celeb a ed John-Ni enbe g Theo em [18]. We need a e y p ecise e sion o i , as ollows: Theo em 2.1. [Sha p John-Ni enbe g] The e a e dimensional cons an s 0≤αn<1< βnsuch ha (2.1) sup Q 1 |Q|ZQ exp αn kbkBMO |b(y)−bQ|dy ≤βn. In ac we can ake αn=1 2n+2 . SHARP BOUNDS FOR GENERAL COMMUTATORS 5 Fo he p oo o his we emi o p. 31-32 o [19] whe e a p oo di e en om he s anda d one can be ound. We de i e om Theo em 2.1 he ollowing Lemma 2.2. ha will be used in he p oo o he main heo em. Fi s ecall ha a weigh wsa is ies he A2condi ion i [w]A2= sup Q1 |Q|ZQ w 1 |Q|ZQ w−1<∞, whe e he sup emum is aken o e all cubes Q∈Rnwi h sides pa allel o he axes. No ice ha [w]A2≥1. I is well known ha i w∈A2 hen b= log w∈BMO. A pa ial con e se also holds, i b∈BMO he e is an s0>0 such ha w=esb ∈Ap,|s| ≤ s0. As a consequence o he Sha p John-Ni enbe g Theo em we can ge a mo e p ecise e sion o his pa ial con e se. Lemma 2.2. Le b∈BMO and le αn<1< βnbe he dimensional cons an s om (2.1). Then s∈R,|s| ≤ αn kbkBMO =⇒es b ∈A2and [es b]A2≤β2 n. P oo . By Theo em 2.1, i |s| ≤ αn kbkBMO and i Qis ixed 1 |Q|ZQ exp(|s||b(y)−bQ|)dy ≤1 |Q|ZQ exp( αn kbkBMO |b(y)−bQ|)dy ≤βn, hus 1 |Q|ZQ exp(s(b(y)−bQ)) dy ≤βn, and 1 |Q|ZQ exp(−s(b(y)−bQ)) dy ≤βn. I we mul iply he inequali ies, he bQpa s cancel ou : 1 |Q|ZQ exp(s(b(y)−bQ)) dy 1 |Q|ZQ exp(s(bQ−b(y))) dy =1 |Q|ZQ exp(sb(y)) dy 1 |Q|ZQ exp(−sb(y)) dy≤β2 n namely es b ∈A2wi h [es b]A2≤β2 n.  We ema k ha i ollows easily om mino modi ica ions o he p oo o Lemma 2.2 ha i 1 < p < ∞ s∈R,|s| ≤ αn kbkBMO min 1,1 p−1=⇒es b ∈Apand [es b]Ap≤βp n, 6 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. whe e as usual [w]Ap= sup Q1 |Q|ZQ w(x)dx 1 |Q|ZQ w(x)−1/(p−1)dxp−1 <∞. 2.2. Sha p e e se H¨olde inequali y o he A2class o weigh s. Recall ha i w∈A2 hen wsa is ies a e e se H¨olde condi ion, namely, he e a e cons an s > 1 and c≥1 such ha o any cube Q (2.2) 1 |Q|ZQ w dx1 ≤c |Q|ZQ w In he usual p oo s, bo h cons an s, cand , depend upon he A2cons an o he weigh . The e is a mo e p ecise e sion o (2.2). Lemma 2.3. Le w∈A2and le w= 1 + 1 2n+5[w]A2. Then 1 |Q|ZQ w wdx1 w≤2 |Q|ZQ w This esul was s a ed and used in [4] bu no p oo was gi en. The au ho men ioned ins ead he celeb a ed wo k [6] whe e no explici s a emen can be ound. We supply in Sec ion 5a p oo aken om [28], whe e a mo e gene al e sion can be ound as well as mo e in o ma ion. 3. Main esul Theo em 3.1. Le Tbe a linea ope a o bounded on L2(w) o any w∈A2. Suppose u he ha he e is an inc easing unc ion ϕ: [1,∞)→[0,∞)such ha (3.1) kTkL2(w)≤ϕ([w]A2). hen he e a e cons an s γnand cnindependen o [w]A2such ha (3.2) k[b, T]kL2(w)≤cnϕ(γn[w]A2) [w]A2kbkBMO. Fo he pa icula case ϕ( ) = c0 , whe e > 0, and c0>0, a simple induc ion a gumen shows ha i kTkL2(w)≤a0[w] A2, hen o each in ege k≥1 he e is a cons an akdepending on k, he ini ial alue a0, and he pa ame e s γnand cnin he heo em, such ha he k h-o de commu a o Tk b de ined ecu si ely by Tk b:= [b, Tk−1 b], obeys he ollowing weigh ed es ima es kTk bkL2(w)≤ak[w] +k A2kbkk BMO. Mo e p ecisely, he sequence {ak}k≥0obeys he ollowing ecu ence equa ion ha can be sol ed easily, ak=cnak−1γ +k−1 n=ck na0γk +(k−2)(k−1) 2 n. SHARP BOUNDS FOR GENERAL COMMUTATORS 7 Using he me hod o p oo o Theo em 3.1 we can ob ain a weigh ed es ima e o he k h-o de commu a o ha wo ks o gene al inc easing unc ion ϕ: [1,∞)→[0,∞). No ice he di e ence in he cons an s wi h wha we jus a gued by induc ion o he pa icula case ϕ( ) = a0 : in he co olla y he cons an is ck na0γ nk!, whe eas in he induc ion a gumen he cons an is ck na0γk +(k−2)(k−1) 2 n. Co olla y 3.2. Le Tbe a bounded linea ope a o on L2(w)wi h w∈A2and (3.3) kTkL2(w)≤ϕ([w]A2). hen he e a e cons an s γnand cnindependen o [w]A2such ha (3.4) kTk bkL2(w)≤ck nk!ϕ(γn[w]A2) [w]k A2kbkk BMO. The cons an s γnand cn ha appea in Co olla y 3.2 a e he same ha appea ed in Theo em 3.1. We i s p esen he p oo o Theo em 3.1, and a e wa ds we discuss he necessa y modi ica ions o ob ain Co olla y 3.2. As an easy consequence o Co olla y 3.2 and he Rubio de F ancia ex apola ion heo em wi h sha p cons an s [12], we ha e he ollowing. Co olla y 3.3. Le Tbe a linea ope a o bounded on L2(w)wi h w∈A2and (3.5) kTkL2(w)≤ϕ([w]A2). Then, o 1< p < ∞, he e a e cons an s γn,p, and cn,p, which only depend on p, and he dimension n, such ha o all weigh s w∈Ap (3.6) kTk bkLp(w)≤√2ck n,p k!ϕγn,p [w]max{1,1 p−1} Ap[w]kmax{1,1 p−1} Apkbkk BMO. In he pa icula case ϕ( ) = a0 he ex apola ed es ima e looks like (3.7) kTk bkLp(w)≤√2a0ck nk!γ nc +k p[w]( +k) max{1,1 p−1} Apkbkk BMO. whe e cpdepends only on p,γnand cna e he cons an s ha appea ed in Theo em 3.1. We will show in Sec ion 4 ha o = 1, ϕ( ) = a0 he powe (1 + k) max{1,1 p−1} canno be dec eased o T=Hand T=Rj he Hilbe and Riesz ans o ms, o all k≥1 and p > 1, he e o e he heo em is op imal in e ms o he a e o he dependence on [w]Ap. In [5], examples o k= 1, o all p > 1, and o T he Hilbe , Beu ling and Riesz ans o ms, we e p esen ed. P oo o Theo em 3.1.We “conjuga e” he ope a o as ollows: i zis any complex numbe we de ine Tz( ) = ezbT(e−zb ). Then, a compu a ion gi es ( o ins ance o ”nice” unc ions), [b, T]( ) = d dzTz( )|z=0 =1 2πi Z|z|=ǫ Tz( ) z2dz , ǫ > 0 by he Cauchy in eg al heo em, see [7], [1]. 8 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. Now, by Minkowski’s inequali y (3.8) k[b, T]( )kL2(w)≤1 2π ǫ2Z|z|=ǫkTz( )kL2(w)|dz|ǫ > 0. The key poin is o ind he app op ia e adius ǫ. To do his we look a he inne no m kTz( )kL2(w) kTz( )kL2(w)=kT(e−zb )kL2(we2Rez b), and y o ind app op ia e bounds on z. To do his we use he main hypo hesis, namely ha Tis bounded on L2(w) i w∈A2wi h kTkL2(w)≤ϕ([w]A2). Hence we should compu e [we2Rez b]A2= sup Q1 |Q|ZQ we2Rez b(x)dx 1 |Q|ZQ w−1e−2Rez b(x)dx. Now, since w∈A2we use Lemma 2.3: i = w= 1 + 1 2n+5[w]A2<2 hen 1 |Q|ZQ w dx1 ≤2 |Q|ZQ w , and simila ly o w−1since w= w−1, 1 |Q|ZQ w− dx1 ≤2 |Q|ZQ w−1. Using his and Holde ’s inequali y we ha e o an a bi a y Q 1 |Q|ZQ w(x)e2Rez b(x)dx 1 |Q|ZQ w(x)−1e−2Rez b(x)dx≤ 1 |Q|ZQ w dx1 1 |Q|ZQ e2Rez ′b(x)dx1 ′1 |Q|ZQ w− dx1 1 |Q|ZQ e−2Rez ′b(x)dx1 ′ ≤41 |Q|ZQ w dx 1 |Q|ZQ w−1dx 1 |Q|ZQ e2Rez ′b(x)dx1 ′1 |Q|ZQ e−2Rez ′b(x)dx1 ′ ≤4 [w]A2[e2Rez ′b] 1 ′ A2 Now, since b∈BMO we a e in a posi ion o apply Lemma 2.2, i |2Rez ′| ≤ αn kbkBMO hen [e2Rez ′b]A2≤β2 n. Hence o hese z, and since 1 < < 2, [we2Rez b]A2≤4 [w]A2β 2 ′ n≤4 [w]A2βn. Using his es ima e o hese z, and obse ing ha ke−zb kL2(we2Rezb)=k kL2(w), kTz( )kL2(w)=kT(e−zb )kL2(we2Rez b)≤ϕ([we2Rez b]A2)k kL2(w)≤ϕ(4[w]A2βn)k kL2(w). SHARP BOUNDS FOR GENERAL COMMUTATORS 9 Choosing now he adius ǫ=αn 2 ′kbkBMO , we can con inue es ima ing he no m in (3.8) k[b, T]( )kL2(w)≤1 2π ǫ2Z|z|=ǫkTz( )kL2(w)|dz| ≤1 2π ǫ2Z|z|=ǫ ϕ(4[w]A2βn)k kL2(w)|dz|=1 ǫϕ(4[w]A2βn)k kL2(w), since |2Rez ′| ≤ 2|z| ′= 2ǫ ′=αn kbkBMO . Finally, o his ǫ, k[b, T]( )kL2(w)≤C22nϕ(4[w]A2βn) [w]A2kbkBMO, because ′= 1 + 2n+5[w]2≈2n[w]2, and αn=1 2n+2 . Obse e ha he op imal adius is essen ially he in e se o [w]2kbkBMO. This p o es he heo em wi h cn∼22nand γn= 4 βn.  P oo o Co olla y 3.2.In his case a compu a ion gi es ( o ins ance o ”nice” unc- ions), see [1] o example o he o iginal pape [7], Tk b( ) = dk dzkTz( )|z=0 =k! 2πi Z|z|=ǫ Tz( ) zk+1 dz ǫ > 0 by he Cauchy in eg al heo em. The same calcula ion as in he case k= 1 gi es he equi ed es ima e, wi h ck n≈22nk, and he same γn= 4βn.  4. Examples In his sec ion, we show ha one can no ha e es ima es be e han Theo em 3.1, Co olla y 3.2, and Co olla y 3.3. We p esen examples which e u n he same g ow h wi h espec o he Apcons an o he weigh ha appea s in ou esul s. Fi s , we discuss he simple case in dimension one. The ollowing example shows ha he quad a ic es ima e o he i s commu a o o he Hilbe ans o m is sha p o p= 2. 4.1. Sha p example o he commu a o o he Hilbe ans o m. Conside he Hilbe ans o m H (x) = p. . ZR (y) x−ydy, and conside he BMO unc ion b(x) = log |x|. We know ha he e is a cons an c such ha (4.1) k[b, H]kL2(w)≤c[w]2 A2 16 DAEWON CHUNG, MAR´ IA CRISTINA PEREYRA AND CARLOS PEREZ. [34] J. Wi we , A sha p es ima es on he no m o Ma ingale ans o m, Ma h. Res. Le . 7(2000) 1-12. Daewon Chung, Depa men o Ma hema ics and S a is ics MSC01 1115, 1 Uni e - si y o New Mexico, Albuque que, NM 87131-0001 E-mail add ess:[email p o ec ed]du Ma ´ ıa C is ina Pe ey a, Depa men o Ma hema ics and S a is ics, MSC01 1115, 1 Uni e si y o New Mexico, Albuque que, NM 87131-0001 E-mail add ess:[email p o ec ed] Ca los P´ e ez, Depa amen o de An´ alisis Ma em´ a ico, Facul ad de Ma em´ a icas, Uni e sidad De Se illa, 41080 Se illa, Spain. E-mail add ess:[email p o ec ed]