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; petrosgala 'at' math.auth.gr

Universidad de Málaga, Análisis Matemático
Campus de Teatinos, 29071 Málaga, Spain; girela 'at' uma.es

Abstract. If 0 < p < ∞ and α > –1, the space of Dirichlet type D_α^p consists of those functions f which are analytic in the unit disc D and have the property that f' belongs to the weighted Bergman space A_α^p. Of special interest are the spaces D_p-1^p (0 < p < ∞) and the analytic Besov spaces Bp = D_p-2^p (1 < p < ∞). Let B denote the Bloch space. It is known that the closure of B^p (1 < p < ∞) in the Bloch norm is the little Bloch space B₀. A description of the closure in the Bloch norm of the spaces H^p ∩ B has been given recently. Such closures depend on p. In this paper we obtain a characterization of the closure in the Bloch norm of the spaces D_α^p ∩ B (1 ≤ p < ∞, α > –1). In particular, we prove that for all p ≥ 1 the closure of the space D_p-1^p ∩ 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 closure in the Bloch norm of the spaces D_α^p ∩ B.

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.

Reference to this article: P. Galanopoulos and D. Girela: The closure of Dirichlet spaces in the Bloch space. Ann. Acad. Sci. Fenn. Math. 44 (2019), 91-101.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4402