Besov spaces induced by doubling weights

Let $1\le p<\infty$, $0<q<\infty$ and $\nu$ be a two-sided doubling weight satisfying $$\sup_{0\le r<1}\frac{(1-r)^q}{\int_r^1\nu(t)\,dt}\int_0^r\frac{\nu(s)}{(1-s)^q}\,ds<\infty.$$ The weighted Besov space $\mathcal{B}_{\nu}^{p,q}$ consists of those $f\in H^p$ such that $$\int_0^1 \left(\int_{0}^{2\pi} |f'(re^{i\theta})|^p\,d\theta\right)^{q/p}\nu(r)\,dr<\infty.$$ Our main result gives a characterization for $f\in \mathcal{B}_{\nu}^{p,q}$ depending only on $|f|$, $p$, $q$ and $\nu$. As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. In particular, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $f\in \mathcal{B}_{\nu}^{p,q}$, then there exist $f_1,f_2\in \mathcal{B}_{\nu}^{p,q} \cap H^\infty$ such that $f=f_1/f_2$. In addition, we give a sufficient and necessary condition guaranteeing that the product of $f\in H^p$ and an inner function belongs to $\mathcal{B}_{\nu}^{p,q}$. Applying this result, we make some observations on zero sets of $\mathcal{B}_{\nu}^{p,p}$.

Our main result gives a characterization for f P B p,q ν depending only on |f |, p, q and ν. As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. For instance, we show the following modification of a classical factorization by F. and R. Nevanlinna: If f P B p,q ν , then there exist f1, f2 P B p,q ν X H 8 such that f " f1{f2. Moreover, we give a sufficient and necessary condition guaranteeing that the product of f P H p and an inner function belongs to B p,q ν . Applying this result, we make some observations on zero sets of B p,p ν .

Introduction and characterizations
Let The Hardy space H 8 is the set of all bounded functions in HpDq. Moreover, we recall that a measurable function f on T belongs to L p pTq for some p P p0, 8q if }f } p L p " 1 2π ż 2π 0 |f pe iθ q| p dθ ă 8.
Alternatively, the Hardy space H p for 0 ă p ă 8 can be characterized as follows: f P H p if and only if f P HpDq, non-tangential limit f pe iθ q exists almost everywhere on T and f pe iθ q P L p pTq.
In particular, }f } H p " }f } L p for 0 ă p ă 8 and f P H p . This is due to Hardy's convexity and the mean convergence theorems. These results and much more can be found in classic book [8] by P. Duren. A function ν : D Ñ r0, 8q is called a (radial) weight if it is integrable over D and νpzq " νp|z|q for all z P D. For 0 ă p, q ă 8 and a weight ν, the weighted mixed norm space A p,q ν consists of those f P HpDq such that }f } q A p,q ν " ż 1 0 M q p pr, f q νprq dr ă 8.
If νpzq " p1´|z|q α for´1 ă α ă 8, then the notation A p,q α is used for A p,q ν . In this note, we study class D of so-called two-sided doubling weights, which originates from the work of J. A. Peláez and J. Rättyä [19,20]. For the definition of D we have to define two wider classes. For a weight ν, set p νpzq " p νp|z|q " ż 1 |z| νpsq ds, z P D.
If a weight ν satisfies the condition p νprq ď C p νp 1`r 2 q for all 0 ď r ă 1 and some C " Cpνq ą 0, then we write ν P p D. Correspondingly, ν P q D if there exist K " Kpνq ą 1 and C " Cpνq ą 1 such that p νprq ě C p νˆ1´1´r K˙, 0 ď r ă 1.
Class D is the intersection of p D and q D. In addition, we define the following subclass of p D: ν P p D p for some p P p0, 8q if the condition is satisfied. As a concrete example, we mention that ν 1 pzq " p1´|z|q α and ν 2 pzq " p1| z|q α´l og e 1´|z|¯β for any β P R belong to D X p D p if and only if´1 ă α ă p´1. Additional information about weights can be found in [18,19,20]. Some basic properties are recalled also in Section 2.
Define the weighted Besov space B p,q ν by B p,q ν " tf : f 1 P A p,q ν u X H p . For´1 ă α ă 8 and νpzq " p1´|z|q α , the notation B p,q α is used for B p,q ν . The space B p,q ν is the main research objective of this note. Hence it is worth pointing out that the definition is rational, which means that H p is not a subset of tf : f 1 P A p,q ν u in general, or conversely. The family of Blaschke products offers examples for the case where f P H 8 and f 1 R A p,q ν ; see for instance [24]. Moreover, it would be natural that certain lacunary series g lie out of H p , while g 1 P A p,q ν . Arguments for this kind of examples can be found in M. Pavlović's book [16], which contains numerous important observations on the topic of this note. The existence of both examples, of course, depends on p, q and ν. In other words, under certain hypotheses for p, q and ν, an inclusion relation between tf : f 1 P A p,q ν u and H p might be valid. However, this is not the case in general.
For 0 ă p ă 8 and f P L p pTq, the L p modulus of continuity ω p pt, f q is defined by We interpret ω p pt, f q " ω p p2π, f q for t ą 2π. It is a well-known fact that, for 0 ă p, q ă 8, 1 ă α ă q´1 and f P H p , the derivative of f belongs to A p,q α if and only if Note that (1.2) is valid also if 0 ă p, q ă 8, ν is a weight and there exists β " βpq, νq ă q´1 such that νprq{p1´rq β is increasing for 0 ď r ă 1. This is due to [17,Theorem 5.1] and its proof. Even though the result is valid also for 0 ă p ă 1, Theorem 1 is more useful for our purposes. In particular, it is worth underlining that the hypothesis 1 ď p ă 8 in Theorems 2 and 3 below is natural.
Theorem 2 gives a practical estimate for }f 1 } q As a by-product of its argument, we deduce that also the converse inequality of (1.2) holds if the norm }f } q H p is added into the right-hand side. Setting Theorem 2 reads as follows.
Theorem 2. Let 1 ď p ă 8, 0 ă q ă 8 and ν P D X p D q . Then there exist positive constants C 1 and C 2 depending only on p, q and ν such that By studying the classical weight νpzq " p1´|z|q α , where´1 ă α ă q´1, we obtain K. M. Dyakonov's [9, Proposition 2.2(a)] as a direct consequence of Theorem 2. Hence it does not come as a surprise that the proofs of [9, Theorem 2.1] and Theorem 2 have some similarities. Nonetheless, it is worth mentioning that the presence of general weights complicates the argument; and consequently, our proof is quite technical. Note also that Theorem 1 plays an essential role in the proof.
Our main result below gives a characterization for functions f in B p,q ν depending only |f |, p, q and ν. This result improves B. Bøe's [6, Theorem 1.1], which concentrates only on the case where 1 ď p, q ă 8,´1 ă α ă q´1 and νpzq " p1´|z|q α . It also generalizes the essential contents of [2, Proposition 2.4] and [9, Proposition 2.2(b)] made by A. Aleman and K. M. Dyakonov, respectively. Theorem 3. Let 1 ď p ă 8, 0 ă q ă 8 and ν P D X p D q . Then there exist positive constants C 1 and C 2 depending only on p, q and ν such that Before we talk about the argument of Theorem 3, recall the inner-outer factorization. An inner function is a member of H 8 having unimodular radial limits almost everywhere on T. For 0 ă p ď 8, an outer function for H p takes the form where φ is a non-negative function in L p pTq and log φ P L 1 pTq. The inner-outer factorization asserts that f P H p can be represented as the product of an inner and outer function; see for instance [8,Theorem 2.8]. It is worth noting that the factorization is unique, and |f pξq| " |O φ pξq| " φpξq (1.5) for almost every ξ P T if O φ is the outer function from the factorization of f . Equation (1.5) is due to the definition of inner functions, Poisson integral formula, harmonicity of log |O φ pzq| and fact that The last inequality in (1.4) can be proved by applying Theorem 2. In the argument of the first inequality, the inner-outer factorization, Schwarz-Pick lemma and an upper estimate for |O 1 φ | from [6] are the main tools. It is worth underlining that this Bøe's idea to make an upper estimate for |f 1 | by using the factorization seems to be quite effective. Another way to prove results like Theorem 3 is to use a modification of Theorem 2 together with the well-known equation ż 2π 0 |f pe iθ q´f pzq| 2 dµ z pθq " ż 2π 0 |f pe iθ q| 2 dµ z pθq´|f pzq| 2 , z P D; but this Dyakonov's method has the obvious defect that it works only when f P H 2 . The advantage of this method in the case where 2 ď p ă 8, 0 ă q ă 8, q{2´1 ă α ă q´1 and νpzq " p1´|z|q α is that F 1 pf q`F 2 pf q in Theorem 3 can be replaced by It is an open problem to prove a corresponding estimate for general weights.
Next we give an example which shows that the hypothesis ν P D X p D q in Theorem 3 for p ě 2 is sharp in a certain sense. Note that the example is a modification of [22,Example 8]. Before the statement we fix some notation. Write f À g if there exists a constant C ą 0 such that f ď Cg, while f Á g is understood analogously. If f À g and f Á g, then we write fg.
We close the section by explaining how the remainder of this note is organized. Auxiliary results on weights are recalled in the next section. The utility of Theorem 3 is demonstrated in Sections 3 and 4. More precisely, in Section 3, we prove the factorization which states that, for any f P B p,q ν , there exist f 1 , f 2 P B p,q ν X H 8 such that f " f 1 {f 2 . Section 4 begins with a result giving a sufficient and necessary condition guaranteeing that the product of f P H p and an inner function belongs to B p,q ν . As a consequence of this theorem, we obtain some results on zero sets of B p,p ν . Sections 5, 6 and 7 consist of the proofs of Theorems 1, 2 and 3, respectively.

Auxiliary results on weights
In this section, we recall some basic properties of weights in p D and q D. These properties are needed in next sections. Another reason for these results is to help the reader to understand the nature of weights in D. We begin with a result which is essentially [19,Lemma 3]; see also [18,Lemma 2.1].
Lemma A. Let ν be a weight. Then the following statements are equivalent: For the point view of our main results Lemma A(iii) is interesting because it states that ν P p D if and only if ν P p D p for some p ą 0. This means that p D " Ť pą0 p D p . Nevertheless, Lemma A(ii) gives maybe the most interesting description for p D. Together with its q D counterpart below it offers a very practical characterization for weights in D. Essentially this characterization says that p ν is normal in the sense of A. L. Shields and D. L. Williams [25].
Lemma B. Let ν be a weight. Then ν P q D if and only if there exist C " Cpνq ą 0 and α " αpνq ą 0 such that Lemma B originates from [20], and it can be proved in a corresponding manner as Lemma A(ii). See in particular the proof of [18, Lemma 2.1].
By the definition of class p D p , it is clear that p D p Ă p D p`ε for any ε ą 0. Next we state [21, Lemma 3], which shows that also the converse inclusion is true for sufficiently small ε " εpν, pq ą 0. The proof of this result is based on integration by parts. Note that p D p pνq in the statement is defined by (1.1).
Lemma C. If 0 ă p ă 8 and ν P p D p , then ν P p D p´ε for any ε P´0, The last result of this section is [22,Lemma 5], which shows that ν P D in the norm }f } A p,q ν can be replaced by p νpzq{p1´|z|q without losing any essential information.
Lemma D. Let 0 ă p, q ă 8 and ν be a weight.
For ν P D, Lemmas A(ii) and B yield In [22], Lemma D is proved by applying this fact together with partial integrations. An alternative way to prove results like Lemma D is to split the integral with respect to dr to infinitely many parts by using a dyadic partition, and then apply (2.1) together with the monotonicity of M q p pr, f q. An advantage of the last method is that M q p pr, f q can be easily replaced by a certain monotonic function gprq. This observation will be utilized several times in the argument of Theorem 2.

Quotient factorization
Recall that if f P H p for some p ą 0, then there exist f 1 , f 2 P H 8 such that f " f 1 {f 2 . This is an important consequence of classical factorization [8, Theorem 2.1] by F. and R. Nevanlinna. The main purpose of this section is to give the following B p,q ν counterpart for the abovementioned result. Note that a part of the next pages is really inspired by [6]. Proposition 6. Let 1 ď p ă 8, 0 ă q ă 8, ν P D X p D q and f P H p be the product of an inner function I and an outer function O φ . Then there exists a constant C " Cpp, q, νq ą 0 such that Before the proof of Proposition 6, we note that the quantities F 1 pf q and F 2 pf q in Theorem 3 are used repeatedly in the future.
Hence it is easy to deduce F 1 pIO mintφ,1u q ď F 1 pf q. Since F 2 pIO mintφ,1u q " F 2 pO mintφ,1u q ď F 2 pO φ q " F 2 pf q can be shown by using a modification of (3.2), the desired estimate (3.4) follows from Theorem 3. This completes the proof.
Now we can easily prove Theorem 5 by using Proposition 6.
Proof of Theorem 5. By the inner-outer factorization, there exist an inner function I and an where f 1 " IO mintφ,1u and f 2 " 1{O maxtφ,1u . Applying Proposition 6 together with the inequalities |O mintφ,1u pzq| ď 1 ď |O maxtφ,1u pzq| and |f 1 2 pzq| ď |O maxtφ,1u pzq| 2 |f 1 2 pzq| " |O 1 maxtφ,1u pzq|, z P D, we can check that f 1 and f 2 belong to B p,q ν X H 8 . Moreover, it is obvious that f 2 is an outer function. Hence the proof is complete. for all z P D. Write z " re it . Raising both sides of (4.1) to power p, integrating from 0 to 2π with respect to dt, then raising both sides to power q{p, integrating from 0 to 1 with respect to νprqdr{p1´rq q and finally splitting the right-hand side into two parts by using well-known inequalities, we obtain Since F 2 pf Iq`}f I} q H p " F 2 pf q`}f } q H p , the assertion follows from Theorem 3.
Recall that a subspace X of H p satisfies the F -property if the hypothesis f I P X, where f P H p and I is an inner function, implies f P X. The F -property for B p,q ν is a direct consequence of Theorem 7. However, it is worth mentioning that if one just aims to prove the F -property for B p,q ν , our argument is not maybe the simplest one, taking into account the length of proofs of Theorem 3 and its auxiliary results. Ideas for an alternative proof can be found, for instance, in [16,Section 5.8.3].
A sequence tz n u Ă D is said to be a zero set of B p,q ν if there exists f P B p,q ν such that tz : f pzq " 0u " tz n u. Here each zero z n is repeated according to its multiplicity and function f is not identically zero. Applying Theorem 7, we make some observations on zero sets of B p,p ν . More precisely, we concentrate on the case where tz n u is separated, which means that there exists δ " δptz n uq ą 0 such that dpz n , z k q ą δ for all n ‰ k, where dpz, wq "ˇˇˇˇz´w 1´zwˇˇˇˇ, z, w P D, is the pseudo-hyperbolic distance between points z and w. Before these results some basic properties of Hardy spaces are recalled. For tz n u Ă D satisfying the Blaschke condition ř n p1´|z n |q ă 8 and a point θ P r0, 2πq, the Blaschke product with zeros tz n u is defined by Bpzq " e iθ ź n |z n | z n z n´z 1´z n z , z P D.
For z n " 0, the interpretation |z n |{z n "´1 is used. By factorization [8, Theorem 2.5] made by F. Riesz, we know that any f P H p for some fixed p P p0, 8s can be represented in the form f " Bg, where B is a Blaschke product and g P H p does not vanish in D. More precisely, Beurling factorization [8, Theorem 2.8] asserts that g is the product of an outer function and a singular inner function where θ P r0, 2πq is a constant and σ a positive measure on T, singular with respect to the Lebesgue measure. Consequently, every zero set of B p,q ν satisfies the Blaschke condition. With these preparations we are ready to state and prove the following result.
Corollary 8. Let 1 ď p ă 8, ν P D X p D p , and assume that tz n u is a finite union of separated sequences and zero set of B p,p ν . Then there exists an outer function O φ P B p,p ν such that ÿ n |O φ pz n q| p p νpz n q p1´|z n |q p´1 ă 8.
Proof. Let tz n u " Ť M j"1 tz j n u, where M P N and each tz j n u is separated. Let B be the Blaschke product with zeros tz n u, S a singular inner function and O φ an outer function such that BSO φ P B p,q ν . By Theorem 7, we know that O φ and BO φ belong to B p,p ν . For w P D and 0 ă r ă 1, set ∆pw, rq " tz : dpz, wq ă ru and Λpw, rq " tz : |w´z| ă rp1´|w|qu.
Since each tz j n u is separated, we find R j , δ j P p0, 1q such that, for a fixed j, discs Λpz j n , R j q are pairwise disjoint and the inclusion ∆pz j n , δ j q Ă Λpz j n , R j q is valid for every n. Hence p ν is essentially constant in each disc ∆pz j n , δ j q by Lemma A(ii). Moreover, |Bpzq| ďˇˇˇˇz j n´z 1´z j n zˇˇˇˇˇď δ j , z P ∆pz j n , δ j q.
Using these facts together with the subharmonicity of |O φ | p , we obtain ÿ n |O φ pz n q| p p νpz n q p1´|z n |q p´1 " where dApzq is the two-dimensional Lebesgue measure. Now it suffices to show that the last integral in (4.2) is finite. Set ψpzq " p νpzq{p1´|z|q for z P D. Note that p νprq -p ψprq for 0 ď r ă 1 by Lemmas A(ii) and B. Moreover, integrating by parts, one can show that ν P p D p if and only if In particular, ψ P D X p D p by the hypotheses of ν. Since Lemma D implies BO φ in B p,p ψ , This completes the proof.
Recall that a sequence tz n u Ă D is said to be uniformly separated if inf nPN ź k‰nˇz k´zn 1´z k z nˇą 0; and a finite union of uniformly separated sequences is called a Carleson-Newman sequence. It is worth mentioning that any Carleson-Newman sequence is a finite union of separated sequences satisfying the Blaschke condition, but the converse statement is not true. For 1 ă p ă 8, p´2 ă α ă p´1 and a Carleson-Newman sequence tz n u, we can give a sufficient and necessary condition for tz n u to be a zero set of B p,p α . This is a straightforward consequence of Theorem 7, Corollary 8 and the reasoning made in paper [4] by N. Arcozzi, D. Blasi and J. Pau.  Proof. Let B be the Blaschke product with zeros tz n u and O φ P B p,p α an outer function satisfying (4.4). Then [14,Theorem 3.5] together with some elementary calculations gives ż |1´z n z| 2 p1´|z|q α`1´p dApzq ": I 1`I2 . (4.5) Following the reasoning in the proof of [4, Proposition 3.2], it is easy to check that I 1 and I 2 are finite. More precisely, estimating in a natural manner, one can show In the argument of I 2 À }O 1 φ } p A p,p α ă 8, [5, Lemma 2.1] and the hypothesis that tz n u is a Carleson-Newman sequence play key roles.
Since O φ P B p,p α and the first integral in (4.5) is finite, BO φ belongs to B p,p α by Theorem 7. Consequently, the implication ð is valid. The converse implication is a direct consequence of Corollary 8. Hence the proof is complete.
It is an open problem to prove a B p,p ν counterpart of Corollary 9. One could try prove such result, for instance, assuming ν P D X p D p and sup 0ďră1 p1´rq p´1 p νprq ż 1 r νpsq p1´sq p´1 ds ă 8.
In this case, the implication ð is the problematic part. An idea to approach this problem is to follow the argument of [4, Proposition 3.2] and aim to apply therein [3, Theorem 3.1] instead of [5, Lemma 2.1]. The down side of this method is that it leads to laborious computations of Bekollé-Bonami weights.
Corollaries 8 and 9 are related to some main results in [15] by J. Pau and J. A. Peláez. In particular, the equivalence piq ô piiq in [15, Theorem 1] follows from Corollary 9 by setting p " 2. Moreover, Corollary 8 shows that the implication piq ñ piiq in [15,Theorem 1] is valid also if tz n u in the statement is a finite union of separated sequences. Applying the last observation, we can also replace a Carleson-Newman sequence in [15, Corollary 1] by a finite union of separated sequences: If 0 ă α ă 1, tz n u is a finite union separated sequences and zero set of B 2,2 α , then ż 2π 0 log˜ÿ n p1´|z n |q α`1 |e iθ´z n | 2¸d θ ă 8.
This result offers a practical way to construct Blaschke sequences which are not zero sets of B 2,2 α ; see [15,Theorem 2] and its proof. Note that (4.2) and (4.5) together with the estimates for I 1 and I 2 are valid also if outer function O φ is replaced by an arbitrary f P H p . Using this observation and Theorem 7, we can rewrite Corollary 9 in the following form. Corollary 10 is a partial improvement of the main result in M. Jevtić's paper [11]. More precisely, this paper contains an extended counterpart of Corollary 10 (in the sense of p and q) with the defect f " 1. It is also worth mentioning that Corollary 10 is not valid if the Carleson-Newman sequence tz n u is replaced by an arbitrary Blaschke sequence. This can be shown by studying the case where f " 1 and B is a Blaschke product with zeros on the positive real axis. More precisely, the counter example follows from [23,Theorem 1], which asserts that all such Blaschke products belong to B p,p α for 1{2 ă p ă 8 and p´3{2 ă α ă 8. Theorem 7 for f " 1 (or Theorem 3 for inner functions) has also extended counterpart [22,Theorem 1].
Theorem E. Let 0 ă p, q ă 8 and ν P D. Then ν P p D q if and only if for all inner functions I. Here the comparison constants may depend only on p, q and ν.
Theorem E confirms that the hypothesis ν P p D q in Theorems 3 and 7 is sharp in a certain sense. Studying the argument of this result in [22], we can also deduce that the proof of Theorem 3 is more straightforward when f is an inner function, and the statement is valid for all 0 ă p ă 8. It is also worth mentioning that results like Theorem E have turned out to be useful in the theory of inner functions. Several by-products of Theorem E can be found in [22,24].

Proof of Theorem 1
Before the proof of Theorem 1 we recall [22,Lemma 6], which is a modification of [1, Lemma 5]. Proof of Theorem 1. Let 4 5 ď s ă 1 and choose n " npsq P Nzt1, 2, 3, 4u such that 1´1 n ď s ă 1´1 n`1 . Set f n pzq " z n for z P D. Sincěˇˇe Using the hypothesis ν P p D together with Lemma A(iv)(ii) in a similar manner as in the proof of [21, Theorem 1], we obtain Finally combining the estimates above and using the inequality we deduce that if ν P p D and (1.2) is satisfied for all f P H p , then ν P p D p . Hence it suffices to prove the converse statement.
Let f P H p , 0 ď θ ă 2π, 1 2 ă r ă 1 and 0 ă h ă 1 2 . Set ρ " r´h and Γ be the contour which goes first rapidly from re iθ to ρe iθ , then along the circle tz : |z| " ρu to ρe ipθ`hq and finally rapidly to re ipθ`hq . Since Note that the deduction above can be found, for instance, in the proof of [8,Theorem 5.4]. By raising both sides of (5.1) to power q, adding sup 0ăhă1´t and then integrating from 1{2 to r with respect to νptq dt{p1´tq q , we obtain ż r 1{2 sup 0ăhă1´tˆż 2π 0 |f pre ipθ`hq q´f pre iθ q| p dθ˙q Letting r Ñ 1´, using the monotone and mean convergence theorems together with the hypothesis f P H p , we deduce ż 1
Hence it suffices to show I À }f 1 } q A p,q ν . Note that the argument of this estimate uses ideas from [22].
If q ď 1, then Lemma F with the choice gpsq " M p ps, f 1 q, Hardy's convexity theorem, Fubini's theorem, the hypothesis ν P p D q and Lemma D give ν for all f P HpDq. Hence the assertion for q ď 1 is proved. If q ą 1, 0 ă ε ă q{p p D q pνq`1q and hpsq " p1´sq q´1´ε q , then Hölder's inequality and Fubini's theorem yield for all f P HpDq. Since ν P p D q´ε by Lemma C, the assertion for q ą 1 follows from Lemma D. This completes the proof. l by Minkowski's inequality, Theorem 1 has the following consequence.
Corollary 11. Let 1 ď p ă 8, 0 ă q ă 8 and ν P D X p D q . Then there exists a constant C " Cpp, q, νq ą 0 such that Note that Corollary 11 is a part of Theorem 2. We state it here as an independent result because it is needed for the proof of Theorem 2.

Proof of Theorem 2
We go directly to the proof of Theorem 2.
Raising both sides to power p, integrating from 0 to 2π with respect to dt, then raising both sides to power q{p and finally integrating from 0 to 1 with respect to νprq dr, we obtain Hence the sub-additivity of gpxq " x q for x ě 0 and Fubini's theorem give Next we show that the weight νprq in the right-hand side can be replaced by p νprq 1´r without losing any essential information.
Using this together with a change of variable and modification of (6.6), we get ω p p1´s, f q q p νpsq p1´sq q`1 ds À ż 1 0 ω p p1´s, f q q νpsq p1´sq q ds`}f } q H p . Proof of Theorem 3. Let f P H p . Then there exist an inner function I and an outer function O φ such that f " IO φ . Hence the Schwarz-Pick lemma, Lemma G and the fact that φpξq " |f pξq| for almost every ξ P T yield f pe iθ q|´ż 2π 0 |f pe is q|dµ z psqˇˇˇˇdµ z pθq`4ˆż 2π 0 |f pe ih q|dµ z phq´|f pzq|˙(

7.1)
for all z P D. Write z " re it . Raising both sides of (7.1) to power p, integrating from 0 to 2π with respect to dt, then raising both sides to power q{p, integrating from 0 to 1 with respect to νprqdr{p1´rq q and finally splitting the right-hand side into two parts, we obtain }f 1 } q A p,q ν À F 1 pf q`F 2 pf q, which is the first inequality in (1.4). Set Γ " Γpz, f q " " θ P r0, 2πq : pe iθ q´f pzqˇˇdµ z pθq, z P D.
Doing a corresponding integration procedure for this estimate as above and applying Theorem 2, we obtain F 1 pf q`F 2 pf q À }f 1 } q A p,q ν`} f } q H p , which is the last inequality in (1.4). This completes the proof. l