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 <p< . Our results N ∞ generalize and unify earlier characterizations obtained by Eoff for α = 0 and α = 1, and by Yanigahara − for the Smirnov class. p Each α is a complete metrizable topological vector space (in fact, even an algebra); it fails to be N locally bounded and locally convex but admits a separating dual. Its bidual will be identified with a p specific nuclear power series space of finite type; this turns out to be the ‘Fr´echet envelope’ of α as N well. θ The generating sequence of this power series space is of the form (n )n N for some 0 <θ<1. For 1 ∈ example, the θs in the interval ( , 1) correspond in a bijective fashion to the Nevanlinna classes α, 2 N α> 1, whereas the θs in the interval (0, 1 ) correspond bijectively to the Hardy–Orlicz spaces LHp, − 2 1 <p< . By the work of Yanagihara, θ = 1 corresponds to +. ∞ 2 N As in the work by Yanagihara, we derive our results from characterizations of coefficient multipliers p from α into various smaller classical spaces of analytic functions on ∆. N 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<p< , consists of all f (∆) such that ∞ ∈H 2π 1/p it p dt f p := sup (log(1 + f(re ) )) k kLH | | 2π 06r<1Z0 is finite. Again we get a complete metrizable topological vector space (also an algebra) p + p with F -norm p (see Stoll [19], where LH is denoted (Log H) ). k·kLH The case p = 1 is a bit delicate. The Nevanlinna class consists of all f (∆) such N ∈H that 2π dt sup log(1 + f(reit) ) < . | | 2π ∞ 06r<1 Z0 This is still an algebra on which a complete metric is defined by 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 <q< . The corresponding weighted Bergman space is defined to q −q ∞ q be β := L (mβ) (∆). This is a closed subspace of L (mβ). We denote the canonical A ∩H q norm (q-norm if q<1) on β by β,q. Again, it will be convenient to include the q A k·k q q classical Hardy spaces H by labelling them 1. Clearly, β embeds continuously into p A− A β , for any 0 <q< and 1 6 p< . N ∞ ∞ p p Point evaluations are easily seen to be continuous on any , so that the dual ( )∗ Nα Nα separates points. This dual has been characterized by Yanagihara [20] in the case in which α = 1, p = 1. Using somewhat different methods, Eoff [7] extended this to − p q characterize the duals (LH )∗ for p>1 and ( )∗ for 1 q< . In this paper, we N0 6 ∞ settle the case of arbitrary p by returning to Yanagihara’s approach. However, we need Nα 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 <p< ). ∞ 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 <q< ; n ∈ Nα Aβ > − ∞ (ii) (λ ) [[ p, q ]] for all β 1 and all 0 <q< ; n ∈ Nα Aβ > − ∞ (iii) (λ ) [[ p,W]] ; and n ∈ Nα (iv) λ = O(exp[ cn(α+2)/(α+2+p)]) for some c>0. n − Here W is the Wiener algebra consisting of all f (∆) whose Taylor coefficients fˆ(n) ∈H form an `1-sequence. It is a Banach algebra with respect to the norm f = fˆ(n) . k k n | | Other Banach algebras, such as the disk algebra and even appropriate F -algebras, can P be used instead. In the case in which p = 1, we are dealing with the function α (α+2)/(α+3), which 7→ maps the interval [ 1, )onto[1 , 1). If α = 1, we are dealing with p 1/(p + 1), and − ∞ 2 − 7→ [1, ) is mapped onto (0, 1 ]. As mentioned before, Eoff [7] has settled this case earlier, ∞ 2 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.