Blaschke Products and Zero Sets in Weighted Dirichlet Spaces
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Blaschke Products and Zero Sets in Weighted Dirichlet Spaces H. Bahajji-El Idrissi1 · O. El-Fallah1 Received: 7 January 2019 / Accepted: 9 October 2019 / © Springer Nature B.V. 2019
Abstract In this paper, we deal with superharmonically weighted Dirichlet spaces Dω . First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for Dω . Our conditions improve Shapiro-Shields condition for Dα , when α ∈ (0, 1). Keywords Blaschke product · Dirichlet space · Capacity Mathematics Subject Classification (2010) 31C25 · 30J10 · 31C15
1 Introduction Let D be the unit disc of the complex plane C and let T := ∂D be the unit circle. Let dA (resp. dm) be the normalized Lebesgue measure on D (resp. T). The Hardy space H 2 is the space of analytic functions f on D such that 2 2 f H 2 := |f (0)| + |f (z)|2 log(1/|z|)dA(z) < ∞. D
A weight ω is a function ω : D → (0, +∞] which is integrable on D with respect to dA. The weighted Dirichlet space Dω associated with ω is defined by 2 2 Dω := f ∈ H : Dω (f ) := |f (z)| ω(z)dA(z) < ∞ . D
Research partially supported by “Hassan II Academy of Science and Technology”. O. El-Fallah
[email protected] H. Bahajji-El Idrissi [email protected] 1
Laboratory of Mathematical Analysis and Applications, Faculty of Sciences, Mohammed 5 University in Rabat, B.P. 1014 Rabat, Morocco
H. Bahajji-El Idrissi, O. El-Fallah
The space Dω will be endowed by the hilbertian norm f 2ω := f 2H 2 + Dω (f ). Let 0 ≤ α < 1 and denote by Dα the standard Dirichlet space which corresponds to the weight ωα (z) = (1 + α)(1 − |z|2 )α . The classical Dirichlet space is D0 and will be denoted by D. In this paper, we are mainly interested in superhamonically weighted Dirichlet spaces. Several results on these spaces can be found in [3, 6, 15–17, 32]. In general the description of zero sets remains an open problem, even for the standard Dirichlet spaces Dα for α ∈ [0, 1). Recall that a sequence Z = (zn )n≥1 ⊂ D is a zero set for Dω if there is a function in Dω that vanishes on Z and nowhere else in D. (1 − |zn |) < ∞. The We say that Z = (zn )n≥1 ⊂ D is a Blaschke sequence if n≥1
associated Blaschke product B is given by B(z) =
|zn | zn − z , zn 1 − zn z
n≥1
with the convention |zn |/zn = −1, if zn = 0. It is known [12] that zero sets for H 2 are the Blaschke sequences and that every Blaschke product is bounded, so it belongs in H 2 . For the classical Dirichlet space, L. Carleson [10] proved that if Z satisfies
1
n≥1
log1−ε 1/(1 − |zn |)
0, such that h ≥ c a.e. on T . lim inf ω(z) > 0.
iv)
Dω ⊂ D .
|z|→1−
The second goal in this paper is to give some sufficient conditions which ensure that a sequence Z is a zero set for Dω . Observe that since Dω ⊂ H 2 , then each zero set for Dω satisfies the Blaschke condition. The converse is in general not true (see Section 6). For the standard Dirichlet spaces there are several
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