Proceedings of the Edinburgh Mathematical Society (2001) 44, 571–583 c
DUALITY FOR SOME LARGE SPACES OF ANALYTIC FUNCTIONS
H. JARCHOW1, V. MONTESINOS2, K. J. WIRTHS3 AND J. XIAO4,5 1Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstr. 190, CH 8057 Z¨urich, Switzerland ([email protected]) 2Departamento de Matematica Aplicada, ETSI Telecomunicacion, Universidad Politecnica de Valencia, C/vera, s/n., E-46071 Valencia, Spain ([email protected]) 3Institut f¨ur Analysis, TU-Braunschweig, Pockelsstr. 14, D-38106 Braunschweig, Germany ([email protected]) 4Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneue Blvd West, Montreal, Quebec, Canada ([email protected]) 5Department of Mathematics, Peking University, Beijing 100871, People’s Republic of China (Received 24 September 1999)
p p Abstract We characterize the duals and biduals of the L -analogues α of the standard Nevanlinna Np classes α, α 1 and 1 p< . We adopt the convention to take to be the classical Smirnov N > − 6 ∞ N 1 class + for p = 1, and the Hardy–Orlicz space LHp (= (Log+H)p)− for 1 Keywords: coefficient multipliers; weighted area Nevanlinna classes; weighted Bergman spaces; nuclear power series spaces AMS 2000 Mathematics subject classification: Primary 46E10; 46A11; 47B38 Secondary 30D55; 46A45; 46E15 1. Introduction We denote by ∆ the unit disk z C : z < 1 in C and by (∆) the space of all { ∈ | | } H analytic functions ∆ C. With respect to uniform convergence on compact subsets of → ∆ (i.e. local uniform convergence), (∆) is a Fr´echet space. H 571 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. 572 H. Jarchow and others 1 Let m be the normalized area measure on ∆; so dm(x +iy)=π− dx dy. We shall work with the probability measures dm (z):=(α + 1)(1 z 2)α dm(z) on (the Borel α −| | sets of) ∆ ( 1 <α< ). Given 1 p< , we define p to consist of all f (∆) − ∞ 6 ∞ Nα ∈H such that log+ f is in Lp(m ). With respect to the F -norm | | α 1/p f := [log(1 + f )]p dm , k| k| α,p | | α Z∆ this is a complete metrizable topological vector space (even an algebra). If p = 1, then we obtain the usual weighted area Nevanlinna class discussed, for example, in [11]. Nα The Hardy–Orlicz space LHp,1 2π dt d (f,g) := sup log(1 + (f g)(reit) ) . N | − | 2π 06r<1 Z0 But d does not generate a linear topology on (see Shapiro and Shields [16]). The N N largest subspace of on which d defines a linear topology is the Smirnov class, denoted N N by +. For more on and +, see Duren’s book [6]. N N N For the purposes of this paper, it is convenient to set 1 + p p LH := and 1 := LH for 1 6 p< . N N− ∞ It is clear that p q for all β α 1 and p q.If(α, p) =(β,q), then these Nα ⊂Nβ > > − > 6 inclusions are clearly proper. Let β> 1 and 0 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. Duality for some large spaces of analytic functions 573 p to use stronger tools. The ( )∗ coincide for all (α, p) on any straight line p = c(α + 2), Nα c>0 a constant, and they are also duals of specific nuclear power series spaces of finite p type, Fθ say. Here, θ varies over the interval (0, 1). Fθ is the ‘Fr´echet envelope’ of α; its p N topology is obtained by convexifying the neighbourhoods of zero in α and then passing 1 N to the completion. If we take into account only the spaces α and LHp, then the full 1 N interval is covered in this way in a one-to-one fashion: θ = 2 corresponds to the Smirnov class [20], θ =(α +2)/(α + 3) corresponds to the scale of spaces 1 (α> 1), and Nα − θ =1/(p + 1) corresponds to the classes LHp (1 2. Main results As in [20], the key result is a characterization of the coefficient multipliers from p into Nα various small spaces of analytic functions. Suppose that X and Y are complete and metrizable topological vector spaces which consist of analytic functions on ∆ and whose topology is finer than that of local uni- form convergence. Let [[X, Y ]] be the collection of all sequences (λn)0∞ in C such that n n n∞=0 λnanz belongs to Y whenever n∞=0 anz is a member of X. This is a linear space whose elements are called the (coefficient) multipliers from X into Y . By the P P Closed Graph Theorem, each (λ ) [[ X, Y ]] gives rise to a continuous linear operator n n ∈ ∞ ∞ Λ : X Y : a zn λ a zn. → n 7→ n n n=0 n=0 X X The case of interest is when X is a space p and Y is a ‘small’ space of analytic functions, Nα such as q . We shall also consider multipliers from X into Y if one of the spaces is as Aβ above, and the other is a complete metrizable sequence space. Such multipliers are defined in an analogous fashion, and they also form a linear space [[X, Y ]] . The first main result of this paper is the following theorem. Theorem 2.1. Let α 1 and 1 p< . For every sequence (λn) in C, the > − 6 ∞ following are equivalent: (i) (λ ) [[ p, q ]] for some β 1 and some 0 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. 574 H. Jarchow and others as well as the case α = 0. It does not seem, however, that her approach can easily be modified to deal with the general case. Theorem 2.1 shows that our multipliers on p are largely independent of the range Nα space. (iv) reveals a certain independence of the domain space as well: p can be replaced Nα by q as long as (α+2)/p =(β+2)/q. We shall see below that certain power series spaces Nβ can be used instead of p. This complements a number of similar results presented in Nα the survey article [2] by Campbell and Leach. p The second result characterizes the duals ( )∗ of our spaces. These spaces are rather Nα small, as is to be expected. Let Fθ be the collection of all sequences (an)n in C such that, 2 θ/(θ+1) 1 for each k N, the sequence ( an exp( n /k))n belongs to ` . In a natural fashion, ∈ | | − Fθ isaFr´echet space (see below). We are interested in the case in which θ =(α +2)/p. Theorem 2.2. Let α, β 1 and 1 p, q< be given. > − 6 ∞ (a) p embeds densely and continuously into F . Nα (α+2)/p p (b) The duals of α and of F(α+2)/p coincide and can be identified with the space of N (α+2)/(α+2+p) all g (∆) whose Taylor coefficients satisfy bn = O(exp[ cn ]). The ∈H n p − action of g on f(z)= ∞ a z in is given by n=0 n Nα P 2π 1 it it g, f = anbn = lim g(re )f(re− )dt. h i r 1 2π n → − 0 X Z p p p (c) F induces on the Mackey topology µ( , ( )∗). (α+2)/p Nα Nα Nα (d) F is the Fr´echet envelope of p. (α+2)/p Nα (e) p is neither locally bounded nor locally convex; in particular, p = F . Nα Nα 6 (α+2)/p For a special case of (a), see [19, Corollaries 5.6 and 4.4]. Our third result is related to earlier investigations in [11] on composition operators with domain a space 1. Given an analytic map ϕ :∆ ∆, we write C for the Nα → ϕ associated composition operator f f ϕ, acting on appropriate spaces of functions. 7→ ◦ Theorem 2.3. With α and p as in Theorem 2.2 and ϕ :∆ ∆ an analytic function, → the following are equivalent statements: (i) C exists as a bounded operator p q for some β 1 and 0 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. Duality for some large spaces of analytic functions 575 Of course, ϕn is the usual multiplicative nth power of ϕ. Nuclear and order bounded operators on F -spaces will be defined later in this paper. Since, for appropriate range spaces, nuclearity implies order boundedness, Theorem 2.3 also provides access to some of the results in [11]. Whereas composition operators with domain +, and even , have already been investigated by Roberts and Stoll [14] and N N in particular by Jaoua [9] and by Choa et al.[3], the general case does not seem to have received attention yet. 3. Nuclear power series spaces of finite type Before we get to the proofs, we recall some basic facts on nuclear power series spaces of finite type. We refer to [10] for any unexplained terminology and notation on topological vector spaces. Let (πn)n be an unbounded increasing sequence of positive numbers. For each ∈N0 0 2 1/2 k N and (an) CN , put qk((an)) := [ ∞ an exp( πn/k)] . It is clear that each ∈ ∈ n=0 | | − F (k) := (a ) N0 : q ((a )) < is a Hilbert space with norm q , that F (k+1) F (k) n C k n P k { ∈ ∞} (k+1) (k) ⊂ (densely) with qk 6 qk+1; in fact, the canonical embedding F , F is even a (k) → nuclear (i.e. trace class) operator. Consequently, F := k F is a nuclear Fr´echet ∈N space with respect to the corresponding projective limit topology: a so-called nuclear T power series space of finite type (see [10, ch. 21]). In the canonical fashion, the dual (k) (k) 0 2 of F can be identified with (F )∗ = (an) CN : ∞ an exp(πn/k) < , and { ∈ n=0 | | ∞} the dual of F can be written F = (F (k)) . By nuclearity, the natural locally convex ∗ k ∗ P inductive limit topology on F ∗ is the strong topology β(F ∗,F)[10, 13.4]. Moreover, S § the strong dual of [F ∗,β(F ∗,F)] is the original space F . Multipliers from F into weighted Bergman spaces can be characterized as follows. 0 Proposition 3.1. Let (λn) CN be given. The following statements are equivalent. ∈ (i) For some 1 β< and 0 Proof. (iv) (iii) follows from the definition of F , whereas (iii) (ii) and (ii) (i) ⇒ ⇒ ⇒ are trivial. So we are left with proving that (i) implies (iv). 1 Fix β > 1. We settle the case q = 2 first. Set κn := (n!)− (β +1)...(β + n + 1), −n 2 so that (κnz )n is an orthonormal basis in β. Use Stirling’s formula to find a constant 2 β+2 A p C = C(β) such that κn 6 C(n +1) for all n. Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. 576 H. Jarchow and others 2 By continuity, the adjoint Λ∗ of the multiplier Λ generated by (λn) maps β into some (k) A (F )∗. The induced operator is also a multiplier, and is given by (λn). Accordingly, ( 0) there is a C>0 such that, for all (an) C N , ∈ ∞ π ∞ λ a 2 exp n C κ a 2. | n n| k 6 n| n| n=0 n=0 X X 2 2 1/2 This is equivalent to [( λ /κ ) exp(π /k)] < .So λ cκn exp( π /(2k)) n | n| n n ∞ | n| 6 − n holds for all n with some constant c, and our claim follows from what was observed above. P To settle the general case we only need to look at 0 The analogy with Theorem 2.1 is by no means accidental. The spaces F of relevance for our topic are obtained by choosing π = nθ/(θ+1) for 0 <θ< . Stoll [19] labelled n ∞ these spaces F and derived several of their properties. Note that (a ) a zn defines θ n 7→ n a continuous embedding of F into (∆) and that (∆) itself can be identified with the θ H H P nuclear power series space of finite type obtained by taking πn = n for each n. Hence, allowing θ = , we can write F = (∆). ∞ ∞ H 4. Proof of Theorem 2.1 We require three lemmas. They are essentially known but of different degrees of difficulty. Lemma 4.1. Let β > 1 and 0 n p Lemma 4.2. Let α 1, 1 p< .Iff(z)= ∞ a z belongs to , then > − 6 ∞ n=0 n Nα (α+2)/(α+2+P p) an = O(exp[o(n )]). Special cases of this are treated in [17] and [19]; the proof of the general case requires only minor changes. The last lemma is crucial and is technically the most difficult one. Up to an inessential modification (we require an extra parameter), an even more precise version was obtained by Beller. We refer to [1] for a proof. Lemma 4.3. Let a>0 and r0 > 0 be given. For 0 f(z) = exp[r/((1 z)a)]. − Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. Duality for some large spaces of analytic functions 577 Then there exists a constant K, depending only on a and r0, such that the Taylor coefficients an of f satisfy 1/(a+1) a/(a+1) a/(a+1) an K exp[r a− n ](n N0). > ∈ (The an are in fact positive: power series expansion!) Proof of Theorem 2.1. For notational convenience, we confine ourselves to the case in which α> 1. The (known) case α = 1 can be settled in a similar fashion; the use − − of the measure dt/2π just requires some natural changes. We start by showing that (i) implies (iv). For this, we look at the functions c(1 s)(α+2)/p f (z) = exp − 1. (s) (1 sz)(2α+4)/p − − By Lemma 4.2.2 in [21], there is a constant K0 = K0(α) such that (1 s)α+2 f p = [log(1 + f )]p dm cp − dm (z) k| (s)k| α,p | (s)| α 6 1 sz 2α+4 α Z∆ Z∆ | − | (1 z 2)α = cp(α + 1)(1 s)α+2 −| | dm(z) cpKp. − 1 sz 2α+4 6 0 Z∆ | − | Let Λ : p q be the multiplier induced by (λ ) and let C>0 be such that Nα →Aβ n Λ(g) 1 for all g p with g C. Hence, if we choose c C/K in our k kβ,q 6 ∈Nα k| k| α,p 6 6 0 definition of f , then Λ(f ) 1 for all 0 ∞ n ∞ n f(s)(z)= an,sz and f(z)= cnz n=0 n=0 X X n we see that an,s = s cn for all n N0 and 0 a C sn exp[L(1 s)(α+2)/(2α+4+p)n(2α+4)/(2α+4+p)]; n,s > 1 − here p (2α+4)/(2α+4+p) L = cp/(2α+4+p) . 2α +4 Next we choose 0 1 (α+2)/(2α+4+p) d Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. 578 H. Jarchow and others 3 p/(α+2+p) and let N0 N be so big that b 6 ( 4 )N0 .ForN > N0 and for all s satisfying p/(α+2+∈p) p/(α+2+p) 2t 3 dN − 1 s bN − , we get (since 1 t>e− for 0 where C := Ld(α+2)/(2α+4+p) 2b (> 0). 2 − By Lemma 4.1 there is a constant C such that λ a C n(β+2)/q for all n and s. 3 | n n,s| 6 3 It follows that C3 (β+2)/q (α+2)/(α+2+p) λN 6 N exp[ C2N ]. | | C1 − for N > N0. From here our claim is immediate. (iii) (ii) (i) are trivial. To prove that (iv) implies (iii) consider any function g(z)= ⇒ n ⇒ p n∞=0anz in α. By Lemma 4.2, there are a constant C0 and a null sequence of positive N (α+2)/(α+2+p) numbers b such that a C0 exp[b n ] for all n. It follows that P n | n| 6 n ∞ ∞ (α+2)/(α+2+p) λ a CC0 exp[(b c)n ] < . | n n| 6 n − ∞ n=0 n=0 X X Remark 4.4. There is no problem in extending Theorem 2.1 to analytic X-valued functions when X is a Banach space, i.e. to functions f :∆ X for which there are xn n → ∈ X such that f(z)= n∞=0 z xn on ∆. It is also possible to obtain a version of Theorem 2.1 when X is a quasi-Banach space. Let A(X) consist of all continuous functions f : ∆¯ X P → that are analytic on ∆ (in the above sense). In a natural fashion, A(X) is a quasi-Banach space. An argument of Eoff [8], which is based on a theorem of Kalton [12], can be used to show that a sequence (x ) is a multiplier from p into A(X) (definition as before) if n Nα and only if there are constants C, c > 0 such that x C exp[ cn(α+2)/(α+2+p)] for all k nk 6 − n. The problem here is to overcome the fact that Cauchy’s formula, on which Lemma 4.1 relies, is not available for analytic functions with values in a quasi-Banach space. Kalton’s theorem just provides an appropriate substitute for Lemma 4.1 for functions in A(X). 5. Proof of Theorem 2.2 p p We begin by showing that α is in fact a linear subspace of F(α+2)/p. Let f α be n N ∈N given, f(z)= n∞=0 anz . By Lemma 4.2, there are a null sequence of scalars sn > 0 and a constant C such that a C exp(s n(α+2)/(α+2+p)). Our claim follows, since, for P | n| 6 n each k N, ∈ ∞ n(α+2)/(α+2+p) ∞ 1 a exp C exp s n(α+2)/(α+2+p) < . | n| − k 6 n − k ∞ n=0 n=0 X X Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. Duality for some large spaces of analytic functions 579 It is clear that the resulting embedding p , F has dense range (polynomials) Nα → (α+2)/p and is continuous (closed graph theorem). Accordingly, F(∗α+2)/p can be identified with p a linear subspace of ( α)∗. N p p n To show that this is all of ( )∗, take any g ( )∗. Clearly, the monomials z Nα ∈ Nα belong to p. Given s>0 we can choose λ>0 in such a way that log(1 + λzn ) s.It Nα | | 6 follows that (zn) is a bounded sequence in p, and so the λ := g(zn) form a bounded Nα n sequence of scalars. n p Take any f(z)= n∞=0 anz from α. Clearly, lim w 1 f fw α,p = 0, where N | |→ k| − k| f (z)=f(wz) for w, z ∆. It follows that w P∈ m n ∞ n g(fw) = lim λnanw = λnanw , m →∞ n=0 n=0 X X and so ∞ n g(f) = lim g(fw) = lim λnanw . w 1 w 1 | |→ | |→ n=0 X p Since g ( α)∗, the right-hand side defines an element of H∞. Moreover, (λn) multi- p∈ N q plies α into H∞ (which embeds into any β). By Theorem 2.1, there is a constant N (α+2)/(α+2+p)A c>0 such that supn λn exp[cn ] < , which signifies that (λn) belongs to (k) | | ∞ (F(α+2)/p )∗ for sufficiently large integers k (see 3 for notation). Therefore, the corre- n § sponding analytic function n∞=0 λnz , which is nothing but g, belongs to F(∗α+2)/p, and n p for f(z)= ∞ a z in we obtain what was asserted: n=0 n NPα 2π P 1 it it anλn = lim f(re )g(re− )dt. r 1 2π n → 0 X Z This proves (a) and (b). Statements (c) and (d) are easy consequences. Let X be any metrizable topological vector space, let τ be its topology and a defining F -norm. k|·k| The sets U(s):= x X : x s , s>0, form a 0-basis for τ. Their convex hulls, { ∈ k| k| 6 } conv U(s), form a 0-basis for the finest locally convex topology on X that is coarser than τ; we label it τ0.IfY is any locally convex space, then every continuous linear map [X, τ] Y is also continuous as a map [X, τ ] Y . In particular, [X, τ] and [X, τ ] have → 0 → 0 the same dual, X∗. τ0 has a countable 0-basis since τ has. It is metrizable whenever X∗ separates the points in X. In that case, it coincides with the Mackey topology µ(X, X∗) determined by the dual pairing X, X∗ . The completion of [X, µ(X, X∗)] is a Fr´echet space, the h i so-called Fr´echet envelope of X. Typically, this envelope is strictly bigger than X:ifτ is complete but not locally convex, then τ0 cannot be complete. p Again let X be , let β(X∗,X) be the usual strong topology on X∗ and let X∗∗ be Nα the dual of [X∗,β(X∗,X)]. We endow X∗∗ with the strong topology β(X∗∗,X∗). It is a simple consequence of our considerations and standard duality theory that X∗∗ can naturally be identified with F(α+2)/p. Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. 580 H. Jarchow and others p p p To prove (e), observe that if is locally bounded, then µ( , ( )∗) is locally Nα Nα Nα bounded too. But this is impossible, since we are dealing with an infinite-dimensional nuclear space. p is properly contained in F : it is in fact easy to construct sequences in Nα (α+2)/p F that do not satisfy the conclusion of Lemma 4.2. Thus p cannot be locally (α+2)/p Nα convex. Remark 5.1. By the extension of Theorem 2.1 mentioned in Remark 4.4, even every continuous linear map from p into a quasi-Banach space X admits an extension to a Nα continuous operator F X. It follows that p even fails to be locally pseudocon- (α+2)/p → Nα vex: no matter how we choose a sequence (s )in(0, 1], p does not admit a 0-basis (U ) n Nα n n consisting of s -convex sets. In other words, p cannot be represented as a projective n Nα limit of quasi-Banach spaces. Since locally bounded spaces are locally s-convex for some 0 ∞ ux = x∗ ,x y for all x X h n i n ∈ n=1 X Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. Duality for some large spaces of analytic functions 581 (convergent series in Y ), where (xn∗ ) and (yn) are sequences in X∗ and Y , respectively, such that, for some s>0, all xn∗ s := supx U(s) xn∗ ,x exist and satisfy | | ∈ |h i| ∞ x∗ y < . | n|sk nkY ∞ n=1 X Since x∗ = sup x∗ ,x : x conv U(s) for each s>0, such an operator is also | n|s {|h n i| ∈ } nuclear as an operator [X, τ ] Y , and it even admits a nuclear extension mapping the 0 → completion of [X, τ0]intoY . Here τ0 is as before. Now let X be a space p, α 1, p 1, and let u be a continuous linear map Nα > − > from p to a Banach space Y . Its continuous extension,u ˜ : F Y , is a nuclear Nα (α+2)/p → operator in the usual sense, since the domain is a nuclear locally convex space. Let ϕ :∆ ∆ be analytic and let β 1, 0 Proof of Theorem 2.3. By Theorem 2.1, (iii), (iv) and (v) are equivalent. We are going to prove (i) (ii) (iii) (i). ⇒ ⇒ ⇒ p q (i) (ii). Suppose that Cϕ : α β exists as a bounded operator for some 0 < ⇒ N →Ap q0 q< . We want to show that C ( ) , for any q q0 Since we may take q0 1, C maps F into , and so > ϕ (α+2)/p Aβ C : p , F q0 , q ϕ Nα → (α+2)/p →Aβ →Aβ is nuclear. (ii) (iii). Recall from 3 that F(α+2)/p is the projective limit of the Hilbert spaces (k) ⇒ § p 1 F . Suppose that Cϕ exists as a map . Then there is a k N such (α+2)/p Nα →Aβ ∈ that C extends to a bounded operator u : F (k) 1 .Fora =(a ) F (k) ϕ (α+2)/p →Aβ n ∈ (α+2)/p ( F (2k)), with ⊂ (α+2)/p ∞ n(α+2)/(α+2+p) C2 = exp , − 2k n=0 X Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 17:12:22, subject to the Cambridge Core terms of use. 582 H. Jarchow and others we have ∞ ∞ n(α+2)/(α+2+p) q (a e )= a exp Cq (a) < ; k n n | n| − 2k 6 2k ∞ n=0 n=0 X X (k) n here en is the nth standard unit vector in F(α+2)/p. But uen = ϕ for each n, so that, by continuity of u, the sequence ( a ϕn ) belongs to `1. This holds for every (a ) for k n kβ,1 n n which ∞ n(α+2)/(α+2+p) a 2 exp < . | n| − k ∞ n=0 X (α+2)/p n 1 Hence sup exp[n p/(2k)] ϕ < , and taking c =(2k +1)− we get what we n k kβ,1 ∞ wanted. (iii) (i). By our hypothesis, ⇒ ∞ n(α+2)/pp c2 := exp ϕn 2 k k k kβ,1 n=0 X is finite for some k. It follows that C f c q (f) for all f F : C maps k ϕ kβ,1 6 k k ∈ (α+2)/p ϕ F , and a fortiori p, continuously into 1 . (α+2)/p Nα Aβ As mentioned above, the 1-part of Theorem 2.3 can also be derived from Theorem 1.4 Nα in [11]. But an operator between Hilbert function spaces is order bounded if and only if it is Hilbert–Schmidt, so that, conversely, Theorem 1.4 in [11] appears as a consequence of Theorem 2.3 as well. Acknowledgements. H.J. was supported by the Swiss National Funds. V.M. was supported in part by project pb96-0758 and by the Universidad Politecnica de Valencia. J.X. was supported by the AvH Foundation of Germany and by the NNSF and SEC of the People’s Republic of China. 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