# OnF-Algebras Mp(1<p<∞) of Holomorphic Functions

Authors: Romeo Meštrović
Journal: The Scientific World Journal
 (2014)

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## Abstract

We consider the classesMp(1<p<∞) of holomorphic functions on the open unit disk 𝔻 in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric ρp defined by ρp(f,g)=f-gp=∫02π‍logp1+Mf-gθdθ/2π1/p, with f,g∈Mp and Mfθ=sup0⩽r<1⁡f(reiθ), becomes an F-space. By a result of Stoll (1977), the Privalov space Np(1<p<∞) with the topology given by the Stoll metric dp is an F-algebra. By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric ρp). Furthermore, we give a characterization of bounded subsets of the spaces Mp. Moreover, we give the examples of bounded subsets of Mp that are not relatively compact.


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## Body

## 1. Introduction and Preliminaries

Let𝔻 denote the open unit disk in the complex plane and let 𝕋 denote the boundary of 𝔻. Let Lq(𝕋)  (0<q≤∞) be the familiar Lebesgue spaces on the unit circle 𝕋.Following Kim ([1, 2]), the class  M consists of all holomorphic functions f on 𝔻 for which
(1)∫02πlog⁡+⁡Mf(θ)dθ2π<∞,
where log⁡+⁡|a|=max⁡⁡{log⁡⁡a,0} and
(2)Mf(θ)=sup⁡0⩽r<1⁡|f(reiθ)|
is the maximal radial function of f. The Privalov class  Np(1<p<∞) consists of all holomorphic functions f on 𝔻 for which
(3)sup⁡0<r<1⁡∫02π(log⁡+⁡|f(reiθ)|)pdθ2π<+∞.
These classes were firstly considered by Privalov in [3, page 93], where Np is denoted as Aq.Notice that forp=1, the condition (3) 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
(4)lim⁡r→1⁡∫02πlog⁡+⁡|f(reiθ)|dθ2π=∫02πlog⁡+⁡|f*(eiθ)|dθ2π<+∞,
where f* is the boundary function of f on 𝕋; that is,
(5)f*(eiθ)=lim⁡r→1-⁡f(reiθ)
is 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 [4, 5]) that
(6)Nr⊂Np(r>p),⋃q>0‍Hq⊂⋂p>1‍Np,⋃p>1‍Np⊂M⊂N+⊂N,
where the above containment relations are proper.The study of the spacesNp(1<p<∞) was continued in 1977 by Stoll [6] (with the notation (log⁡+H)α in [6]). Further, the topological and functional properties of these spaces were studied in [4, 5, 7–14]; typically, the notation varied and these spaces are called the Privalov spaces in [12–15].It is well known [16, page 26] that a function f∈N+ if and only if f=IF, where I is an inner function on 𝔻 and F is an outer function given by
(7)F(z)=exp⁡⁡(∫02πeit+zeit-zlog⁡⁡|F*(eit)|dt2π),
where log⁡⁡|F*|∈L1(𝕋).Privalov [3, page 98] showed that f∈Np if and only if f=IF, where I is an inner function on 𝔻 and F is an outer function as given above with log⁡+|f*|∈Lp(𝕋).Stoll [6, Theorem 4.2] showed that the space Np (with the notation (log⁡+H)α in [6]) with the topology given by the metric dp defined by
(8)dp(f,g)=(∫02π(log⁡⁡(1+|f*(eiθ)-g*(eiθ)|))pdθ2π)1/p,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhf,g∈Np
becomes an F-algebra. Recall that the function d1=d defined on the Smirnov class N+ by (8) with p=1 induces the metric topology on N+. Yanagihara [17] showed that, under this topology, N+ is an F-space.Furthermore, in connection with the spacesNp(1<p<∞), Stoll [6] (also see [7] and [12, Section 3]) also studied the spaces Fq(0<q<∞) (with the notation F1/q in [6]), consisting of those functions f holomorphic on 𝔻 for which
(9)lim⁡r→1⁡(1-r)1/qlog⁡+⁡M∞(r,f)=0,
where
(10)M∞(r,f)=max⁡|z|≤r⁡|f(z)|.
Stoll [6, Theorem 3.2] proved that the space Fq with the topology given by the family of seminorms {∥·∥q,c}c>0 defined for f∈Fq as
(11)|∥f∥|q,c=∑n=0∞|f^(n)|e-cn1/(q+1)<∞,
for each c>0, where f^(n) is the nth Taylor coefficient of f, becomes a countably normed Fréchet algebra. By a result of Eoff [7, Theorem 4.2], Fp is the Fréchet envelope  of  Np, and hence Fp and Np have the same topological duals.Here, as always in the sequel, we will need some of Stoll’s results concerning the spacesFq only with 1<q<∞, and hence we will assume that q=p>1 is any fixed number.The study of the classM has been extensively investigated by Kim in [1, 2], Gavrilov and Zaharyan [18], and Nawrocky [19]. Kim [2, Theorems 3.1 and 6.1] showed that the space M with the topology given by the metric ρ defined by
(12)ρ(f,g)=∫02πlog⁡(1+M(f-g)(θ))dθ2π,f,g∈M
becomes an F-algebra. Furthermore, Kim [2, Theorems 5.2 and 5.3] gave an incomplete characterization of multipliers of M into H∞. Consequently, the topological dual of M is not exactly determined in [2], but, as an application, it was proved in [2, Theorem 5.4] (also cf. [19, Corollary 4]) that M is not locally convex space. Furthermore, the space M is not locally bounded ([2, Theorem 4.5] and [19, Corollary 5]).Although the classM is essentially smaller than the class N+, Nawrocky [19] showed that the class M and the Smirnov class N+ have the same corresponding locally convex structure which was already established by Yanagihara for the Smirnov class in [17, 20]. More precisely, it was proved in [19, Theorem 1] that the Fréchet envelope of the class M can be identified with the space F+ of holomorphic functions on the open unit disk 𝔻 such that
(13)|∥f∥|c:=∑n=0∞|f^(n)|e-cn<∞,
for each c>0, where f^(n) is the nth Taylor coefficient of f. Notice that F+ coincides with the space F1 defined above. It was shown in [17, 21] that F+ is actually the containing Fréchet space for N+. Moreover, Nawrocky [19, Theorem 1] characterized the set of all continuous linear functionals on M which by a result of Yanagihara [17] coincides with those on the Smirnov class N+.Motivated by the mentioned investigations of the classesM and N+, and the fact that the classes Np(1<p<∞) are generalizations of the Smirnov class N+, in Section 2, we consider the classes Mp(1<p<∞) as generalizations of the class M. Accordingly, the class  Mp(1<p<∞) consists of all holomorphic functions f on 𝔻 for which
(14)∫02π(log⁡+⁡Mf(θ))pdθ2π<∞.
Obviously,
(15)⋃p>1Mp⊂M.
Following [2], by analogy with the space M, the space Mp is equipped with the topology induced by the metric ρp defined as
(16)ρp(f,g)=∥f-g∥p=(∫02π‍log⁡p⁡(1+M(f-g)(θ))dθ2π)1/p,
with f,g∈Mp.In Section2, we give the integral limit criterion for a function f holomorphic on the disk 𝔻 to belong to the class Mp (Lemma 3). Furthermore, we prove that the space Mp is closed under integration (Theorem 4).In Section3 we study and compare the uniform convergence on compact subsets of 𝔻 and the convergences induced by the metrics ρp and dp in the space Mp, respectively. It is proved (Theorem 11) that Mp=Np for each p>1.It is proved in Section4 that the space of all polynomials on ℂ is a dense subset of Mp (Theorem 13). Hence, Mp is a separable metric space. We show that the space Mp with the topology given by the metric ρp becomes an F-space (Theorem 15). As an application, we prove that the metric spaces (Mp,ρp) and (Np,dp) have the same topological structure (Theorem 16). Consequently, we obtain a characterization of continuous linear functionals on Mp (Theorem 17). Notice that Theorem 17 with p=1 characterizes the set of all continuous linear functionals on the space M, which is in fact the Nawrocky result [19, Theorem 1] mentioned above.In Section5 we obtain a characterization of bounded subsets of the spaces Mp(=Np) (Theorem 19). It is also given another necessary condition for a subset of Mp (Np) to be bounded (Theorem 22). Finally, we give the examples of bounded subsets of Mp that are not relatively compact (Theorem 24).


## 2. The ClassesMp(1<p<∞)

Recall that, for a fixed1<p<∞, the class Mp consists of all holomorphic functions f on 𝔻 for which
(17)∫02π(log⁡+⁡Mf(θ))pdθ2π<∞.
Combining the inequalities log⁡(|a|+1)≤log⁡+|a|+log⁡2 and (|b|+|c|)p≤2p-1(|b|p+|c|p), we obtain log⁡p(|a|+1)≤2p-1((log⁡+|a|)p+(log⁡2)p)(a,b,c∈ℂ). The last inequality implies the fact that the condition (17) is equivalent to
(18)∥f∥p:=(∫02π‍(log⁡(1+Mf(θ)))pdθ2π)1/p<∞.Lemma 1.
The function∥·∥p defined on Mp by (18) satisfies the following conditions:
(19)∥f+g∥p≤∥f∥p+∥g∥p∀f,g∈Mp,(20)∥fg∥p≤∥f∥p+∥g∥p∀f,g∈Mp.
Hence, Mp is an algebra with respect to the pointwise addition and multiplication of functions.Proof.
Combining the inequality(21)log⁡⁡(1+M(f+g)(θ))≤log⁡⁡(1+Mf(θ))+log⁡⁡(1+Mg(θ)),f,g∈Mp
with Minkowski’s integral inequality (with the power p), we immediately obtain (19). Similarly, combining the inequality
(22)log⁡⁡(1+M(fg)(θ))≤log⁡⁡(1+Mf(θ))+log⁡⁡(1+Mg(θ)),f,g∈Mp
with Minkowski’s integral inequality (with the exponent p), we obtain (20).Theorem 2.
The functionρp defined on Mp as
(23)ρp(f,g)=∥f-g∥p=(∫02πlog⁡p⁡(1+M(f-g)(θ))dθ2π)1/p,f,g∈Mp
is a translation invariant metric on Mp. Further, the space Mp is a complete metric space with respect to the metric ρp.Proof.
If we suppose thatρp(f,g)=0, for some f,g∈Mp, then by (23) it follows that M(f-g)(θ)=0 for almost every θ∈[0,2π]. Hence, f*(eiθ)=g*(eiθ) for almost every eiθ∈𝕋, and, by Riesz uniqueness theorem, we infer that f(z)=g(z) for all z∈𝔻. As, by (19), the triangle inequality is satisfied, it follows that ρp is a metric on Mp. Finally, by the obvious inequality
(24)ρp(f+h,g+h)=ρp(f,g),f,g,h∈Mp,
we see that ρp is a translation invariant metric. This concludes the proof.For simplicity, here as always in the sequel, we shall writeMp instead of the metric space (Mp,ρp). For a function f holomorphic in 𝔻 and for any fixed 0≤ρ<1, denote by fρ the function defined on 𝔻 as fρ(z)=f(ρz), z∈𝔻. Furthermore, for a given holomorphic function f on 𝔻, let
(25)Mfρ(θ)=sup⁡0≤r≤ρ⁡|fr(θ)|=sup⁡0≤r≤ρ⁡|f(reiθ)|,0≤ρ<1.Lemma 3.
A functionf holomorphic on the unit disk 𝔻 belongs to the class Mp if and only if it satisfies
(26)lim⁡ρ→1⁡∫02π(log⁡+Mfρ(θ))pdθ2π=∫02π(log⁡+Mf(θ))pdθ2π<∞.Proof.
The condition (26) implies that f∈Mp. Conversely, assume that f∈Mp. Then
(27)Mfρ(θ)⟶Mf(θ)asρ⟶1for  almost  everyθ∈[0,2π].
Since, by the assumption, f∈Mp; that is, ∫02π‍(log⁡+Mf(θ))p(dθ/2π)<∞, using (27) and applying the Lebesgue dominated convergence theorem, we obtain
(28)lim⁡ρ→1∫02π‍(log⁡+Mfρ(θ))pdθ2π=∫02π‍(log⁡+Mf(θ))pdθ2π,
which completes the proof.Theorem 4.
The spaceMp is closed under integration.Proof.
For a given functionf∈Mp, define
(29)F(z)=∫0zf(z)dz=∫0rf(teiθ)eiθdt.
It follows that |F(reiθ)|≤Mf(θ), and thus MF(θ)≤Mf(θ) for almost every θ∈[0,2π]. Therefore F∈Mp, as desired.


## 3. Convergences in the SpaceMp

Theorem 5.
For each functionf∈Mp, fρ→f in Mp as ρ→1-.Proof.
Assume thatf∈Mp. Since f∈N+, by Fatou's theorem, the radial limit f*(eiθ)=lim⁡r→1-f(reiθ) exists for almost every θ∈[0,2π]. Hence, for such a fixed θ, the function t↦f(teiθ) is a continuous on [0,1], and thus it is uniformly continuous on [0,1]. Therefore, for such a θ, we have
(30)M(f-fρ)(θ)⟶0asρ⟶1-.
By the inequality
(31)log⁡⁡(1+M(f-fρ)(θ))≤log⁡⁡(1+Mf(θ))+log⁡⁡(1+Mfρ(θ))≤2log⁡⁡(1+Mf(θ)),
in view of the fact that (18) is satisfied for f∈Mp, we obtain
(32)log⁡p(1+M(f-fρ)(θ))≤2plog⁡p(1+Mf(θ))∈L1(𝕋).
From this and (30), by the Lebesgue dominated convergence theorem, we obtain
(33)∫02π(log⁡⁡(1+M(f-fρ)(θ)))pdθ2π⟶0,hhhhhhhhhhhhhhhhhhhhhhasρ⟶1-.
That is, fρ→f in Mp as ρ→1-.For the proof of completeness of the metric space(Mp,ρp) we will need the following lemmas.Lemma 6.
If{fn} is a Cauchy sequence in Mp, then (fn)ρ→fn in Mp as ρ→1-, where this convergence is uniform with respect to n∈ℕ.Proof.
Suppose that{fn} is an arbitrary Cauchy sequence in Mp. Then for a given ε>0 there is a k∈ℕ such that
(34)ρp(fn,fm)<ε3∀n,m≥k.
So by the triangle inequality, for each n≥k, we have
(35)ρp(fn,(fn)ρ)≤ρp(fn,fk)+ρp(fk,(fk)ρ)+ρp((fk)ρ,(fn)ρ)≤2ρp(fn,fk)+ρp(fk,(fk)ρ)<2ε3+ρp(fk,(fk)ρ).
By Theorem 5, there exists 0<ρ0<1  sufficiently near to 1, for which
(36)ρp(fl,(fl)ρ)<ε3for  eachρ0<ρ<1,for  eachl=1,…,k.
Hence, by (35), we immediately obtain
(37)ρp(fn,(fn)ρ)<εfor  eachρ0<ρ<1,for  eachn∈ℕ.
This completes proof of Lemma 6.Lemma 7.
For anyp>1, Mp⊆Np and
(38)dp(f,g)≤ρp(f,g)foreachf,g∈Mp,
where dp is the metric of Np defined by (8).Proof.
The inclusionMp⊆Np is obvious, and (38) follows by the definition of the metrics dp and ρp.Lemma 8.
The convergence with respect to the metricdp of the space Np is stronger than the metric of uniform convergence on compact subsets of the disk 𝔻.Proof.
The assertion immediately follows from the inequality on [5, page 898], which implies that, for any function f∈Np and 0≤r<1, we have
(39)max⁡|z|=r⁡|f(z)|≤exp⁡⁡((1+r1-r)1/pdp(f,0)).Lemma 9.
If{fn} is a Cauchy sequence in the space Mp, then {fn} converges uniformly on compact subsets of 𝔻 to some holomorphic function f on 𝔻.Proof.
From the inequality (38) of Lemma 7, it follows that {fn} is a Cauchy sequence in Np. Therefore, there exists f∈Np such that fn→f in Np, and so, by Lemma 8, fn→f uniformly on compact subsets of 𝔻.The following result is a maximal theorem of Hardy and Littlewood.Lemma 10 (see [16, page 11]).
Let1<p≤+∞ and let φ be a function in the Lebesgue space Lp(𝕋). Let
(40)u(r,θ)=12π∫02π1-r21-2rcos⁡⁡(θ-t)+r2φ(t)dt,0≤r<1
be the Poisson integral of the function φ. Define
(41)U(θ)=sup⁡0≤r<1⁡|u(r,θ)|,θ∈[0,2π].
Then U∈Lp(𝕋) and there is a constant Ap depending only on p such that
(42)∥U∥Lp≤Ap∥φ∥Lp,
where ∥·∥Lp is the usual norm of the space Lp(𝕋).We are now ready to state the following result.Theorem 11.
M
p
=
N
p for each p>1; that is, the spaces Mp and Np coincide.Proof.
By Lemma7, Mp⊆Np for each p>1. For the proof of the converse of this inclusion, assume that f∈Np. We will show that f∈Mp. As noticed in Section 1, f can be factorized as
(43)f(z)=I(z)F(z),z∈𝔻,
where I(z) is the inner function and F(z) is an outer function for the class Np; that is,
(44)F(z)=ωexp⁡⁡(12π∫02πeit+zeit-zlog⁡⁡|f*(eit)|dt),
where ω is a constant of unit modulus. Furthermore, log⁡+|f*|∈Lp(𝕋). As |I(z)|≤1, for each z∈𝔻, the previous factorization and the fact that F∈Mp immediately imply that f∈Mp. Since
(45)ℜeit+zeit-z=1-r21-2rcos⁡⁡(θ-t)+r2,z=reiθ,
from (44), we immediately obtain
(46)log⁡⁡|F(reiθ)|=12π∫02π1-r21-2rcos⁡⁡(θ-t)+r2log⁡⁡|f*(eit)|dt,0≤r<1,
whence it follows that, for 0≤r<1,
(47)log⁡+⁡|F(reiθ)|=(12π∫02π‍1-r21-2rcos⁡(θ-t)+r2log⁡⁡|f*(eit)|dt)+≤12π∫02π1-r21-2rcos⁡⁡(θ-t)+r2log⁡+⁡|f*(eit)|dt.
The above inequality yields
(48)log⁡+⁡MF(θ)≤sup⁡0≤r<1⁡(log⁡+⁡|F(reiθ)|)≤sup⁡0≤r<1(12π∫02π1-r21-2rcos⁡⁡(θ-t)+r2×log⁡+⁡|f*(eit)|dt∫12π).
From the above inequality and the fact that log⁡+|f*|∈Lp(𝕋), we conclude by Lemma 10 that log⁡+MF(θ)∈Lp(𝕋). This means that F∈Mp and therefore f∈Mp. Thus Np⊆Mp, and therefore Mp=Np. This completes the proof.Corollary 12.
Letf∈Mp. Then
(49)∫02π(log⁡+Mf(θ))pdθ≤Cp∫02π(log⁡+⁡|f*(eiθ)|)pdθ,
where Cp is a nonnegative constant depending only on p.Proof.
LetF be the outer factor in the canonical factorization of f∈Mp. From the proof of Theorem 11, we see that for the functions U(θ)=log⁡+MF(θ) and φ(θ)=log⁡+|f*(eiθ)| the inequality (42) can be applied from Lemma 10. The obtained inequality is in fact (49) with F instead of f. Since |f(z)|≤|F(z)|, for each z∈𝔻, it follows that Mf(θ)≤MF(θ) at almost every θ∈[0,2π]; thus (49) is obviously satisfied.


## 4.Mp as an F-Algebra

Theorem 13.
The space of all polynomials overℂ is a dense subset of Mp. Hence, Mp is a separable metric space.Proof.
Suppose thatf∈Mp. Since, for a fixed 0≤ρ<1, fρ is a holomorphic function on the closed unit disk 𝔻¯:|z|≤1, by Runge's theorem, fρ can be uniformly approximated by polynomials on 𝔻¯. This together with the fact that, by Theorem 5, fρ→f in Mp as ρ→1- yields that the space of all polynomials over ℂ is a dense subset of Mp. Therefore, the set of all polynomials whose coefficients have rational real parts and rational imaginary parts becomes a countable dense subset of Mp. This concludes the proof.Theorem 14.
M
p is a complete metric space.Proof.
Let{fn} be a Cauchy sequence in Mp. Then since Np is complete, there is a f∈Np such that fn→f in Np. Since, by Theorem 11, Mp=Np, it follows that f∈Mp, and thus it remains to show that fn→f in Mp. By Theorem 5 and Lemma 6, there exist 0<r<1 and n1∈ℕ such that
(50)ρp(fr,f)<ε3,ρp(fn,(fn)r)<ε3for eachn≥n1.
Since, by Lemma 9, a sequence {fn} converges uniformly on each closed disk |z|≤ρ<1 to some function f, it follows that there exists n2∈ℕ such that
(51)ρp((fn)r,fr)<ε3for eachn≥n2.
Taking n0=max⁡{n1,n2}, by (50) and (51), the triangle inequality implies that
(52)ρp(fn,f)<ε∀n≥n0.
This shows that fn→f in Mp, which completes the proof.Theorem 15.
M
p with the topology given by the metric ρp defined by (23) becomes an F-space.Proof.
By [22, page 51], it suffices to show the following properties:(i)
ρp is an additive-invariant metric,(ii)
for any fixedf∈Mp, c↦cf is a continuous map from ℂ into Mp,(iii)
for any fixedc∈ℂ, f↦cf is a continuous map from Mp into Mp, and(iv)
Mp is a complete metric space.The assertion (i) follows from Theorem 2.
By the Lebesgue dominated convergence theorem, we have(53)ρp(cf,0)=(∫02π‍log⁡p⁡(1+|c|Mf(θ))dθ2π)1/p⟶0hhhhhhhhhhhhhhhhhhhhhhhhasc⟶0.
Letk∈ℕ such that |c|≤k. Then the triangle inequality yields
(54)ρp(cf,0)≤ρp(kf,0)≤kρp(f,0),
whence we see that f↦cf is a continuous map from Mp into Mp.
The assertion (iv) is in fact the assertion of Theorem14.
This concludes the proof.We are now ready to prove that the (metric) spaces(Mp,ρp) and (Np,dp) have the same topological structure.Theorem 16.
For eachp>1, the classes Mp and Np coincide, and the metric spaces (Mp,ρp) and (Np,dp) have the same topological structure.Proof.
Consider the identity mapj:Mp→Np. Then, by the inequality (38) of Lemma 7, j is continuous. Since, by Theorem 11, Mp=Np, j maps Mp onto Np. Since Mp and Np are both F-spaces, it follows, by the open mapping theorem [23, Corollary 2.12 (b)], that the inverse map j-1 of j is continuous. Hence, j is a homeomorphism, and so the metrics dp and ρp induce the same topology on Np and Mp, respectively.As an application of Theorem16 and using the characterization of topological dual of the space Fp (which is by [7, Theorem 4.2] the Fréchet envelope of Np) given by Stoll [6, Theorem 3.3] (cf. also [12, Theorem 3.5] and [13, Theorem 2]), we immediately get the following result.Theorem 17.
Ifγ is a continuous linear functional on Mp, then there exists a sequence {γn}n of complex numbers with γn=O(exp⁡(-cn1/(p+1))), for some c>0, such that
(55)γ(f)=∑n=0∞anγn,
where f(z)=∑n=0∞‍anzn∈Mp, with convergence being absolute. Conversely, if {γn} is a sequence of complex numbers for which
(56)γn=O(exp⁡⁡(-cn1/(p+1))),
then (55) defines a continuous linear functional on Mp.Corollary 18.
M
p is an F-algebra.Proof.
By Theorem15, Mp becomes an F-space. As Np is an F-algebra, by Theorem 16, the multiplication is also continuous on Mp. Hence, Mp is an F-algebra.


## 5. Bounded Subsets ofMp

It is proved in Section4 (Theorem 16) that the spaces Mp and Np coincide and have the same topological structure. Since Np and Mp are not Banach spaces, it is of interest to obtain a characterization of bounded subsets of these spaces in terms of both metrics dp and ρp.Recall that, for a functionf∈Np, its boundary function f* is defined as the radial limit f*(eiθ)=lim⁡r→1-f(reiθ) which exists for almost every eiθ∈𝕋.The following result gives a characterization of bounded subsets ofNp(=Mp). Recall that the assertion (i)⇔(iii) is analogous to Theorem 1 in [21] that describes bounded subsets of N+.Theorem 19.
For given setL⊂Mp, the following conditions are equivalent:(i)
L is a bounded subset of Mp;(ii)
for allε>0 there exists δ>0 such that
(57)∫E(log⁡+⁡Mf(θ))pdθ2π<ε∀f∈L,
for every measurable set E⊂𝕋 with the Lebesgue measure |E|<δ;(iii)
for allε>0 there exists δ>0 such that
(58)∫E(log⁡+⁡|f*(eiθ)|)pdθ2π<ε∀f∈L,
for each measurable set E⊂𝕋 with the Lebesgue measure |E|<δ.Proof.
(ii)⇒(iii). It follows that from the obvious inequality |f*(eiθ)|≤Mf(θ), f∈Mp, for almost every θ∈[0,2π].
(iii)⇒(i). Let
(59)V={g∈Np:dp(g,0)<η}
be an arbitrary neighborhood of zero in Np. Choose sufficiently small ε>0 such that
(60)log⁡p⁡(1+ε)+2p-1log⁡p⁡2δ+2p-1ε<ηp.
Now it follows that there exists δ, 0<δ<ε, such that (iii) holds. Choose an n∈ℕ for which 1/n<δ. Set
(61)Ek={eiθ:θ∈[2(k-1)πn,2kπn)},k=1,2,…,n.
Then |Ek|=1/n<δ, and thus by (iii) we have
(62)∫02π‍(log⁡+⁡|f*(eiθ)|)pdθ2π=∑k=1n∫Ek<nε‍∀f∈L.
By (62) and Chebyshev’s inequality, we conclude that for every function f∈Np there exists a measurable set Ef⊂𝕋 depending on f such that
(63)|𝕋∖Ef|<δ,(log⁡+⁡|f*(eiθ)|)p≤nεδon⁡Ef.
From (63), we obtain
(64)|f*(eiθ)|≤exp⁡⁡(nεδ)1/p=K(δ)=Kon⁡Ef.
Choose α such that 0<α<ε/δ. Then using the inequality
(65)log⁡p⁡(1+|a|)≤2p-1((log⁡+⁡|a|)p+log⁡p2),
(60) and (iii), for every f∈L, we obtain
(66)(dp(αf,0))p=∫02πlog⁡p⁡(1+|αf*(eiθ)|)dθ2π=∫Ef+∫𝕋∖Ef≤∫Eflog⁡p⁡(1+ε)dθ2π+2p-1(∫𝕋∖Eflog⁡p⁡2dθ2π+∫𝕋∖Ef(log⁡+⁡|f*(eiθ)|)pdθ2π)≤log⁡p⁡(1+ε)+2p-1log⁡p⁡2δ+2p-1ε<ηp.
Therefore, dp(αf,0)<η, from which it follows that αL⊂V. Hence, L is a bounded subset of Np.
(i)⇒(ii). Assume that L is a bounded subset of Mp. Then for any given η>0 there is a α0=α0(η), 0<α0<1, such that
(67)(ρp(αf,0))p=∫02πlog⁡p⁡(1+|α|Mf(θ))dθ2π<ηp
for each f∈L and |α|≤α0. It follows that
(68)∫02π(log⁡+⁡|α|Mf(θ))pdθ2π<ηphhhhhforeachf∈L,|α|≤α0.
Since
(69)log⁡+⁡Mf(θ)≤log⁡+⁡α0Mf(θ)+log⁡⁡1α0,
we obtain
(70)(log⁡+⁡Mf(θ))p≤2p-1((log⁡+⁡α0Mf(θ))p+(log⁡⁡1α0)p).
For given ε>0, choose η>0 satisfying
(71)η<ε1/p2,
and α0=α0(η) satisfying (67) and so also satisfying (68). Next, take δ>0 such that
(72)δlog⁡p⁡1α0<ε2p.
Then for each set E⊂𝕋 with |E|<δ, by (68)–(72), for every f∈L, we obtain
(73)∫E(log⁡+⁡Mf(θ))pdθ2π≤2p-1(∫E‍(log⁡+⁡α0Mf(θ))pdθ2π+∫Elog⁡p⁡1α0dθ2π)≤2p-1ηp+2p-1|E|log⁡p⁡1α0≤ε.
Therefore, the condition (ii) of the theorem is satisfied, which concludes the proof.Remark 20.
Note that the condition (ii) from Theorem19 in fact means that the family {(log⁡+Mf(θ))p:f∈L} is uniformly integrable on 𝕋. The same assertion is also valid for the condition (iii). On the other hand, from the proof of Theorem 19, we see that (ii) implies that the family {(log⁡+Mf(θ))p:f∈L} forms a bounded subset of the space L1(𝕋); that is, there holds
(74)limsup⁡f∈L⁡∫02π(log⁡+⁡Mf(θ))pdθ2π<+∞.
Similarly, it follows from (iii) that the family {(log⁡+|f*(eiθ)|)p:f∈L} is bounded in L1(𝕋).Corollary 21.
IfL is a subset of Mp for which the family
(75){(log⁡+⁡|f*(eiθ)|)p:f∈L}
is uniformly integrable, then the family
(76){(log⁡+⁡|f(reiθ)|)p:f∈L,0≤r<1}
is also uniformly integrable.Proof.
The condition of Corollary21 and (iii)⇒(ii) of Theorem 19 immediately yield that the family {(log⁡+Mf(θ))p:f∈L} is uniformly integrable on the circle 𝕋. This fact and the obvious inequality |f(reiθ)|≤Mf(θ), f∈Mp, 0≤r<1, for almost every θ∈[0,2π], imply that the family {(log⁡+|f(reiθ)|)p:f∈L,0≤r<1} is uniformly integrable.The following result gives a necessary condition for a subset ofMp(=Np) to be bounded.Theorem 22.
LetL be a subset of Mp. If L is bounded in Mp, then
(77)M∞(r,f)≤Kexp⁡⁡(ω(r)(1-r)1/p)foreachf∈L,
where M∞(r,f)=max⁡0≤θ<2π|f(reiθ)|, K is a positive constant, and ω(r), 0≤r<1, is a positive continuous function that does not depend on f∈L and for which ω(r)↓0 as r→1.Proof.
By the inequqlity (5.4) from the proof of Theorem 5.2 in [4], for all f∈Np, we have
(78)(log⁡+⁡|f(reiθ)|)p≤12π∫02π1-r21-2rcos⁡⁡(θ-t)+r2(log⁡+⁡|f*(eit)|)pdt.
As, by the assumption, L is a bounded subset of Np, by Theorem 19 (iii), for all ε>0 there exists δ=δ(ε)>0, such that
(79)∫02π(log⁡+⁡|f*(eiθ)|)pdθ2π<ε2∀f∈L
and for every measurable set E⊂𝕋 with the Lebesgue measure |E|<δ.
Further, from the proof of (iii)⇒(i) of Theorem 19, we see that for each f∈Np there is a measurable set Ef⊂𝕋 depending on f for which
(80)|𝕋∖Ef|<δ,(log⁡+⁡|f*(eiθ)|)p≤nεδ
for almost every eiθ∈Ef. From (78)–(80), we obtain
(81)(log⁡+⁡|f(reiθ)|)p=∫Ef+∫Ef≤nεδ+11-rε2,
whence it follows that
(82)(1-r)(log⁡+⁡M∞(r,f))p≤(1-r)nεδ+ε2.
Choose a sequence {εk} of positive numbers such that εk↓0. For each k∈ℕ, let rk>0 be a number such that
(83)(1-rk)nεδk+εk2<εk,
where εk=δ(εk) and
(84)rk-1<rk<1,rk↑1ask⟶∞.
Put
(85)ω1(r)=εkforrk≤r<rk+1,k=1,2,….
From (82), (83), and (85) we obtain
(86)(log⁡+⁡M∞(r,f))p≤ω1(r)1-r∀0≤r<1.
Since
(87)ω1(r)⟶0asr⟶1,
we conclude that there exists a continuous function ω2(r) satisfying
(88)ω1(r)≤ω2(r),ω2(r)↓0asr→1.
Therefore,
(89)(log⁡+⁡M∞(r,f))p≤ω2(r)1-rfor each0≤r<1,
whence by setting
(90)ω(r)=(ω2(r))1/pfor each0≤r<1,
we obtain
(91)M∞(r,f)≤exp⁡⁡(ω(r)(1-r)1/p)∀f∈L.
This concludes the proof.Remark 23.
The condition of Theorem22 is not a sufficient condition for a set L⊂Mp to be bounded. To show this, define
(92)fn(z)=anzn,an=exp⁡⁡(λnn1/(p+1)),
where
(93)λn=n-1/2(p+1).
Then as in the proof of Lemma 1 in [21] it is easy to verify that the set L={fn}⊂Mp satisfies the condition of Theorem 22. Since
(94)log⁡⁡|fn*(eiθ)|=n1/2(p+1),
we see that L is not bounded in Mp.Theorem 24.
There exist bounded subsets ofMp that are not relatively compact.Proof.
Define a sequence{hn} of functions on [0,2π] as
(95)hn(t)=1+sin⁡(nt),t∈[0,2π],
and set
(96)fn(z)=exp⁡⁡(12π∫02πeit+zeit-zhn(t)dt)=exp⁡⁡(1-izn),z∈𝔻.
Obviously, {fn}⊂Np and for each measurable set E⊂𝕋 we have
(97)∫02πhn(t)dt=2π,0≤∫Ehn(t)dt≤2|E|,
where |E| denotes the Lebesgue measure of E. From this and Theorem 19, we see that the set L={fn} is bounded in Np.
Now suppose thatE is relatively compact. This means that there exists a subsequence {fnk} of {fn} and a function f∈Np such that
(98)dp(fnk,f)⟶0ask⟶∞,
and thus
(99)fnk(z)⟶f(z),uniformly  on  each  closed  disk|z|≤r<1.
Therefore, by (96), it follows that f(z)≡e on 𝔻. On the other hand, from (98), it follows that
(100)fnk*(eiθ)⟶f*(eiθ)in  measure  on𝕋.
Therefore,
(101)log⁡+⁡|fnk*(eiθ)|=1+sin⁡(nkθ)⟶log⁡+⁡|f*(eiθ)|=1hhhhhhhhhhhhhhhhhhhhhhhhhhhin  measure  on𝕋.
This contradiction shows that L is not relatively compact in Np.



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*Source: 901726-2014-01-28.xml*