Weighted lp estimates for the ∂-equation on convex domains of finite type

Publ. Mat. 48 (2004), 139–157 WEIGHTED LpESTIMATES FOR THE ∂-EQUATION ON CONVEX DOMAINS OF FINITE TYPE Heungju Ahn Abstract We prove non-isotropic Lp(1 ≤p≤ ∞) estimates with weights for solutions of the Cauchy-Riemann equation on bounded convex domains of finite type in Cnusing the integral kernel method. We also give an example which guarantees the optimum of the estimates. 1. Introduction and statement of results Let Ω ⊂Cnbe a smoothly bounded convex domain of finite type m with a defining function ρ. In this paper we treat a certain weighted Lpestimates for the ∂-equation on Ω. The notation δ(ζ) will stand for the distance from ζto the boundary of Ω, bΩ, which is up to constants |ρ(ζ)|. With solutions using the integral kernel introduced by Cumenge [Cum01a] we can prove the following theorem. Theorem 1.1. For the domain Ωas above the equation ∂u =fhas a solution uin Ωsuch that for 1≤p < ∞ (1) ZΩ δ(ζ)α−1|u(ζ)|pdV (ζ)≤Cp,αZΩ δ(ζ)α−1+p||f(ζ)||pdV (ζ), α > 0 and (2) sup ζ∈Ω δ(ζ)α−1|u(ζ)| ≤ Cαsup ζ∈Ω δ(ζ)α||f(ζ)||, α > 1, if fis a smooth (n, 1)-form with ∂f = 0 and the right hand sides of (1) and (2) are finite. Here the non-isotropic norm of forms, || · || =|| · ||Ω is defined by ||f(ζ)|| = supv∈Cn\{0}|f(ζ)(v)|/k(ζ, v), where k(ζ, v)−1is a weighted boundary distance of ζ∈Ωin the direction v. 2000 Mathematics Subject Classification. Primary: 32W05, 32A26. Key words. Cauchy-Riemann equation, Lpestimates, convex domain, finite type. The author was partially supported by the Post-doctoral Fellowship Program of Korea Science and Engineering Foundation and the Post-doctoral Fellowship of University of Padua in Italy. 140 H. Ahn Remark. (i) The non-isotropic norm || · || was first introduced by Bruna, Charpentier and Dupain [BCD98]. The definition of the quantity k(ζ, v) is rather complicate even though k(ζ, v)−1is a natural weighted boundary distance, so we postpone the precise definition in the next section. (ii) Theorem 1.1 implies that for given a smooth (0,1)-form fwith ∂f = 0 one can find a solution for the ∂-equation on Ω and that solution satisfies the inequalities (1) and (2). If Ω is strongly pseudoconvex, (1) and (2) were proved by AhnCho [AC02] and Dautov-Henkin [DH79]. When the domain is convex of finite type, Cumenge [Cum01a] proved (1) in case p= 1. Therefore it seems natural to extend Cumenge’s result to other Lp-norms, 1≤p≤ ∞. There are a number of papers related to the ∂-equation on convex domains of finite type m. We mention a few of them, which are closely related to our work; Diederich-Fischer-Fornæss [DFF99], Cumenge [Cum01a] independently proved 1/m-H¨older estimates and Cumenge [Cum01a], Fischer [Fis01] obtained the best possible Lpestimates with respect to the isotropic norm. Diederich-Mazzilli [DM01] (resp. Cumenge [Cum01b]) obtained the characterization of the zero sets of functions in the Nevanlinna (resp. Nevanlinna-Djrbachian) classes using non-isotropic L1(bΩ) (resp. weighted nonisotropic L1(Ω)) estimates for the solution of the ∂-equation on Ω. Here we briefly sketch the methods used to prove Theorem 1.1. First, to obtain the solution of ∂on Ω we use an integral representation introduced by Berndtsson-Andersson [BA82]. In the integral representation the key part is a kernel with a weight containing the Bergman kernel of Ω. Second, for the estimates of the kernel and integrals relevant to the kernel we use non-isotropic polydiscs and ε-extremal coordinates of McNeal on Ω [McN94]. Last, in order to pass into Lpestimates from L1estimates we employ a variation of H¨older’s inequality used in [AC02]. 2. Preliminaries 2.1. Integral kernels and solution operators. Let B(z, ζ) be the Bergman kernel for the domain Ω. Then B(z, ζ) is holomorphic in zand antiholomorphic in ζ. Moreover B(z, ζ)∈C∞(Ω×Ω\{(ζ,ζ), ζ ∈bΩ}) and the boundary behavior of B(z, ζ) is now well understood by [McN94]. We define Q=Q(z, ζ) = 1 B(ζ,ζ) n X j=1 Z1 0 ∂B ∂zj (zt, ζ)dtdzj, zt=ζ+t(z−ζ). Weighted LpEstimates for the ∂-Equation 141 For sufficiently large integer Nto be determined and fixed later we define the weighed kernel on (z, ζ)∈Ω×Ω\ {(ζ,ζ), ζ ∈bΩ}, K(z, ζ)= n−1 X k=0 cn,k,NB(z, ζ) B(ζ,ζ)N−k∂z|z−ζ|2∧(∂ζQ)k∧(d∂z|ζ−z|)n−k−1 |ζ−z|2n−2k = n−1 X k=0 cn,k,N K(k)(z, ζ), where cn,k,N =−(−1)n(n−1)/2N nand K(k)(z, ζ) is the k-th term in the summation. For f∈C1 (n,1)(Ω) with ∂f = 0, if we define (3) u(z) = CZΩ K(z, ζ)∧f(ζ), z ∈Ω then it is known that ∂u =f[Cum01a]. 2.2. McNeal’s result on the geometry of convex domains of finite type. We adapt to the notation of [Cum01a] and [McN94]. 2.2.1. Weighted boundary distance. Define the radius of the largest complex disc centered at zin the direction vthat fits in the domain {z:ρ(z)< ρ(ζ) + ε}, τ(z, v, ε) = sup{r > 0 : |ρ(z+λv)−ρ(z)| ≤ ε, |λ| ≤ r, λ ∈C}. Introduce a weighted boundary distance k(z, v, ε) = δ(z)/τ(z, v, ε) and write k(z, v) when ε=δ(z)/2. 2.2.2. Non-isotropic polydisc. If {v1,...,vn}is an ε-extremal basis of McNeal at z(see [McN94] and [BCD98] for the precise definition), then the non-isotropic polydisc at zwith radius εis defined by P(z, ε) =    w=z+ n X j=1 wjvj,|wj| ≤ cτ(z, vj, ε)   , where cis chosen so that w∈P(z, ε) implies |ρ(w)−ρ(z)|< ε. The properties of τwhich were proved by McNeal [McN94] are the following: (4) τ(z, v1, ε)≈ε.τ(z, vn, ε)≤ · · · ≤ τ(z, v2, ε).ε1 m and for 0 < ε1≤ε2 (5) ε1 ε2τ(z, v, ε2).τ(z, v, ε1).ε1 ε21 m τ(z, v, ε2). 142 H. Ahn Proposition 2.1 ([McN94]). (i) For all C > 0,vol P(z, Cε)≈vol P(z, ε)uniformly in z,εwith constants depending on C. (ii) vol P(z, ε)≈vol P(ζ, ε)if P(z, ε)∩P(ζ, ε)6=∅. (iii) τ(ζ, v, ε)≈τ(z, v, ε)for ζ∈P(z, ε). (iv) If {v1,...,vn}is an ε-extremal basis at z, then we have vol P(z, ε)≈ Qn i=1 τ(z, vi, ε)2. 2.2.3. Tent and quasi distance. For z∈Ω close to the boundary and η > 0, T(z, η) = P(π(z), η)∩Ω is called the tent at zof radius η, where π(z) is the projection of zto the boundary. The quasi distance of McNeal is defined as follows: M(z, ζ)≈ M(ζ, z) = inf{η:ζ∈P(z, η)} for |ζ−z|  1 and zclose to bΩ. For v∈Cnwith |v|= 1, ϕ∈C∞(Ω) let Dvϕdenote the directional derivatives of ϕin the direction v. From now on set η=η(z, ζ) = |ρ(z)|+|ρ(ζ)|+M(z, ζ). Then the following proposition is proved by McNeal [McN94]. Proposition 2.2. For every p∈bΩthere exists a neighborhood Uof p such that for ζ, z ∈U∩Ω,µ, ν ∈N,v, v0∈Cnwith |v|=|v0|= 1, (i) |Dµ vDν v0B(z, ζ)| ≤ C(µ, ν)τ(ζ, v, η)−µτ(ζ, v0, η)−ν(vol Tζ,z)−1. Here vol Tζ,z is the volume of the smallest tent containing both z,ζ. (ii) For ζ∈U∩Ω,B(ζ,ζ)≥(vol P(ζ, δ))−1,δ=δ(ζ) = |ρ(ζ)|/2. 2.2.4. Coverings. There exists a constant β > 1 such that M(z, ζ)< ε ⇒ζ∈P(z, βε), z ∈U∩Ω,0< ε 1, where Uis some neighborhood defined in Proposition 2.2. For the integral estimates we define the covering of W∩Ω C0(z) = P(z, β d(z)) ∩W∩Ω, C`(z) = {ζ∈Ω∩W: 2`−1d(z)≤ M(z, ζ)<2`d(z)}, ` ≥1, where W= 1/2U. 3. Kernel estimates 3.1. Estimate of the term K(k)(z, ζ), 0 ≤k≤n−1. Now we want to write all forms with respect to extremal coordinates of McNeal at ζ or z. Let {e(`) j=e(`) j(ζ),1≤j≤n}be a β2`δ-extremal basis at ζand (e` j(z))jaβ2`d-extremal basis at z, respectively. If vj=v(`) j(ζ) is the Weighted LpEstimates for the ∂-Equation 143 j-th component of their coordinates, we denote L(`) j=∂/∂vjand L(`)∗ j which is the dual of L(`) j. To simplify notations, in any ambiguous case, we write L(z) j,L(ζ) j,L∗(z) j,L∗(ζ) jfor L(`)(z) j,L(`)(ζ) j,L(`)∗(z) j,L(`)∗(ζ) j, 1≤j≤n, where the superscripts z,ζmean the derivations act on the variables z,ζ, respectively. To save us from confusion, we denote dist(ζ, Ω) and dist(z, bΩ) by δ=δ(ζ) and d=d(z), respectively. First we estimate K(k)(z, ζ)∧f(ζ), 1 ≤k≤n−1. Let R= R1 0∂zB(zt, ζ)dt. Then computing the k-th exterior product we obtain ∂ζQk=ck ∂ζB(ζ,ζ) B(ζ,ζ)k+1 ∧R∧(∂ζR)k−1+∂ζRk B(ζ,ζ)k. Using this and expressing all forms in terms of L(ζ) jand L(z) j’s, we have K(k)(z, ζ)∧f(ζ) = B(z, ζ)N−k B(ζ,ζ)N 1 |ζ−z|2n−2kH1+H2, where H1=1 B(ζ,ζ)X I,J L(z) i0|ζ−z|2L(ζ) j1B(ζ,ζ)Z1 0L(z) i1B(zt, ζ)dt × k Y ν=2 Z1 0L(ζ) jνL(z) iνB(zt, ζ)dtf(ζ)(e(`) jn) ×AIJS(z, ζ)L∗(ζ) S∧L∗(ζ) J∧L∗(z) I H2=X I,J L(z) i0|ζ−z|2 k Y ν=1 Z1 0L(ζ) jνL(z) iνB(zt, ζ)dtf(ζ)(e(`) jn) ×AIJS(z, ζ)L∗(ζ) S∧L∗(ζ) J∧L∗(z) I. Here I=I(k) = Ik∪Ik0,Ik={i0, i1,...,ik}and Ik0={ik+1,...,in−1}; J=J(k) = Jk∪Jk0,Jk={j1, j2,...,jk}and Jk0={jk+1,...,jn}, iν, jν∈S={1,2,...,n},L∗(z) I=L∗(z) i0∧ · · · ∧ L∗(z) in−1, etc. and AIJS is uniformly bounded on Ω ×Ω. Note that if k= 1, then the product term Qk ν=2(...) of H1does not appear. Next we note the following inequality (6) |f(ζ)(e(`) j(ζ))| ≤ ||f(ζ)|| δ(ζ) τ(ζ, e(`) j(ζ), δ).||f(ζ)||. 144 H. Ahn Then it is easy to see that K(0)(z, ζ)∧f(ζ)≤ ||f(ζ)||K(0)(z, ζ) ≤ ||f(ζ)|| |B(z, ζ)| B(ζ,ζ)N1 |ζ−z|2n−1. First we estimate |B(z, ζ)|/B(ζ,ζ), R1 0L(z) i1B(zt, ζ)dt and R1 0L(ζ) jnuL(z) iνB(zt, ζ)dt on the neighborhood Uof p∈bΩ, where Uis the neighborhood defined in Proposition 2.2. By Proposition 2.2 (ii), (4) and (5) the following estimates can be proved: |B(z, ζ)| B(ζ,ζ)≤δ(ζ) η(z, ζ) Z1 0 L(z) i1B(zt, ζ)dt .vol P(ζ, δ)τ(ζ, e(`) i1, δ)−1 Z1 0 L(ζ) jνL(z) iνB(zt, ζ)dt .vol P(ζ, δ)τ(ζ, e(`) jν, δ)τ(ζ, e(`) iν, δ)−1. (For details see [Cum01a].) Combining above estimates and the inequality (6) we have for 1 ≤k≤n−1 K(k)(z, ζ)∧f(ζ) .X Ik,Jk δ(ζ)N−k η(z, ζ)N−k δ(ζ)||f(ζ)|| |ζ−z|2n−2k−1 1 Qk ν=1 τ(ζ, e(`) jν, δ)τ(ζ, e(`) iν, δ) 1 τ(ζ, e(`) jn, δ) and K(0)(z, ζ)∧f(ζ).δ(ζ)N−1 η(z, ζ)N δ(ζ)||f(ζ)|| |ζ−z|2n−1. For the simplification of notation we write for z, ζ ∈U K(0) +(z, ζ) = δ(ζ)N−1 η(z, ζ)N 1 |ζ−z|2n−1 K(k) +(z, ζ) = δ(ζ)N−k η(z, ζ)N−k 1 |ζ−z|2n−2k−1 ×1 Qk ν=1 τ(ζ, e(`) jν, δ)τ(ζ, e(`) iν, δ) 1 τ(ζ, e(`) jn, δ). Weighted LpEstimates for the ∂-Equation 145 3.2. Estimates of K(k) +(z, ζ) on coverings {C`(z)}and {C`(ζ)}. We only consider the estimates of K(k) +(z, ζ) for k= 1, n −1. In the integral estimates other cases can be reduced to cases k= 1, n −1. If k= 0, we can directly estimate integrals on the neighborhood W. First assume that ζ∈Uis fixed and we will estimate K(k) +(z, ζ) on C`(ζ). By the definition of C`(ζ), M(z, ζ) and (5), it is easy to see that η(z, ζ)≈2`δ(ζ), z ∈ C`(ζ) τ(ζ, e(`) j(ζ), δ)&2−`τ(ζ, e(`) j(ζ), β2`δ). Note that τ(`) jn(ζ)≤τ(`) 2(ζ). Therefore we have for z∈ C`(ζ) K(n−1) +(z, ζ).τ(`) 2(ζ) 2(N−3n+2)`|ζ−z|Qn j=1 τ(`) j(ζ)2 (7) K(1) +(z, ζ).X I10,J10 τ(`) i0Qn−1 ν=2 τ(`) iν(ζ)τ(`) jν(ζ) 2(N−4)`Qn j=1 τ(`) j(ζ)2|ζ−z|2n−3,(8) where τ(`) j(ζ) = τ(ζ, e(`) j(ζ), β2`δ(ζ)). Next assume that z∈Uis fixed and we will estimate K(k)(z, ζ) on C`(z). By Proposition 2.1 (ii) and (iii), we have τ(ζ, e(`) j, δ)&δ 2`dτ(ζ, e(`) j, β2`δ)≈δ 2`dτ(z, e(`) j(z), β2`d). Since η(z, ζ)≈2`d(z), ζ∈ C`(z) we have (9) K(n−1) +(z, ζ).δ 2`dN−3n+2 1 |ζ−z| τ(`) 2(z) Qn j=1 τ(`) j(z)2 and K(1) +(z, ζ). n X i,j,p=1 i6=p δ 2`dN−41 |ζ−z|2n−3 1 τ(`) j(z)τ(`) i(z)τ(`) p(z) .X I10,J10 δ 2`dN−41 |ζ−z|2n−3 τ(`) i0(z)Qn−1 ν=2 τ(`) iν(z)τ(`) jν(z) Qn j=1 τ(`) j(z)2. (10) 146 H. Ahn 4. Integral estimates In this section we verify preliminary integral estimates that are an essential step to prove our Theorem 1.1. Lemma 4.1. Let α > 0and ε > 0with α−1−ε > −1. Then for k= 0,1,...,n−1we have ZΩ d(z)α−1−εK(k) +(z, ζ)dV (z)≤Cα,εδ(ζ)α−ε−1 (11) ZΩ δ(ζ)−εK(k) +(z, ζ)dV (ζ)≤Cεd(z)−ε.(12) Proof: We prove (11) and (12) for k= 0,1, n −1. The other cases can be reduced to the cases k= 0,1, n −1. Since the only singularity is of the form |ζ−z|−j, we may assume that z, ζ ∈W.Wcan be covered by ∪`C`(z) and ∪`C`(ζ) so basically we have to deal with the domain of the form C`(z) or C`(ζ). (i) By the estimate (7) we have for any integer `≥0 (13) ZC`(ζ) d(z)α−1−εK(n−1) +(z, ζ)dV (z) .τ(`) 2(ζ) 2(N−3n+2)`Qn j=1 τ(`) j(ζ)2ZP(ζ,β2`δ) d(z)α−1−ε |ζ−z|dV (z). To obtain a desired estimate, we use the system of coordinate associated to the basis (e(`) 1(ζ),...,e(`) n(ζ)). We set wk=hζ−z, e(`) ki,1≤k≤n, t1=−ρ(z), t2= Im w1 (14) and for 2 ≤k≤n, t2k−1= Re wk, t2k= Im wk. Weighted LpEstimates for the ∂-Equation 147 Since τ(`) 1(ζ)≈2`δand d(z).2`dfor z∈P(ζ, β2`d), by the coordinates change (14) we have ZP(ζ,β2`δ) d(z)α−1−ε |ζ−z|dV (z) .Z|tj|<2`δ, j=1,2 |wj|<τ(`) j, j≥2 tα−1−ε 1dt1dt2dV (w2,...,wn) |w2| .1 α−ε(2`δ)α−ε+1 Z|wj|<τ(`) j, j≥2 dV (w2,...,wn) |w2| .1 α−ε(2`δ)α−ε−1 n Y j=1 τ(`) j(ζ)2 τ(`) 2(ζ). (15) If we choose an integer Nso that N−3n+3−α > 0, then (13) and (15) give ZW∩Ω d(z)α−1−εK(n−1) +(z, ζ)dV (z) .X ` 2−(N−3n+3−α+ε)`δ(ζ)α−ε−1.δ(ζ)α−ε−1. To prove (11) for k= 1 we have to consider the integral ZP(ζ,β2`δ) d(z)α−1−ε |ζ−z|2n−3dV (z). To change coordinates we again use the coordinates (14). Here we may assume that i0< i1,ν= min(i0, j1), µ= max(i0, j1) and ri1=|wi1|. We first consider t1,t2ν,t2i2−1,t2i2variables and then we integrate with the remaining (2n−4) variables, t0. Since τ(`) 1≈2`δand 2`δ.τ(`) µ, we 154 H. Ahn where we set s=r/(1 − |z1|2)1/m.To calculate the upper bound of (22) we need the following lemma: Lemma 6.2 ([Rud80]).For z∈Bn={z∈Cn:|z|<1},creal, η > −1, define Jc,η(z) = ZBn (1 − |ζ|2)η |1−ζ·z|n+1+η+cdV (ζ). When c < 0, then Jc,η is bounded in Bn. When c > 0, then Jc,η(z)≈ (1 − |z|2)−c. Finally, J0,η ≈−log(1 − |z|2). From (22), (23) and by Lemma 6.2 it follows that if γ > α, then kfkp p,γ,Em.Z|z1|<1 dA(z1) |1−z1|dp−γ+1−p/m−2/m = lim r→1−Z|z1|<1 dA(z1) |1−z1r|dp−γ+1−p/m−2/m .1 since dp −γ+ 1 −p/m −2/m < 2. Let v(z1, z2) = z2/(1 −z1)d. Then it is clear that ∂v =fon Em. On the other hand, we have (24) kvkp p,α =Z|z1|<1 dA(z1) |1−z1|dp Z|z2|<bz1 |z2|p(1−|z1|2−|z2|m)α−1dA(z2). By polar coordinate change, we have J(z1) = Z|z2|<bz1 |z2|p(1 − |z1|2− |z2|m)α−1dA(z2) = 2π(1 − |z1|2)α−1Zbz1 0 rp+1 1−rm 1− |z1|2α−1 dr = 2π(1 − |z1|2)α−1+ (p+2) mZ1 0 (1 −sm)α−1sp+1 ds &(1 − |z1|2)α−1+ (p+2) m. (25) Weighted LpEstimates for the ∂-Equation 155 From (24) and (25) we have kvkp p,α &Z|z1|<1 (1 − |z1|2)α−1+ (p+2) m |1−z1|dp dA(z1) = lim r→1−Z|z1|<1 (1 − |z1|2)α−1+ (p+2) m |1−z1r|dp dA(z1) ≈lim r→1−log 1 1−r2=∞, (26) by Lemma 6.2. Thus, v /∈Lp α(B2). Next we consider the inner product hh, viαfor every h∈L2 α(B2)∩ O(B2). By Fubini’s theorem, we have hh, viα=ZEm h(ζ)v(ζ)|ρ(ζ1, ζ2)|α−1dV (ζ) =Z|ζ1|<1 dA(ζ1) (1 −ζ1)dZ|ζ2|<bζ1 ζ2h(ζ1, ζ2)(1 − |ζ1|2− |ζ2|m)α−1dA(ζ2). Putting ζ2=reiθ, we see Z|ζ2|<bζ1 ζ2h(ζ1, ζ2)(1 − |ζ1|2− |ζ2|m)α−1dA(ζ2) =Zbζ1 0Z2π 0 r2eiθh(ζ1, reiθ)(1 − |ζ1|2−rm)α−1dθ dr =Zbζ1 0 r2(1 − |ζ1|2−rm)α−1Z2π 0 eiθh(ζ1, reiθ)dθdr =Zbζ1 0 r2(1 − |ζ1|2−rm)α−1·0·dr = 0, since h(ζ1,·) is holomorphic. Thus vis orthogonal to L2 α(B2)∩ O(Em), i.e., vis the canonical solution for ∂u =f. To complete our theorem we need another well-known theorem on the boundedness of the weighted Bergman projections on Em. Proposition 6.3 ([Cho], [LS92]).Let Bα:L2 α(Em)→L2 α(Em)∩O(Em) be the orthogonal projection, α > 0. Then Bα:Lp α(Em)→Lp α(Em)∩ O(Em)is a bounded operator for every 1<p<∞. Assume that ∂u =f∈Lp γ(Em,|| · ||) and u∈Lp α(Em), γ > α. Then by the Proposition 6.3, v=u−Bα(u) and it would be in Lp α(Em). 156 H. Ahn By (26) this is impossible. Hence there is no solution uin Lp α(Em) to the equation ∂u =f. If r < p, i.e. dr −α−r/2<2, then it also follows by a similar calculation to the above that f∈Lr α(Em,|| · ||) and no solution uto ∂u =fbelongs to Lp α(Em). References [AC02] H. Ahn and H. R. Cho, Optimal non-isotropic Lpestimates with weights for ∂in strictly pseudoconvex domains, Kyushu J. Math. 56(2) (2002), 447–457. [BA82] B. Berndtsson and M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble) 32(3) (1982), 91–110. [BCD98] J. Bruna, P. Charpentier and Y. Dupain, Zero varieties for the Nevanlinna class in convex domains of finite type in Cn,Ann. of Math. 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Stoll, Projections on spaces of holomorphic functions on certain domains in C2,Complex Variables Theory Appl. 17(3–4) (1992), 223–233. [McN94] J. D. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109(1) (1994), 108–139. Weighted LpEstimates for the ∂-Equation 157 [Rud80] W. Rudin,“Function theory in the unit ball of Cn”, Grundlehren der Mathematischen Wissenschaften 241, Springer-Verlag, New York-Berlin, 1980. Department of Pure and Applied Mathematics University of Padova Via Belzoni 7 35131 Padova Italy E-mail address:[email protected] Primera versi´o rebuda el 29 d’abril de 2003, darrera versi´o rebuda el 20 de novembre de 2003.