Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 44, 2019, 91–101 THE CLOSURE OF DIRICHLET SPACES IN THE BLOCH SPACE Petros Galanopoulos and Daniel Girela Aristotle University of Thessaloniki, Department of Mathematics 54124, Thessaloniki, Greece; [email protected] Universidad de Málaga, Análisis Matemático Campus de Teatinos, 29071 Málaga, Spain; [email protected] Abstract. 0 1 p If <p< ∞ and α > − , the space of Dirichlet type Dα consists of those functions f which are analytic in the unit disc D and have the property that f ′ belongs to the p p 0 weighted Bergman space Aα. Of special interest are the spaces Dp−1 ( <p< ∞) and the analytic p = p 1 Besov spaces B Dp−2 ( <p< ∞). Let B denote the Bloch space. It is known that the closure p of B (1 <p< ∞) in the Bloch norm is the little Bloch space B0. A description of the closure in the Bloch norm of the spaces Hp ∩B has been given recently. Such closures depend on p. In this paper p 1 we obtain a characterization of the closure in the Bloch norm of the spaces Dα ∩B ( ≤ p < ∞, 1 1 p α > − ). In particular, we prove that for all p ≥ the closure of the space Dp−1 ∩B coincides with that of H2 ∩B. Hence, contrary with what happens with Hardy spaces, these closures are independent of p. We apply these results to study the membership of Blaschke products in the p closure in the Bloch norm of the spaces Dα ∩B. 1. Introduction and main results Let D = {z ∈ C: |z| < 1} denote the open unit disc in the complex plane C, ∂D will be the unit circle. Also, dA will denote the area measure on D, normalized so D 1 1 that the area of is 1. Thus dA(z)= π dxdy = π rdrdθ. The space of all analytic functions in D will be denoted by Hol(D). We also let Hp (0 < p ≤ ∞) be the classical Hardy spaces. We refer to [9] for the notation and results regarding Hardy spaces. The space BMOA consists of those functions f ∈ H1 whose boundary values have bounded mean oscillation on ∂D. The “little oh” version of BMOA is the space VMOA. We refer to [15] for the theory of BMOA-functions. p For 0 <p< ∞ and α > −1 the weighted Bergman space Aα consists of those f ∈ Hol(D) such that 1/p def 2 α p kfk p = (α + 1) (1 −|z| ) |f(z)| dA(z) < ∞. Aα ˆ D p p The unweighted Bergman space A0 is simply denoted by A . We refer to [10, 19, 31] p for the notation and results about Bergman spaces. The space of Dirichlet type Dα D ′ p (0 <p< ∞ and α > −1) consists of those f ∈ Hol( ) such that f ∈ Aα. In other https://doi.org/10.5186/aasfm.2019.4402 2010 Mathematics Subject Classification: Primary 30H30; Secondary 46E15. Key words: Bloch space, Dirichlet spaces, Besov spaces, weighted Bergman spaces, closure in the Bloch norm, Blaschke product. This research is supported by grants from “El Ministerio de Economía y Competitividad, Spain” MTM2014-52865-P; and from “La Junta de Andalucía” (FQM210). 92 Petros Galanopoulos and Daniel Girela D p words, a function f ∈ Hol( ) belongs to Dα if and only if 1/p def 2 α ′ p kfk p = |f(0)| + (α + 1) (1 −|z| ) |f (z)| dA(z) < ∞. Dα ˆ D p p If α>p − 1 then it is well known that Dα = Aα−p (see, e.g., [11, Theorem 6]). p p 1 For 1 <p< ∞, the space Dp−2 is the analytic Besov space B . The space B requires a special definition: it is the space of all functions f ∈ Hol(D) such that f ′′ ∈ A1. Although the corresponding semi-norm is not conformally invariant, the space itself is. Another possible definition (with a conformally invariant semi-norm) is given in the fundamental article [3], where B1 was denoted by M. The spaces Bp, 1 ≤ p< ∞, form a nested scale of conformally invariant spaces which are contained in VMOA and show up naturally in different settings (see [3], [8] and [30]). In 2 2 particular, D0 = B is the classical Dirichlet space D. Finally, we recall that a function f ∈ Hol(D) is said to be a Bloch function if def 2 ′ kfkB = |f(0)| + sup(1 −|z| )|f (z)| < ∞. z∈D The space of all Bloch functions will be denoted by B. It is a non-separable Banach space with the norm k·kB just defined. A classical source for the theory of Bloch functions is [1]. The closure of the polynomials in the Bloch norm is the little Bloch space B0 which consists of those f ∈ Hol(D) with the property that lim (1 −|z|2)|f ′(z)| =0. |z|→1 It is well known that ∞ p ∞ H ( BMOA ( ∩0<p<∞H , H ( BMOA ( B, VMOA ( B0 ( B. Anderson, Clunie and Pommerenke [1, p. 36] raised the question of determining ∞ the closure of H in B. They remarked that this closure strictly contains B0 but is not identical with B. The problem is still open. However, Jones gave an unpublished description of the closure of BMOA in B (see [2, Theorem 9]). Given f ∈ B and ε> 0, we define 2 ′ Ωε(f)= {z ∈ D: (1 −|z| )|f (z)| ≥ ε}. Then a Bloch function f is in the closure of BMOA in the Bloch norm if and only if 2 −1 for every ε > 0 the Borel measure (1 −|z| ) χΩε(f)(z) dA(z) is a Carleson measure in D. As usual, for a Borel subset E of D, χE denotes the characteristic function of E. A proof of Jones’ description is provided by Ghatage and Zheng [14]. This study has been broaden to determine the closure in the Bloch norm of other subspaces of B. For simplicity, if X is a subspace of the Bloch space we shall let CB(X) denote the closure in the Bloch norm of the space X. dA(z) Tjani [26] proved that if f ∈ B, then f ∈ B if and only if 2 2 < ∞ for 0 Ωε(f) (1−|z| ) every ε> 0. Since all Besov spaces contain the polynomials and´ are contained in B0, we have p (1.1) CB(B )= B0, 1 ≤ p< ∞. This was observed in [29] where the closures in the Bloch norm of other conformally invariant spaces were also described. Bao and Göğüş [5] have recently characterized 2 the closure in the Bloch norm of the space Dα ∩ B (−1 <α ≤ 1). The closure of Dirichlet spaces in the Bloch space 93 Monreal Galán and Nicolau [22] described the closure in the Bloch norm of B∩Hp, for 1 <p< ∞. Galanopoulos, Monreal Galán and Pau [13] have extended this result to the whole range 0 <p< ∞. Let us fix some notation. Given a Lebesgue measurable subset Ω of D, we let Ah(Ω) be the hyperbolic area of Ω, that is, dA(z) Ah(Ω) = 2 2 . ˆΩ (1 −|z| ) Also, for fixed a > 1 and for ξ ∈ ∂D, we let Γa(ξ) = {z ∈ D: |z − ξ| < a(1 −|z|)} be the Stolz angle with vertex at ξ. Putting [22, Theorem 1] and [13, Theorem 1] together yields the following result. Theorem A. Let 0 <p< ∞ and a> 1. A Bloch function f is in the closure in p the Bloch norm of B ∩ H if and only if for every ε> 0 the function Fε(f) defined by 1/2 D Fε(f)(ξ)= Ah (Γa(ξ) ∩ Ωε(f)) , ξ ∈ ∂ , belongs to Lp(∂D), that is, dA(z) p/2 2 2 |dξ| < ∞. ˆ D ˆ (1 −|z| ) ∂ Γa(ξ)∩Ωε(f) It is well known that there exists a positive constant C such that 2 |f(z)| ≤ Ckfk log , (z ∈ D), for every f ∈ B, B 1 −|z| p (see [1, p. 13]). Then it follows trivially that B ⊂ Aα whenever 0 <p< ∞ and p α> −1. So the question of characterizing CB(Aα ∩ B) is trivial: p (1.2) CB(Aα ∩ B)= CB(B)= B, 0 <p< ∞, α> −1. The main object of this paper is to characterize the closure in the Bloch norm p p p of the spaces Dα ∩ B. As we mentioned above, if p − 1 <α then Dα = Aα−p. Thus, using (1.2) we obtain p (1.3) CB(Dα ∩ B)= B, 0 <p< ∞, p − 1 < α. p p p If −1 < α ≤ p − 2 then we have that Dα ⊂ Dp−2 = B ⊂ B, and then (1.1) implies that p p p CB(Dα ∩ B)= CB(Dα) ⊂ CB(B )= B0. p p Now it is clear that the polynomials lie in Dα and then it follows that B0 ⊂ CB(Dα). Consequently, we have p (1.4) CB(Dα ∩ B)= B0, 0 <p< ∞, α ≤ p − 2. If remains to consider the case where p − 2 < α ≤ p − 1 and we shall pay a p special attention to the case α = p − 1 because the spaces Dp−1 are the closest ones p to Hardy spaces among all the Dα-spaces. By the Littlewood–Paley identity, we have 2 2 D1 = H . We have also [21] p p H ( Dp−1, for 2 <p< ∞, and [11, 27] p p Dp−1 ( H , for 0 <p< 2. p p A number of similarities and differences between the spaces H and Dp−1 are pre- sented in [4, 16, 17, 23, 27]. As in the case of Hardy spaces, there is no inclusion p relation between the spaces Dp−1 and the Bloch space.