Radial growth of the derivatives of analytic functions in Besov spaces

Abstract: For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc D = {z ∈ C : |z| < 1} and satisfy ∫ D(1 − |z| 2)p−2|f ′(z)|p dA(z) < ∞. The space B2 reduces to the classical Dirichlet space D. It is known that if f ∈ D then |f ′(reiθ)| = o[(1 − r)−1/2], for almost every θ ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.


Introduction
Let D = {z ∈ C : |z| < } denote the open unit disc in the complex plane C and let Hol(D) be the space of all analytic functions in D. Also, dA will denote the normalized area measure on D: dA(z) = π dx dy = π r dr dθ. For ≤ r < and f analytic in D we set We refer to [8] for the theory of Hardy spaces. The weighted Bergman space A p α ( < p < ∞, α > − ) consists of those f ∈ Hol(D) such that The unweighted Bergman space A p is simply denoted by A p . We refer to [9,20,30] for the notation and results about Bergman spaces.
For < p < ∞, the analytic Besov space B p is de ned as the set of all functions f analytic in D such that f ∈ A p p− . Thus a function f ∈ Hol(D) belongs to B p if and only if ρ p (f ) < ∞, where All B p spaces ( < p < ∞) are conformally invariant with respect to the semi-norm ρ B p de ned by [3, p. 112] or [6, p. 46]) and Banach spaces with the norm · B p de ned by f B p = |f ( )| + ρ B p (f ). The space B requires a special de nition: it is the space of all analytic functions f in D for which f ∈ A . Although the corresponding semi-norm is not conformally invariant, the space itself is. Another possible de nition (with a conformally invariant semi-norm) is given in [3], where B was denoted by M. It is well known that the spaces B p form and ascending chain and that they are all contained in the space BMOA [14] and, hence, in the Bloch An important and well-studied case is the classical Dirichlet space B , usually denoted by D, of analytic functions whose image has a nite area, counting multiplicities. We mention [3,6,7,21,22,29,30] as references to nd a lot of information on Besov spaces.
Obtaining results about the behaviour along radii of distinct classes of analytic functions in the unit disc has shown to be an important question in complex analysis with applications to a good number of problems over the years. One of the best known results in this line is due to Rudin [25] who showed the existence of an H ∞ -function f for which the radial variation V(f , θ) = |f (re iθ )|dr is in nite for every θ except possibly for those θ in a set of the rst category and of measure zero.
By the de nition, it is clear that, for f ∈ Hol(D) and < p < ∞, the following two assertions hold: The authors of this paper have recently proved that (1.2) and (1.3) are sharp in a very strong sense connecting this with (1.1). Indeed, for < q < p < ∞ they have constructed a function f ∈ B p ∩ H ∞ with Mp(r, f ) "as big as possible" and having "bad integrability properties of order q along all the radii". To be precise, they have proved the following result in [5,Theorem 1].
Theorem A. Suppose that < q < p < ∞ and let ϕ be a positive increasing function de ned in [ , ) satisfying Then there exists a function f ∈ B p ∩ H ∞ \ B q with the following two properties: Theorem A was applied in [5] to obtain results on multipliers and superposition operators acting on Besov spaces. Our aim in this work is to obtain sharp estimates on the growth of the derivatives of B p -functions on "almost every radius". In [10], [13], [18] and [26] it was proved by distinct methods that if f ∈ D = B then Hallenbeck and Samotij [18] proved that this estimate is sharp in a very strong sense. Indeed, they proved the following result.
Theorem B. Let ε : ( , ) → R be a positive function such that ε(r) → , as r → . Then there exists a function g ∈ D for which Our main result is this paper is the following extension of the above results to all B p spaces, < p < ∞.
(ii) Suppose that < p < ∞ and let ε : ( , ) → R be a positive function such that ε(r) → , as r → . Then there exists a function g ∈ B p ∩ H ∞ , given by a power series with Hadamard gaps, for which Section 2 will contain the proof of Theorem 1.1 and we will apply this result to the study of pointwise multipliers from B p into B q ( < q < p < ∞) in Section 3. In Section 4 we will present other results in the spirit of Theorem 1.1 (ii), showing the sharpness of (1.2) and (1.3) in a way di erent from that in Theorem A.
Let us nish this section mentioning that, as usual, throughout the paper we shall be using the convention that C = C(p, α, q, β, . . . ) will denote a positive constant which depends only upon the displayed parameters p, α, q, β . . . (which most of the times will be omitted) but not necessarily the same at di erent occurrences. Moreover, for two real-valued functions E , E we write E E , or E E , if there exists a positive constant C independent of the arguments such that E ≤ CE , respectively E ≥ CE . If we have E E and E E simultaneously then we say that E and E are equivalent and we write E E . Also, if < p < ∞, p will stand for its conjugate exponent, that is, p + p = .

Proof of Theorem 1.1
Proof of Theorem 1.1(i). Take p ∈ ( , ∞) and f ∈ B p . A well known result of Hardy and Littlewood [19] (see also, e. g., [8,Theorem 5 .6], [11,Theorem 6], or [30,Theorem 4. 28]) shows that and then it follows that for almost every θ ∈ [ , π]. Take θ for which (2.1) is true and ϵ > . Then there exists r ∈ ( , ) such that Using Hölder's inequality, we see that for r < r < , This and (2.2) give a fact, which given that Now we turn to prove part (ii) of our theorem. This is the extension of Theorem B to all B p spaces. Our proof is di erent from the given for p = in [18] and we believe that it is somewhat simpler. We shall use the following result which is the Lemma in p. 339 of [23].

Lemma C. Let S k be a sequence of positive numbers such that
Clearly, it is enough to prove that there exist g ∈ B p , a constant C > , and an increasing sequence We have that g is an analytic function in D which is given by a power series with Hadamard gaps (by (a)). Then (2.4) implies that g ∈ B p (see, e. g., [6, p. 55]). Also, (a) implies that ∞ n= |an| < ∞ and, hence, g ∈ H ∞ . Consequently, De ne also where, We have If we set α j = λ − p j j , using (a), we see that Then, using Lemma C, we obtain Using again Lemma C, it follows that (2.7) Using (2.5), (2.6) and (2.7), the de nitions of the sequences {λ j } and {r j }, and (b), we deduce that there exists a constant C > such that, if |z| = r k and k is big enough, then This gives (2.3), nishing the proof.
Let us remark that Hallenbeck y Samotij in their work [18] actually studied the radial growth of the successive derivatives f (k) (k ∈ N) of functions f in the Dirichlet space. The arguments that we have used to prove Theorem 1.1 can be adapted to extend it to all B p spaces. Namely, we can prove the following result. We omit the details of the proof.

An application of Theorem 1.1 to multipliers
For g ∈ Hol(D), the multiplication operator Mg is de ned by If X and Y are two spaces of analytic function in D (which will always be assumed to be Banach or F-spaces continuously embedded in Hol(D)) and g ∈ Hol(D) then g is said to be a (pointwise) multiplier from X to The space of all multipliers from X to Y will be denoted by M(X, Y) and M(X) will stand for M(X, X). Using the closed graph theorem we see that if g ∈ M(X, Y) then Mg is a bounded operator from X into Y.
It is well known that if X is nontrivial then M(X) ⊂ H ∞ (see, e. g., [ The spaces of multipliers M(B p , B q ) have been studied in a good number of papers (see [4,12,15,16,27,28,31] and the references therein). In particular, the following result holds.
This was proved in [12] using, among other facts, a decomposition theorem for Besov spaces and Khinchine's inequality. The authors of this paper gave in [5] a new proof using Theorem A. Next we are going to give a new proof Theroem D using Theorem 1.1.
Proof of Theorem D. Since B ⊂ B q for all q > , it su ces to consider the case < q < p < ∞. So, assume this and take g ∈ M(B p , B q ). Since the constant function belongs to B p , we see that g ∈ B q . Also, the inclusion B q ⊂ B p and (3.1) implies that g ∈ H ∞ . Thus, we have g ∈ B q ∩ H ∞ .
Use Theorem 1.1 (ii) to pick a function f ∈ B p ∩ H ∞ , given by a power series with Hadamard gaps, with lim sup Since g ∈ B q and f ∈ H ∞ we see that This and (3.3) imply that or, equivalently, We will show that g ≡ arguing by contradiction. So, assume that g ≢ . Since g ∈ H ∞ and g ≢ , using Fatou's theorem and the Riesz uniqueness theorem [8], we see that g has a nite non-zero radial limit at almost every point e iθ of ∂D. Then it follows that there exist C > , r ∈ ( , ), and a measurable set E ⊂ [ , π] whose Lebesgue measure |E| is positive such that |g(re iθ )| ≥ C, r < r < , θ ∈ E.

This and (3.4) readily imply that
Since f is given by a power series with Hadamard gaps, using an extension of Lemma This and (3.5) yield that f ∈ B q . Using Theorem 1.1(i), we see that this in contradiction (3.2). This nishes the proof.

Further results
This section is devoted to prove another result in the line of those in Theorem A and in Theorem 1.1(ii). Clearly, (4.1) implies that ∞ k= a k < ∞. Now, use a well known theorem of Dini (see [24, p. 293]), to pick an increasing sequence of positive numbers {c k } ∞ k= with lim{c k } = ∞ and ∞ k= a k c k < ∞.
It is easy to see that the function ϕ de ned by satis es the conclusion of the lemma.
Proof of Theorem 4.1. It is clear that we may assume without loss of generality that ϕ(r) → ∞, as r → and, bearing in mind Lemma 4.2, it su ces to prove that there exist f ∈ B p and a constact C > such that lim sup Take a sequence {r k } ∞ k= ⊂ ( , ) with the following properties: (a) {r k } is strictly increasing and lim{r k } = .
Note that (b) implies − r k = ( − r k+ ) + (r k+ − r k ) > ( − r k+ ) + ( − r k ) and, hence, − r k − r k+ > or, equivalently, − r k+ < ( − r k ). This gives and then we see that with A = p− − p− . Using the fact that ϕ is increasing and (4.4), we obtain De ne now Here, as usual, for x ∈ R, [x] denotes the integral part of x. De ne also Bearing in mind the de nition of the sequence {n k } and (4.3), we see that f is an analytic function in D which is given by a power series with Hadamard gaps. Furthermore, Then, we see that f ∈ B p . Using Hölder's inequality, (4.5), (4.3), and the de nition of the sequence {n k }, we see also that Then it follows that min Bearing in mind the de nition of n k , it is clear that there exists C > such that I ≥ Cϕ(r k ).
Using again the de nition of the sequence of exponents {n j } and property (d), we see that n j ( − r j )ϕ(r j ) n j+ ( − r j+ )ϕ(r j+ ) ≈ ϕ(r j ) ϕ(r j+ ) → , as j → ∞ and then Lemma C implies that II = k− j= n j ϕ(r j )( − r j )r n j k = o(ϕ(r k )).
This, together with the bounds obtained for I and II, implies that lim inf k→∞ min |z|=r k |f (z)| ϕ(r k ) ≥ C.