Let Np(1<p<∞) be the Privalov class of holomorphic functions on the open unit disk D in the complex plane. The space Np equipped with the topology given by the metric dp defined by dp(f,g)=(∫02π(log(1+|f∗(eiθ)-g∗(eiθ)|))p(dθ/2π))1/p, f,g∈Np, becomes an F-algebra. For each p>1, we also consider the countably normed Fréchet algebra Fp of holomorphic functions on D which is the Fréchet envelope of the space Np. Notice that the spaces Fp and Np have the same topological duals. In this paper, we give a characterization of bounded subsets of the spaces Fp and weakly bounded subsets of the spaces Np with p>1. If (Fp)∗ denotes the strong dual space of Fp and Npw∗ denotes the space Sp of complex sequences γ={γn}n satisfying the condition γn=Oexp-cn1/(p+1), equipped with the topology of uniform convergence on weakly bounded subsets of Np, then we prove that Fp∗=Npw∗ both set theoretically and topologically. We prove that for each p>1Fp is a Montel space and that both spaces Fp and (Fp)∗ are reflexive.

1. Introduction and Preliminaries

Let D denote the open unit disk in the complex plane and let T denote the boundary of D.

The Privalov class Np (1<p<∞) consists of all holomorphic functions f on D for which (1)sup0<r<1∫02πlog+freiθpdθ2π<+∞.These classes were firstly considered by Privalov in [1, page 93], where Np is denoted as Aq.

Notice that, for p=1, condition (1) defines the Nevanlinna class N of holomorphic functions in D. Recall that the Smirnov class N+ is the set of all functions f holomorphic on D such that (2)limr→1∫02πlog+freiθdθ2π=∫02πlog+f∗eiθdθ2π<+∞,where f∗ is the boundary function of f on T; that is, (3)f∗eiθ=limr→1-freiθis the radial limit of f which exists for almost every eiθ. We denote by Hq(0<q≤∞) the classical Hardy space on D. It is known (see [2, 3]) that (4)Nr⊂Npr>p,⋃q>0Hq⊂⋂p>1Np,⋃p>1Np⊂N+⊂N,where the above containment relations are proper.

Notice that Privalov in [1, page 98] established the inner-outer factorization theorem for the spaces Np(1<p<∞) (for another proof see [4]). The study of the spaces Np(1<p<∞) on the unit disk was continued in 1977 by Stoll [5] (with the notation (log+H)α in [5]). The topological and functional properties of these spaces were extensively studied in [6]. Different topologies on the spaces Np were considered and compared in [7] with related applications. Complex-linear isometries of Np are investigated in [8]. Motivated by some results of Matsugu [9], in [10] the structure of closed weakly dense ideals in Privalov spaces Np(1<p<∞) was studied. Notice that the structure of maximal ideals of the algebras Np and their Fréchet envelopes Fp(1<p<∞) was investigated in [11]. The interpolation problems for the spaces Np are treated in [12].

Stoll [5, Theorem 4.2] showed that the space Np (with the notation (log+H)α in [5]) with the topology given by the metric dp defined by (5)dpf,g=∫02πlog1+f∗eiθ-g∗eiθpdθ2π1/p,f,g∈Np,becomes an F-algebra.

Recall that the function d1=d defined on the Smirnov class N+ by (5) with p=1 induces the metric topology on N+. Yanagihara [13] showed that, under this topology, N+ is an F-space.

Recently, in [14] the author of this note characterized some topological properties of the spaces Np(1<p<∞). For these purposes, the fact that the metric ρp defined on Np as (6)ρpf,g=∫02πlogp1+Mf-gθdθ2π1/p,with f,g∈Np and (7)Mfθ=sup0⩽r<1freiθ,induces on the space Np the same topological structure as the initial metric dp given by (5) (see [14, Theorem 16]) is used.

Furthermore, in connection with the spaces Np(1<p<∞), Stoll [5] (also see [15] and [10, Section 3]) also studied the spaces Fq(0<q<∞) (with the notation F1/q in [5]), consisting of those functions f holomorphic on D for which (8)limr→11-r1/qlog+M∞r,f=0,where (9)M∞r,f=maxz≤rfz.Stoll [5, Theorem 3.2] proved that the space Fq with the topology given by the family of seminorms ·q,cc>0 defined for f∈Fq as (10)fq,c≔∑n=0∞anexp-cn1/q+1<∞is a countably normed Fréchet algebra.

Here, as always in the sequel, we will need some Stoll’s results concerning the spaces Fq only with 1<q<∞, and, hence, we will assume that q=p>1 be any fixed number.

Theorem 1 (see [<xref ref-type="bibr" rid="B5">5</xref>, Theorem 2.2]).

Suppose that f(z)=∑n=0∞anzn is a holomorphic function on D. Then the following statements are equivalent:

f∈Fp.

There exists a sequence {cn}n of positive real numbers with cn→0 such that (11)an≤expcnn1/p+1,n=0,1,2,….

For any c>0, (12)fp,c≔∑n=0∞anexp-cn1/p+1<∞.

Remark 2.

Notice that, in view of Theorem 1 ((a)⇔(c)), by (10) the family of seminorms ·p,cc>0 on Fp is well defined.

Recall that a locally convex F-space is called a Fréchet space, and a Fréchet algebra is a Fréchet space that is an algebra in which multiplication is continuous.

Notice that the space Np is not locally convex (see [15, Theorem 4.2] and [16, Corollary]), and, hence, Np is properly contained in Fp. Moreover, Np is not locally bounded (see [17, Theorem 1.1]). The most important connection between spaces Np and Fp is given by the following result.

Theorem 3 (see [<xref ref-type="bibr" rid="B5">5</xref>, Theorem 4.3]).

For any fixed p>1 the following assertions hold:

Np is a dense subspace of Fp.

The topology on Fp defined by the family of seminorms (10) is weaker than the topology on Np given by the metric dp defined by (5).

Remark 4.

For p=1, the space F1 has been denoted by F+ and has been studied by Yanagihara in [13, 18]. It was shown in [13, 18] that F+ is actually the containing Fréchet space for N+; that is, N+ with the initial topology embeds densely into F+, under the natural inclusion, and F+ and the Smirnov class N+ have the same topological duals.

Observe that the space Fp topologised by the family of seminorms ·p,cc>0 given by (10) is metrizable by the metric λp defined as (13)λpf,g=∑n=1∞2-nf-gp,1/np/p+11+f-gp,1/np/p+1,f,g∈Fp.The following Stoll’s result describes the topological dual of the space Fp.

Theorem 5 (see [<xref ref-type="bibr" rid="B5">5</xref>, Theorem 3.3]).

If γ is a continuous linear functional on Fp, then there exists a sequence {γn}n of complex numbers with γn=O(exp-cn1/(p+1)), for some c>0, such that (14)γf=∑n=0∞anγn,where f(z)=∑n=0∞anzn∈Fp, with convergence being absolute. Conversely, if {γn}n is a sequence of complex numbers for which (15)γn=Oexp-cn1/p+1,then (14) defines a continuous linear functional on Fp.

Let us recall that if X=(X,τ) is an F-space whose topological dual (the set of all continuous linear functionals on X) X∗ separates the points of X, then its Fréchet envelope X^ is defined to be the completion of the space (X,τc), where τc is the strongest locally convex (necessarily metrizable) topology on X that is weaker than τ. In fact, it is known that τc is equal to the Mackey topology of the dual pair (X,X∗), that is, to the unique maximal locally convex topology on X for which X still has dual space X∗ (see [19, Theorem 1]). For each metrizable locally convex topology τ on X, (X,τ) is a Mackey space; that is, τ coincides with the Mackey topology of the dual pair (X,X∗) (see [20, Corollary 22.3, page 210]).

Eoff [15, the proof of Theorem 4.2] showed that the topology of Fp, p>1 (resp., F1=F+), is stronger than that of the Fréchet envelope of Np (resp., N+). As an immediate consequence of this result, we obtain the following statements.

Theorem 6 (see [<xref ref-type="bibr" rid="B15">15</xref>, Theorem 4.2, the case <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M172">
<mml:mi>p</mml:mi>
<mml:mo>></mml:mo>
<mml:mn mathvariant="normal">1</mml:mn></mml:math>
</inline-formula>]).

For each p>1, Fp is the Fréchet envelope of Np.

Theorem 7 (see [<xref ref-type="bibr" rid="B21">21</xref>, Theorem 2]; also see [<xref ref-type="bibr" rid="B14">14</xref>, Theorem 17]).

The spaces Np and Fp have the same dual spaces in the sense that every continuous linear functional on Fp (given by (14)) is restricted to one on Np, and every continuous linear functional on Np extends continuously to one on Fp.

Hence, the dual spaces of Np and Fp can be identified with the collection Sp of complex sequences {γn} satisfying the growth condition (15).

Remark 8.

Theorem 7 is proved in [21, Theorem 1] directly, by using the characterization of multipliers from Nq to H∞. Notice also that the dual space of the Smirnov class N+ is described by Yanagihara in [13] and applying another method by McCarthy in [22].

Remark 9.

Recall that we may introduce the weak topology on Np in the usual way. The basic weak neighborhoods of zero are defined by Vφ1,…,φn;ɛ=f∈Np:|φi(f)|<ɛ,i=1,…,n, where ɛ>0 and n∈N are arbitrary, and φ1,…,φn are arbitrary continuous linear functionals on Np. The weak topology of Np is locally convex, and, hence, by [20, Corollary 17.3, page 154], a linear functional on Np is weakly continuous if and only if it is continuous with respect to the initial metric topology dp.

In Section 2, we give a characterization of bounded subsets of the spaces Fp(1<p<∞) (Theorem 10). As an application, we obtain a characterization of weakly bounded subsets of the spaces Np (Theorem 11). In Section 3, we prove that Fp∗=Npw∗ both set theoretically and topologically (Theorem 12). Here (Fp)∗ denotes the strong dual space of Fp, and Npw∗ is the space Sp of complex sequences γ={γn}n satisfying the growth condition (15), equipped with the topology of uniform convergence on weakly bounded subsets of Np. Finally, we prove that Fp is a Montel space (Theorem 13) and that both spaces Fp and (Fp)∗ are reflexive (Theorem 14).

2. Bounded Subsets of the Spaces <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M210">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>F</mml:mi></mml:mrow>
<mml:mrow>
<mml:mi>p</mml:mi></mml:mrow>
</mml:msup></mml:mrow>
</mml:math></inline-formula>

The following result characterizes bounded sets of the space Fp.

Theorem 10.

Let p>1 and let E be a subset of Fp. Then the following assertions are equivalent:

E is a bounded subset of Fp.

E is a relatively compact subset of Fp.

There exists a constant A>0 depending on E and a sequence {cn} of positive real numbers such that cn↓0 and (16)an≤Aexpcnn1/p+1foreachfunctionfz=∑n=0∞anzn∈E.

Proof.

(ii)⇒(i): it follows immediately from the fact that every relatively compact set in a topological vector space is bounded.

(iii)⇒(i): for given c>0 choose a positive integer m such that cn<c/2 for each n>m. Then by condition (iii) for every function f(z)=∑n=0∞anzn∈E we have (17)fp,c=∑n=0∞anexp-cn1/p+1≤A∑n=0∞expcn-cn1/p+1≤A∑n=0mexpcn-cn1/p+1+∑n=m+1∞exp-c2n1/p+1=Kc=K,where K is a constant depending only on c. Hence, E is a bounded set in every normed space (Fp,·p,c), c>0, and, thus, E is a bounded subset of Fp.

(i)⇒(iii): suppose that E is a bounded set in Fp. For arbitrary c>0 and η>0 the set V defined as (18)V=g∈Fp:gp,c<ηis a neighbourhood of 0 in the space Fp. Since, by the assumption, E is a bounded set in Fp, there exists α>0 such that αE⊂V. This yields (19)∑n=0∞αanexp-cn1/p+1<ηforeachfz=∑n=0∞anzn∈E.From (19) we find that (20)an≤ηαexpcn1/p+1foreachn∈N.Then putting (21)an∗=supanf:fz=∑n=0∞anfzn∈Einto (20), we find that (22)an∗≤ηαexpcn1/p+1foreachn∈N.Since c>0 is arbitrary, from (22) we immediately have (23)an∗=Oexpon1/p+1,which immediately yields (16).

(iii)⇒(ii): since in a metric space compactness and sequential compactness are equivalent, it is necessary to show that a set E satisfying condition (16) is a relatively compact subset of Fp; that is, for every sequence {fn}⊂E there exists a subsequence of {fn} which is convergent in Fp. We will inductively construct such a subsequence of a sequence {fn} in the following way. Take (24)fnz=∑k=0∞aknzkforeachn∈N.Since, by assumption (16), (25)a0n≤Aexpc0foreachn∈N,it follows that there is a subsequence {fn(0)} of {fn} (fn(0) denotes the nth term of this subsequence) such that the appropriate subsequence {a0n(0)} of a sequence {a0n} is convergent; assume that (26)limn→∞a0n0=a0.Take fi0=f0(0); that is, denote by fi0 the first term of the obtained subsequence {fn(0)}.

Since, by assumption (16), we have (27)a1n≤Aexpc1foreachn∈N,it follows that there exists a subsequence {fn(1)} of {fn(0)} (fn(1) denotes the nth term of this subsequence) such that the corresponding subsequence {a1n(1)} of a sequence {a0n(0)} is convergent; assume that (28)limn→∞a1n1=a1.Put fi1=f1(1); that is, denote by fi1 the first term of the obtained subsequence {fn(1)} which is different from fi0.

By continuing the above diagonal procedure and taking into account that in view of (16) and (24) (29)akn≤Aexpckk1/p+1foreachn∈N,after s+1 steps we obtain a subsequence {fn(s)} of {fn(s-1)} (fn(s) denotes the nth term of this subsequence) such that the corresponding subsequence {asn(s)} of {a(s-1)n(s-1)} converges; assume that (30)limn→∞asns=as.Take fis=fs(s); that is, denote by fis the first term of the obtained subsequence {fn(s)} which is different from {fn(j)} for all j=0,1,…,s-1.

In this way we have obviously constructed a subsequence {fin} of a sequence {fn} such that (31)finz=∑k=0∞aknnzkforeachn∈N.Furthermore, by the above construction, {akn(k)}n is a convergent sequence for any fixed k∈N; assume that (32)limn→∞aknk=akforeachfixedk∈N.Let f be a function defined as (33)fz=∑k=0∞akzk,where the coefficients ak are defined by (26) and (32). Since by (29) and (30) we have (34)ak≤Aexpckk1/p+1foreachk∈N,it follows from Theorem 1 that f belongs to the space Fp.

It remains to show that fin→f in Fp as n→∞. Let c>0 be any fixed positive real number. Choose a nonnegative integer k1=k1(c) such that (35)ck<c2foreachk>k1,and also choose a nonnegative integer k2=k2(c) for which (36)∑k=k2+1∞exp-c2k1/p+1<ɛ4A.Take k0=max{k1,k2}. For every k∈{0,1,…,k0} choose nk∈N such that (37)aknn-ak<ɛ2k0+1expck1/p+1foreachn>nk.Take m=max{nk:k=0,1,…,k0}. Then from (32), (33), (35), (36), and (37) it follows that for each n>m there holds (38)fin-fp,c=∑k=0∞aknn-akexp-ck1/p+1=∑k=0k0aknn-akexp-ck1/p+1+∑k=k0+1∞aknn-akexp-ck1/p+1≤ɛ2k0+1k0+1+2A∑k=k0+1∞expck-ck1/p+1≤ɛ2+2A∑k=k2+1∞exp-c2k1/p+1≤ɛ2+ɛ2=ɛ.Therefore, fin→f in the space (Fp,·p,c). From this and the fact that c>0 is arbitrary we conclude that fin→f in Fp. Hence, E is a relatively compact subset of Fp. This completes the proof.

Theorem 11.

Let E be a subset of Np. E is a weakly bounded subset of Np if and only if there is a constant A>0 depending on E and a sequence {cn} of positive real numbers with cn↓0 such that (39)an≤Aexpcnn1/p+1foreachfz=∑n=0∞anzn∈E.

Proof.

The proof follows immediately from Theorem 7, the equivalence (i)⇔(iii) of Theorem 10, and the well known fact that in every locally convex topological vector space a set is bounded if and only if it is weakly bounded (see, e.g., [23, page 68]).

3. The Dual Spaces of the Spaces <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M342">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>F</mml:mi></mml:mrow>
<mml:mrow>
<mml:mi>p</mml:mi></mml:mrow>
</mml:msup></mml:mrow>
</mml:math></inline-formula> and <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M343">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>N</mml:mi></mml:mrow>
<mml:mrow>
<mml:mi>p</mml:mi></mml:mrow>
</mml:msup></mml:mrow>
</mml:math></inline-formula>

Let X be a topological vector space over the field C. Consider its strong dual space X∗ which consists of all continuous linear functionals f:X→C and is equipped with the usual strong topology β(X∗,X). Recall that in the case when X is a locally convex space, the strong topology β(X∗,X) on X∗ coincides with the topology of uniform convergence on bounded subsets in X. In this case the topology β(X∗,X) coincides with the topology of uniform convergence on bounded sets in X, that is, with the topology on X∗ generated by the seminorms of the form (40)φB=supx∈Bφx,φ∈X∗,where B runs over the family of all bounded sets in X (see, e.g., [24, § 21, page 21]).

The space X∗ is a locally convex topological vector space, so one can consider its strong dual space X∗∗≔(X∗)∗ which is called strong bidual space for X. More precisely, the strong bidual space X∗∗ for X consists of all continuous linear functionals h:X∗→C and is equipped with the strong topology β(X∗∗,X∗). Furthermore, X is called reflexive if the evaluation map J:X→(X)∗∗ defined as J(x)(f)=f(x) (x∈X, f∈X∗) is surjective and continuous (in this case J is an isomorphism of topological vector spaces) (see [20, page 189]). Accordingly, X is a reflexive space if and only if X coincides with the continuous dual of its continuous dual space X∗, both as linear space and as topological space.

In particular, in the case of locally convex space Fp, the strong topology β((Fp)∗,Fp) coincides with the topology on (Fp)∗=Sp (see Theorem 7) generated by the family of seminorms of the form (41)γB=supf∈Bγf,γ∈Fp∗=Sp,where B is an arbitrary bounded subset of Fp.

Furthermore, denote by Npw∗ the space Sp of complex sequences γ={γn}n satisfying the growth condition (15), equipped with the topology of uniform convergence on weakly bounded subsets of Np. (Let us recall that a subset E⊂Np is weakly bounded if it is bounded with respect to the weak topology on Np described in Remark 9.) More precisely, this topology on the space Sp is defined by the family of seminorms of the form (42)γE′=supf∈Eγf,γ∈Np∗=Sp,where E is an arbitrary weakly bounded subset of Np.

Notice that (see Theorems 5 and 7) for γ={γn}n=0∞∈Sp and f(z)=∑n=0∞anzn∈Fp (or f∈Np) (43)γf=∑n=0∞anγn.

Then we have the following result which is analogous to Yanagihara’s result [25, Theorem 3] concerning the spaces F+ and N+.

Theorem 12.

For each p>1Fp∗=Npw∗ both set theoretically and topologically.

Proof.

Proof follows immediately from Theorem 7, the equivalence (i)⇔(iii) of Theorems 10 and 11 in view of (41)–(43).

A locally convex topological vector space is called barrelled if every closed, absolutely convex, absorbing set is a neighborhood of zero. It is known that every F-space is barreled (see [24, page 263 § 21, the assertion (3)]). The most important fact about barrelled spaces is that every pointwise bounded family of continuous linear functionals is equicontinuous. Furthermore, each barrelled space is a Mackey space (see, e.g., [20, pages 171–173]). Recall that a Montel space is a barrelled topological vector space in which every closed bounded set is compact. It is known that every Montel space is reflexive (e.g., see [24, page 369]).

Theorem 13.

Fp is a Montel space for each p>1.

Proof.

As noticed above, every F-space is barreled (see [24, page 263 § 21, 5.(3)]). Furthermore, by Theorem 10, every closed bounded subset of Fp is compact. Therefore, Fp is a Montel space.

Theorem 14.

The space Fp is reflexive. Hence, (Fp)∗ is reflexive.

Proof.

The first assertion follows from Theorem 13 and the fact that every Montel space is reflexive (see [24, page 369, the assertion (1)]). The second assertion follows from the first assertion and the fact that the strong dual of a reflexive F-space is also reflexive (see [20, page 303, the assertion (5)]).

Theorem 15.

For each p>1Fp∗∗=Fp.

Proof.

The assertion follows immediately from Theorem 14.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

PrivalovI. I.MeštrovićR.PavićevićŽ.Remarks on some classes of holomorphic functionsMochizukiN.Algebras of holomorphic functions between Hp and N*EoffC. M.A representation of Nα+ as a union of weighted Hardy spacesStollM.Mean growth and Taylor coefficients of some topological algebras of analytic functionsMeštrovićR.MeštrovićR.PavićevićŽ.Topologies on some subclasses of the Smirnov classIidaY.MochizukiN.Isometries of some F-algebras of holomorphic functionsMatsuguY.Invariant subspaces of the Privalov spacesMeštrovićR.PavićevićŽ.Weakly dense ideals in Privalov spaces of holomorphic functionsMeštrovićR.Maximal ideals in some F-algebras of holomorphic functionsMeštrovićR.ŠušićJ.Interpolation in the spaces Np(1<p<∞)YanagiharaN.Multipliers and linear functionals for the class N^{+}MeštrovićR.On F-algebras Mp(1<p<∞) of holomorphic functionsEoffC. M.Fréchet envelopes of certain algebras of analytic functionsMeštrovićR.The failure of the Hahn-Banach properties in Privalov spaces of holomorphic functionsMeštrovićR.PavićevićŽ.A topological property of Privalov spaces on the unit diskYanagiharaN.The containing Fréchet space for the class N+ShapiroJ. H.Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spacesKelleyJ. E.NamiokaI.MeštrovićR.SubbotinA. V.Multipliers and linear functionals for Privalov spaces of holomorphic functions in the discMcCarthyJ. E.Common range of co-analytic Toeplitz operatorsRudinW.KötheG.YanagiharaN.The second dual space for the space N+