Besov Spaces Induced by Doubling Weights

Let 1⩽p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\leqslant p<\infty $$\end{document}, 0<q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q<\infty $$\end{document}, and ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} be a two-sided doubling weight satisfying sup0⩽r<1(1-r)q∫r1ν(t)dt∫0rν(s)(1-s)qds<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{0\leqslant r<1}\frac{(1-r)^q}{\int _r^1\nu (t)\,dt}\int _0^r\frac{\nu (s)}{(1-s)^q}\,ds<\infty . \end{aligned}$$\end{document}The weighted Besov space Bνp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{\nu }^{p,q}$$\end{document} consists of those f∈Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in H^p$$\end{document} such that ∫01∫02π|f′(reiθ)|pdθq/pν(r)dr<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^1 \left( \int _{0}^{2\pi } |f'(re^{i\theta })|^p\,d\theta \right) ^{q/p}\nu (r)\,dr<\infty . \end{aligned}$$\end{document}Our main result gives a characterization for f∈Bνp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathcal {B}_{\nu }^{p,q}$$\end{document} depending only on |f|, p, q, and ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}. 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∈Bνp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathcal {B}_{\nu }^{p,q}$$\end{document}, then there exist f1,f2∈Bνp,q∩H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1,f_2\in \mathcal {B}_{\nu }^{p,q} \cap H^\infty $$\end{document} such that f=f1/f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=f_1/f_2$$\end{document}. Moreover, we give a sufficient and necessary condition guaranteeing that the product of f∈Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in H^p$$\end{document} and an inner function belongs to Bνp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{\nu }^{p,q}$$\end{document}. Applying this result, we make some observations on zero sets of Bνp,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{\nu }^{p,p}$$\end{document}.

Alternatively, the Hardy space H p for 0 < p < ∞ can be characterized as follows: f ∈ H p if and only if f ∈ H(D), the nontangential limit f (e iθ ) exists almost everywhere on T, and f (e iθ ) ∈ L p (T). In particular, f H p = f L p for 0 < p < ∞ and f ∈ H p . This is due to Hardy's convexity and the mean convergence theorems. These results and much more can be found in the classic book [8] by P. Duren If ν(z) = (1 − |z|) α for −1 < α < ∞, then the notation A p,q α is used for A p,q ν . In this paper, 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 If a weight ν satisfies the condition ν(r ) C ν( 1+r 2 ) for all 0 r < 1 and some C = C(ν) > 0, then we write ν ∈ D. Correspondingly, ν ∈Ď if there exist K = K (ν) > 1 and C = C(ν) > 1 such that Class D is the intersection of D andĎ. In addition, we define the following subclass of D: ν ∈ D p for some p ∈ (0, ∞) if the condition is satisfied. As a concrete example, we mention that ν 1 (z) = (1−|z|) α and ν 2 (z) = (1− |z|) α log e 1−|z| β for any β ∈ R belonging to D ∩ D p if and only if −1 < α < p − 1. Additional information about weights can be found in [18][19][20]. Some basic properties are recalled also in Sect. 2 [24]. Moreover, it would be natural that certain lacunary series g lie out of H p , while g ∈ A p,q ν . Arguments for these kinds of examples can be found in M. Pavlović's book [16], which contains numerous important observations on the topic of this paper. 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 { f : f ∈ A p,q ν } and H p might be valid. However, this is not the case in general.
Then ν ∈ D q if and only if there exists a constant C = C( p, q, ν) > 0 such that Note that (1.2) is valid also if 0 < p, q < ∞, ν is a weight, and there exists β = β(q, ν) < q − 1 such that ν(r )/(1 − r ) β is increasing for 0 r < 1. This is due to [17,Theorem 5.1] and its proof. Even though the result is valid also for 0 < p < 1, Theorem 1 is more useful for our purposes. In particular, it is worth underlining that the hypothesis 1 p < ∞ in Theorems 2 and 3 below is natural.
Theorem 2 gives a practical estimate for f 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.
Then there exist positive constants C 1 and C 2 depending only on p, q, and ν such that By studying the classical weight ν(z) = (1 − |z|) α , where −1 < α < 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 Theorem 3 Let 1 p < ∞, 0 < q < ∞, and ν ∈ D ∩ 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 ∞ having unimodular radial limits almost everywhere on T. For 0 < p ∞, an outer function for H p takes the form where φ is a non-negative function in L p (T) and log φ ∈ L 1 (T). The inner-outer factorization asserts that f ∈ 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 for almost every ξ ∈ T if O φ is the outer function from the factorization of f . Equation (1.5) is due to the definition of inner functions, the Poisson integral formula, the harmonicity of log |O φ (z)| and the 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, the Schwarz-Pick lemma, and an upper estimate for |O φ | from [6] are the main tools. It is worth underlining that Bøe's idea to make an upper estimate for | f | 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 but this method of Dyakonov has the obvious defect that it works only when f ∈ H 2 . The advantage of this method in the case where 2 It is an open problem to prove a corresponding estimate for general weights.
Next we give an example that shows that the hypothesis ν ∈ D ∩ D q in Theorem 3 for p 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 g if there exists a constant C > 0 such that f Cg, while f g is understood analogously. If f g and f g, then we write f g.
The existence of such f is guaranteed by [7,Theorem 5]. Then which originates from [13].
We close the section by explaining how the remainder of this paper is organized. Auxiliary results on weights are recalled in the next section. The utility of Theorem 3 is demonstrated in Sects. 3 and 4. More precisely, in Sect. 3, we prove the factorization that states that, for any f ∈ B p,q Section 4 begins with a result giving a sufficient and necessary condition guaranteeing that the product of f ∈ 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 D andĎ. These properties are needed in subsequent sections. Another reason for these results is to help the reader to understand the nature of weights in D. We begin with a result that is essentially [19,Lemma 3]; see also [18,Lemma 2.1].
Lemma A Let ν be a weight. Then the following statements are equivalent: From the point view of our main results, Lemma A(iii) is interesting because it states that ν ∈ D if and only if ν ∈ D p for some p > 0. This means that D = p>0 D p . Nevertheless, Lemma A(ii) gives maybe the most interesting description for D. Together with itsĎ counterpart below it offers a very practical characterization for weights in D. Essentially this characterization says that ν is normal in the sense of A. L. Shields and D. L. Williams [25].

Lemma B Let ν be a weight. Then ν ∈Ď if and only if there exist C
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 D p , it is clear that D p ⊂ D p+ε for any ε > 0. Next we state [21, Lemma 3], which shows that also the converse inclusion is true for sufficiently small ε = ε(ν, p) > 0. The proof of this result is based on integration by parts. Note that D p (ν) in the statement is defined by (1.1).
The last result of this section is [22,Lemma 5], which shows that ν ∈ D in the norm f A p,q ν can be replaced by ν(z)/(1 − |z|) without losing any essential information.
Lemma D Let 0 < p, q < ∞ and ν be a weight.
For ν ∈ 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 into infinitely many parts by using a dyadic partition, and then apply (2.1) together with the monotonicity of M q p (r , f ). An advantage of the last method is that M q p (r , f ) can be easily replaced by a certain monotonic function g(r ). This observation will be utilized several times in the argument of Theorem 2.

Quotient Factorization
Recall that if f ∈ H p for some p > 0, then there exist f 1 , f 2 ∈ H ∞ 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 to the above-mentioned result.
Before the proof of Proposition 6, we note that the quantities F 1 ( f ) and F 2 ( f ) in Theorem 3 are used repeatedly hereafter.
Proof Let us begin by noting that |O φ (e iθ )| = φ(e iθ ), |O max{φ,1} (e iθ )| = max{φ(e iθ ), 1}, and for all z ∈ D and almost every θ ∈ [0, 2π). Using these facts together with Jensen's inequality [10, Chapter I, Lemma 6.1] and the definition of outer functions, we obtain Consequently, the obvious inequality (3.1) Write z = re it . Raising both sides of (3.1) 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 ν(r )dr Then elementary calculations yield (3.2) Consequently, we obtain by arguing as above using Jensen's inequality. It follows that Hence it is easy to deduce 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 outer function
for all z ∈ 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 ν(r )dr/(1 − r ) q and finally splitting the right-hand side into two parts by using well-known inequalities, we obtain Since the assertion follows from Theorem 3.
Recall that a subspace X of H p satisfies the F-property if the hypothesis f I ∈ X , where f ∈ H p and I is an inner function, implies f ∈ 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 maybe not 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 {z n } ⊂ D is said to be a zero set of B 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 {z n } is separated, which means that there exists δ = δ({z n }) > 0 such that d(z n , z k ) > δ for all n = k, where is the pseudo-hyperbolic distance between points z and w. Before these results some basic properties of Hardy spaces are recalled. For {z n } ⊂ D satisfying the Blaschke condition n (1 − |z n |) < ∞ and a point θ ∈ [0, 2π), the Blaschke product with zeros {z n } is defined by For z n = 0, the interpretation |z n |/z n = −1 is used.
where θ ∈ [0, 2π) is a constant and σ is 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.  Since each {z j n } is separated, we find R j , δ j ∈ (0, 1) such that, for a fixed j, discs (z j n , R j ) are pairwise disjoint and the inclusion (z j n , δ j ) ⊂ (z j n , R j ) is valid for every n. Hence ν is essentially constant in each disc (z j n , δ j ) by Lemma A(ii). Moreover, Using these facts together with the subharmonicity of |O φ | p , we obtain where d A(z) is the two-dimensional Lebesgue measure. Now it suffices to show that the last integral in (4.2) is finite. Set ψ(z) = ν(z)/(1 − |z|) for z ∈ D. Note that ν(r ) ψ(r ) for 0 r < 1 by Lemmas A(ii) and B. Moreover, integrating by parts, one can show that ν ∈ D p if and only if In particular, ψ ∈ D ∩ D p by the hypotheses of ν.
This completes the proof.
Recall that a sequence {z n } ⊂ D is said to be uniformly separated if 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 < p < ∞, p − 2 < α < p − 1, and a Carleson-Newman sequence {z n }, we can give a sufficient and necessary condition for {z n } 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. (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 It is an open problem to prove a B p, p ν counterpart of Corollary 9. One could try to prove such a result, for instance, assuming ν ∈ D ∩ D p and sup 0 r <1 In this case, the implication ⇐ is the problematic part. An idea to approach this problem is to follow the argument of [ This result offers a practical way to construct Blaschke sequences that 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 ∈ 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 {z n } 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 < p < ∞ and p − 3/2 < α < ∞. Theorem E confirms that the hypothesis ν ∈ 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 < p < ∞. 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]. Since Using the hypothesis ν ∈ 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 . Note that the argument of this estimate uses ideas from [22].
If q 1, then Lemma F with the choice g(s) = M p (s, f ), Hardy's convexity theorem, Fubini's theorem, the hypothesis ν ∈ D q , and Lemma D give Since ν ∈ D q−ε by Lemma C, the assertion for q > 1 follows from Lemma D. This completes the proof.
by Minkowski's inequality, Theorem 1 has the following consequence.
Corollary 11 Let 1 p < ∞, 0 < q < ∞, and ν ∈ D ∩ D q . Then there exists a constant C = C( p, q, ν) > 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.
Proof of Theorem 2 Let 0 r < 1 and 0 t < 2π . Since 2π 0 e iθ dθ (e iθ − re it ) 2 = 0, Cauchy's integral formula gives 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 ν(r ) dr, we obtain Next we show that the weight ν(r ) in the right-hand side can be replaced by ν(r ) 1−r without losing any essential information. Set ψ(z) = ν(z)/(1 − |z|) for z ∈ D, and recall that ν(r ) ψ(r ) for 0 r < 1 by Lemmas A(ii) and B. In particular, ψ belongs to class D, and thus, there exist Doing a corresponding integration procedure for this estimate as above and applying Theorem 2, we obtain which is the last inequality in (1.4). This completes the proof.